Preface

Chapter 8 **A Perturbation Theory for Nonintegrable Equations with Small**

**Horizontal Porous Layer Using a Multidomain Spectral**

Osman A.I. Noreldin, Precious Sibanda and Sabyasachi Mondal

Chapter 9 **Weakly Nonlinear Stability Analysis of a Nanofluid in a**

Chapter 10 **Small-Angle Scattering from Mass and Surface Fractals 169**

**Dispersion 133** Georgy Omel'yanov

**VI** Contents

**Collocation Method 149**

Eugen Mircea Anitas

Modeling and simulating biological and physical systems are nowadays active branches of sci‐ ence. The diversity and complexity of behaviors and patterns present in the natural world have their reciprocity in the life systems. Bifurcations, solitons and fractals are some of these ubiqui‐ tous structures that can be indistinctively identified in many models with the most diverse appli‐ cations, from microtubules with an essential role in the maintenance and the shaping of cells, to the nano/microscale structure in disordered systems determined with small-angle scattering techniques. This book collects several works in this direction, giving an overview of some mod‐ els and theories that are useful for the study and analysis of complex biological and physical systems. It can provide a good guidance for physicists with interest in biology, applied research scientists, and postgraduate students.

The first section of the book presents different biological models with a wide variety of applica‐ tions. In Chapter 1, Zdravkovic presents three nonlinear mechanical models to explain the dy‐ namics of solitons in microtubules. In Chapter 2, Liu et al. study the behavior of two delayed epidemic spreading models on scale-free networks. In Chapter 3, Salleh and Rahim investigate the existence and stability of equilibria in a nutrient-prey-predator model with intratrophic pre‐ dation. In Chapter 4, Gul and Bernhard apply global sensitivity analysis to a multicompartment, lumped-parameter model of an arm artery to identify the bifurcation parameters of the arm ar‐ teries. The last chapter of this section, Chapter 5 by Maldonado, introduces the idea of biological hypercomputation and analyzes the relationship between matter, energy and information.

The second section of the book presents some physical models showing soliton and fractal behav‐ iors. In Chapter 6, Kalashnikov and Sorokin present the concept of dissipative soliton and its full life cycle as a self-organized object. In Chapter 7, Luke discusses the use of solitons for particle models in the nonlinear Klein-Gordon equation. In Chapter 8, Omel'yanov considers the problem of propagation and interaction of solitons in the generalized KdV equation. In Chapter 9, Noreldin at al. perform a weakly nonlinear stability analysis of the flow of a nanofluid in a porous medium with stress-free boundary conditions. Finally, in the last chapter, Chapter 10, Anitas introduces the concepts of mass and surface fractals in the context of small-angle scattering techniques.

As the editor of this book, I would like to thank all the authors who have contributed to this volume as well as the reviewers for their assessment. Also, I must express my gratitude to the InTechOpen editorial staff for their invitation asking me to be the editor for the second time. With particular help from Ms. Kristina Kardum, the Publishing Process Manager (PPM), we have arrived to convert in this new InTechOpen book. Finally, at this moment where life is a nonsense time flow, I want to dedicate all this effort to the memory of my father, Ricardo López-Barasoain (1935–2015), and to my mother, Amelia Ruiz-Gastón (1935–present), from Villafranca, Navarra, Spain. Of course, the rest of my family and all my friends and advisers are not forgot‐ ten in this dedicatory final paragraph.

> **Ricardo López-Ruiz** University of Zaragoza, Spain

**Section 1**

**Complexity in Biological Systems**

**Complexity in Biological Systems**

**Chapter 1**

Provisional chapter

**Mechanical Models of Microtubules**

Mechanical Models of Microtubules

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Microtubules are the major part of the cytoskeleton. They are involved in nuclear and cell division and serve as a network for motor proteins. The first model that describes nonlinear dynamics of microtubules was introduced in 1993. Three nonlinear models are described in this chapter. They are longitudinal U-model, representing an improved version of the first model, radial φ-model and new general model. Also, two mathematical procedures are explained. These are continuum and semi-discrete approximations. Continuum approximation yields to either kink-type or bell-type solitons, while semidiscrete one predicts localized modulated waves moving along microtubules. Some

DOI: 10.5772/intechopen.71181

possible improvements and suggestions for future research are discussed.

Keywords: microtubules, partial and ordinary differential equations, kink solitons,

A cell is defined as eukaryotic if it has a membrane-bound nucleus. Such cells are generally larger and much more sophisticated than prokaryotic ones. Microtubules (MTs) are the basic components of cytoskeleton existing in eukaryotes [1]. They are long structures that spread between a nucleus and a cell membrane. MTs play an essential role in the shaping and the maintenance of cells and are involved in cell division. Also, they represent a network for motor proteins. These proteins move with a velocity of 0:1 2μm=s [2] carrying a certain cargo such as mitochondrion. All eukaryotic cells produce two kinds of tubulin proteins. These are α and β tubulins, or monomers, and they spontaneously arrange head to tail forming biologically functional subunit that we call a heterodimer, or a dimer for short. When intracellular conditions favor assembly, the dimers assemble into long structures called protofilaments (PFs). Microtubules are usually

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

http://dx.doi.org/10.5772/intechopen.71181

Slobodan Zdravković

Slobodan Zdravković

Abstract

breathers

1. Introduction

formed of 13 PFs, as shown in Figure 1.

#### **Chapter 1**

Provisional chapter

## **Mechanical Models of Microtubules**

Mechanical Models of Microtubules

## Slobodan Zdravković

Additional information is available at the end of the chapter Slobodan Zdravković

http://dx.doi.org/10.5772/intechopen.71181 Additional information is available at the end of the chapter

#### Abstract

Microtubules are the major part of the cytoskeleton. They are involved in nuclear and cell division and serve as a network for motor proteins. The first model that describes nonlinear dynamics of microtubules was introduced in 1993. Three nonlinear models are described in this chapter. They are longitudinal U-model, representing an improved version of the first model, radial φ-model and new general model. Also, two mathematical procedures are explained. These are continuum and semi-discrete approximations. Continuum approximation yields to either kink-type or bell-type solitons, while semidiscrete one predicts localized modulated waves moving along microtubules. Some possible improvements and suggestions for future research are discussed.

DOI: 10.5772/intechopen.71181

Keywords: microtubules, partial and ordinary differential equations, kink solitons, breathers

#### 1. Introduction

A cell is defined as eukaryotic if it has a membrane-bound nucleus. Such cells are generally larger and much more sophisticated than prokaryotic ones. Microtubules (MTs) are the basic components of cytoskeleton existing in eukaryotes [1]. They are long structures that spread between a nucleus and a cell membrane. MTs play an essential role in the shaping and the maintenance of cells and are involved in cell division. Also, they represent a network for motor proteins. These proteins move with a velocity of 0:1 2μm=s [2] carrying a certain cargo such as mitochondrion.

All eukaryotic cells produce two kinds of tubulin proteins. These are α and β tubulins, or monomers, and they spontaneously arrange head to tail forming biologically functional subunit that we call a heterodimer, or a dimer for short. When intracellular conditions favor assembly, the dimers assemble into long structures called protofilaments (PFs). Microtubules are usually formed of 13 PFs, as shown in Figure 1.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

systems, are modeled by nonlinear "springs". This means that expressions for energy of biological systems require nonlinear terms, which brings about nonlinear partial differential equations (PDEs) explaining nonlinear dynamics of these systems. This is the topic of the present chapter.

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 5

The word soliton was introduced in 1965 to designate solitary waves describing the propagation of excitations in continuous media with nonlinearity and dispersion [14]. The first qualitative description of solitary waves dates back to 1834 when hydrodynamic engineer John Scott Russell observed them on a surface in a shallow channel [15]. The wave was so stable that the engineer followed it about 1 or 2 miles. From then, there has been tremendous interest for various kinds of solitons in many branches of physics [15–19]. In this chapter, the terms soliton and solitary waves are treated as synonyms, which is commonly accepted in literature. Solitons are localized waves possessing some interesting properties. The most important is their stability in a sense that they conserve their shape and energy after mutual interaction. In other words, they can pass through one another without annihilation. This was experimentally

To model complex MT dynamics, we should introduce some simplifications. To the best of the author's knowledge, all the models introduced so far have only one degree of freedom per dimer. Hence, for the models explained in this chapter, elementary subunits of PFs are dimers and they perform either longitudinal or angular oscillations and the appropriate models can be

The longitudinal contacts along PFs are much stronger than those between adjacent PFs [21, 22], which allows us to construct a simplified Hamiltonian of MT, which is, practically, Hamiltonian for a single PF only. However, the influence of the neighboring PFs is taken into consideration through the electric field. Namely, each dimer exists in the electric field coming from the dimers belonging to all PFs. Also, the nearest neighbor approximation is assumed.

The first model that describes nonlinear dynamics of MTs is a longitudinal one. It was introduced in 1993 by Satarić et al. [23]. According to the model, the dimers perform angular oscillations but a coordinate u, describing the dimer's displacement, is a projection of the top of the dimer on the direction of PF. Therefore, the displacements are radial but the used coordinate is longitudinal. There is a real longitudinal model assuming longitudinal displacements of the dimers that we call as Z-model [24]. Both U- and Z-models bring about equal

Somewhat improved and more general version of the first nonlinear model is what we call as U-model [25] and this will be explained in the following paragraphs. Both models are based on the fact mentioned above that the dimers are electric dipoles and that the whole MT can be regarded as ferroelectric [23, 26], which means that the interaction between a single dimer and its surrounding can be modeled by W-potential [23, 27]. This yields to the following Hamilto-

crucial differential equations and the latter one will not be studied here.

We will see that, in case of MTs, the solutions of these nonlinear PDEs are solitary waves.

observed in neurons [20].

3. U-model

nian for MT [23, 25, 28]

called as longitudinal or angular (radial), respectively.

Figure 1. A tubulin dimer, a protofilament and a microtubule [3].

Hence, MTs are long cylindrical polymers whose lengths vary from a few hundred nanometers up to meters in long nerve axons [4]. Each dimer is an electric dipole whose mass and length are <sup>m</sup> <sup>¼</sup> <sup>1</sup>:<sup>8</sup> � <sup>10</sup>�<sup>22</sup> kg and <sup>l</sup> <sup>¼</sup> 8 nm, respectively [5]. The component of its electric dipole moment in the direction of PF is <sup>p</sup> <sup>¼</sup> 337Debye <sup>¼</sup> <sup>1</sup>:<sup>13</sup> � <sup>10</sup>-27 Cm [6]. Consequently, MT as a whole appears to be a giant dipole with negatively charged end coinciding with biologically positive end (more active) and vice versa. This is the reason why an intrinsic electric field exists within MT.

MTs in non-neuronal cells are unstable structures. They exhibit dynamic instability behavior existing in phases of elongation (polymerization) or rapid shortening (depolymerization). This size fluctuation has been called as dynamic instability [7, 8]. Notice that the shrinkage rate is bigger than the growth rate (see Ref. [9] and references therein). MTs grow steadily at positive end, corresponding to the β– subunit, and then shrink rapidly by loss of tubulin dimers at the negative end, corresponding to the α–monomer. Many anticancer drugs, for example, taxol (paclitaxel), prevent growth and shrinkage of MTs and thus prevent cell proliferation [10].

MTs existing in neuronal cells are stable and, consequently, neurons, once formed, do not divide [4]. This stability is crucial as there are evidences that neuronal MTs are responsible for processing, storage and transduction of biological information in a brain [4, 11].

It was mentioned that MTs represent the traffic road for motor proteins. Some more information can be found in Ref. [9] and in an exhaustive review paper [12]. It suffices now to state that the cellular motors with dimensions of less than 100 nm convert chemical energy into useful work. These small machines have the fundamental role of dissipation in biological systems, which has been confirmed by both the theoretical and the experimental investigations [13]. The molecular motors dissipate continuously and operate as irreversible systems [13].

It is clear that any molecular motor, to start moving, should obtain a certain signal. One of the promising dynamical mechanisms for intracellular signaling is solitary waves, which is explained in this chapter.

#### 2. Mechanical models

MTs, as well as all biological systems, are nonlinear in nature. Strong covalent chemical bonds are usually modeled by linear "springs", while weak chemical interactions, existing in all biological systems, are modeled by nonlinear "springs". This means that expressions for energy of biological systems require nonlinear terms, which brings about nonlinear partial differential equations (PDEs) explaining nonlinear dynamics of these systems. This is the topic of the present chapter. We will see that, in case of MTs, the solutions of these nonlinear PDEs are solitary waves.

The word soliton was introduced in 1965 to designate solitary waves describing the propagation of excitations in continuous media with nonlinearity and dispersion [14]. The first qualitative description of solitary waves dates back to 1834 when hydrodynamic engineer John Scott Russell observed them on a surface in a shallow channel [15]. The wave was so stable that the engineer followed it about 1 or 2 miles. From then, there has been tremendous interest for various kinds of solitons in many branches of physics [15–19]. In this chapter, the terms soliton and solitary waves are treated as synonyms, which is commonly accepted in literature.

Solitons are localized waves possessing some interesting properties. The most important is their stability in a sense that they conserve their shape and energy after mutual interaction. In other words, they can pass through one another without annihilation. This was experimentally observed in neurons [20].

To model complex MT dynamics, we should introduce some simplifications. To the best of the author's knowledge, all the models introduced so far have only one degree of freedom per dimer. Hence, for the models explained in this chapter, elementary subunits of PFs are dimers and they perform either longitudinal or angular oscillations and the appropriate models can be called as longitudinal or angular (radial), respectively.

The longitudinal contacts along PFs are much stronger than those between adjacent PFs [21, 22], which allows us to construct a simplified Hamiltonian of MT, which is, practically, Hamiltonian for a single PF only. However, the influence of the neighboring PFs is taken into consideration through the electric field. Namely, each dimer exists in the electric field coming from the dimers belonging to all PFs. Also, the nearest neighbor approximation is assumed.

#### 3. U-model

Hence, MTs are long cylindrical polymers whose lengths vary from a few hundred nanometers up to meters in long nerve axons [4]. Each dimer is an electric dipole whose mass and length are <sup>m</sup> <sup>¼</sup> <sup>1</sup>:<sup>8</sup> � <sup>10</sup>�<sup>22</sup> kg and <sup>l</sup> <sup>¼</sup> 8 nm, respectively [5]. The component of its electric dipole moment in the direction of PF is <sup>p</sup> <sup>¼</sup> 337Debye <sup>¼</sup> <sup>1</sup>:<sup>13</sup> � <sup>10</sup>-27 Cm [6]. Consequently, MT as a whole appears to be a giant dipole with negatively charged end coinciding with biologically positive end (more active) and vice versa. This is the reason why an intrinsic electric field exists

Figure 1. A tubulin dimer, a protofilament and a microtubule [3].

4 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

MTs in non-neuronal cells are unstable structures. They exhibit dynamic instability behavior existing in phases of elongation (polymerization) or rapid shortening (depolymerization). This size fluctuation has been called as dynamic instability [7, 8]. Notice that the shrinkage rate is bigger than the growth rate (see Ref. [9] and references therein). MTs grow steadily at positive end, corresponding to the β– subunit, and then shrink rapidly by loss of tubulin dimers at the negative end, corresponding to the α–monomer. Many anticancer drugs, for example, taxol (paclitaxel), prevent growth and shrinkage of MTs and thus prevent cell proliferation [10].

MTs existing in neuronal cells are stable and, consequently, neurons, once formed, do not divide [4]. This stability is crucial as there are evidences that neuronal MTs are responsible for

It was mentioned that MTs represent the traffic road for motor proteins. Some more information can be found in Ref. [9] and in an exhaustive review paper [12]. It suffices now to state that the cellular motors with dimensions of less than 100 nm convert chemical energy into useful work. These small machines have the fundamental role of dissipation in biological systems, which has been confirmed by both the theoretical and the experimental investigations [13]. The

It is clear that any molecular motor, to start moving, should obtain a certain signal. One of the promising dynamical mechanisms for intracellular signaling is solitary waves, which is exp-

MTs, as well as all biological systems, are nonlinear in nature. Strong covalent chemical bonds are usually modeled by linear "springs", while weak chemical interactions, existing in all biological

processing, storage and transduction of biological information in a brain [4, 11].

molecular motors dissipate continuously and operate as irreversible systems [13].

within MT.

lained in this chapter.

2. Mechanical models

The first model that describes nonlinear dynamics of MTs is a longitudinal one. It was introduced in 1993 by Satarić et al. [23]. According to the model, the dimers perform angular oscillations but a coordinate u, describing the dimer's displacement, is a projection of the top of the dimer on the direction of PF. Therefore, the displacements are radial but the used coordinate is longitudinal. There is a real longitudinal model assuming longitudinal displacements of the dimers that we call as Z-model [24]. Both U- and Z-models bring about equal crucial differential equations and the latter one will not be studied here.

Somewhat improved and more general version of the first nonlinear model is what we call as U-model [25] and this will be explained in the following paragraphs. Both models are based on the fact mentioned above that the dimers are electric dipoles and that the whole MT can be regarded as ferroelectric [23, 26], which means that the interaction between a single dimer and its surrounding can be modeled by W-potential [23, 27]. This yields to the following Hamiltonian for MT [23, 25, 28]

$$H\_{\rm u} = \sum\_{n} \left[ \frac{m}{2} \dot{u}\_n^2 + \frac{k\_{\rm u}}{2} (u\_{n+1} - u\_n)^2 - \frac{1}{2} A u\_n^2 + \frac{1}{4} B u\_n^4 - Q E u\_n \right] \tag{1}$$

traveling wave uð Þ ξ is a solution which depends upon x and t only through a unified variable ξ as ξ ¼ κx � ωt, where κ and ω are constants. If we substitute the variables x and t by ξ we

> 2 κ2 <sup>A</sup> , <sup>σ</sup> <sup>¼</sup> qE

Eq. (4) becomes the appropriate one in Ref. [23] for α<sup>u</sup> ¼ �1. Therefore, the U-model is more general than its predecessor introduced in Ref. [23]. It is crucial that the parameter α<sup>u</sup> can be determined together with the function Ψ for known or estimated σ and ru. This is because α<sup>u</sup> has very important physical meaning. The first term in Eq. (3) is the inertial term and it is coming from the kinetic energy in Hamiltonian (1), while the second one is the elastic one. Therefore, positive α<sup>u</sup> means that the inertial term is bigger than the elastic one and vice versa. Eq. (4) has already been solved using different mathematical procedures like standard procedure [23, 27, 29, 30] and method of factorization [31, 32]. There exists a group of procedures where the function Ψ is represented as a serious expansion over other known function like

coefficients Ak. A common example for Φ is a solution of Riccati equation, which is either tangent or tangent hyperbolic. As only the latter function may have physical meaning, we call the method as tangent hyperbolic function method (THFM) [25, 33–35] and extended or modified extended THFM [36]. The function Φ can also be one of Jacobian elliptic functions

It is very likely that the most general procedure is the simplest equation method (SEM) [39–41] and its simplified version called as modified simplest equation method (MSEM) [42]. Accor-

where A0, Ak and Bk are coefficients that should be determined and Φ<sup>0</sup> represents the first derivative. In general, the function Φ ¼ Φð Þ ξ is known and represents a solution of a certain ODE of lower order than the equation that should be solved. A commonly used example is the

and concentrate our attention on the leading terms [42]. One can easily show that N ¼ 1 for

AkΦ<sup>k</sup> <sup>þ</sup> Bk

Φ0 Φ

<sup>Φ</sup><sup>0</sup> <sup>þ</sup> <sup>Φ</sup><sup>2</sup> � <sup>2</sup>a<sup>Φ</sup> � <sup>b</sup> <sup>¼</sup> <sup>0</sup>, a, b <sup>¼</sup> const (7)

.

� �<sup>k</sup> ! (6)

, c ¼ const, into Eq. (4)

<sup>Ψ</sup> <sup>¼</sup> <sup>A</sup><sup>0</sup> <sup>þ</sup><sup>X</sup>

To determine the positive integer <sup>N</sup> in Eq. (6), we should plug <sup>Ψ</sup> <sup>¼</sup> <sup>c</sup>=ξ<sup>p</sup>

Eq. (4) as the leading terms are proportional to ξ�ð Þ <sup>p</sup>þ<sup>2</sup> and ξ�3<sup>p</sup>

N

k¼1

. The function Φ is usually known and we plug Ψ into Eq. (4) and determine the

<sup>α</sup>uΨ<sup>00</sup> � <sup>r</sup>uΨ<sup>0</sup> � <sup>Ψ</sup> <sup>þ</sup> <sup>Ψ</sup><sup>3</sup> � <sup>σ</sup> <sup>¼</sup> <sup>0</sup> (4)

A ffiffiffiffiffiffiffiffiffi

<sup>A</sup>=<sup>B</sup> <sup>p</sup> , <sup>r</sup><sup>u</sup> <sup>¼</sup> γω

<sup>A</sup> (5)

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 7

straightforwardly transform Eq. (3) into the following ODE

<sup>Ψ</sup>, <sup>α</sup><sup>u</sup> <sup>¼</sup> <sup>m</sup>ω<sup>2</sup> � kul

where u<sup>0</sup> � du=dξ and

<sup>Ψ</sup> <sup>¼</sup> <sup>P</sup> N k¼0

AkΦ<sup>k</sup>

Riccati equation [40]

[37] and, even, unknown [38].

ding to SEM, the series expansion is [39–41].

u ¼

ffiffiffiffi A B r

where dot means the first derivative with respect to time while the integer n determines the position of the considered dimer in PF. The first term obviously represents a kinetic energy of the dimer of mass m. The second one is interaction between the neighboring dimers belonging to the same PF in the nearest neighboring approximation and k is an intradimer stiffness parameter. The next two terms represent the W-potential energy mentioned earlier, where the parameters A and B should be determined or, at least, estimated and are assumed to be positive. We should point out that the double-well potential is rather common in physics [27, 29, 30]. The very last term is coming from the fact that the dimer is the electric dipole existing in the field of all other dimers, where Q > 0 represents the excess charge within the dipole and E > 0 is internal electric field. The last three terms together can be regarded as unsymmetrical W-potential.

Our final goal is the function unð Þt , describing nonlinear dynamics of MT. This function is a solution of so-called dynamical equation of motion, which can be obtained from Eq. (1). To derive it, we introduce generalized coordinates qn and pn defined as qn ¼ un and pn ¼ mdun=dt. Using well-known Hamilton's equations of motion dpn=dt ¼ �dH=dq<sup>n</sup> and dqn=dt ¼ dH=dpn, we obtain the following discrete differential equation that should be solved

$$m\ddot{u}\_n = k\_u(u\_{n+1} + u\_{n-1} - 2u\_n) + Au\_n - Bu\_n^{-3} + QE - \gamma \dot{u}\_n \tag{2}$$

The last term is a viscosity force with γ being a viscosity coefficient [23]. Therefore, nonlinear dynamics of MTs has been described by Eq. (2). Obviously, nonlinearity is coming from the fourth degree term in the W-potential.

It was explained earlier that we used some approximations to derive Eq. (2). However, we need one more to solve it. We now explain two mathematical methods for solving this equation. Practically, these two approaches are two approximations. They are continuum and semidiscrete approximations. We will see that the different mathematical procedures yield to different solutions. Therefore, the function unð Þt depends not only on the physical system but also on the used mathematical method.

Let us explain the continuum approximation first. A question if MTs are discrete or continuum systems was studied in Ref. [31], where it was shown that the continuum approximation is valid. The continuum approximation means a transition unðÞ!t u xð Þ ; t , which allows a series expansion of the terms un�1, that is, un�<sup>1</sup> ! <sup>u</sup> � <sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>x</sup> <sup>l</sup> <sup>þ</sup> <sup>1</sup> 2 ∂2u <sup>∂</sup>x<sup>2</sup> l 2 , where l is the dimer's length explained earlier. In fact, PF can be seen as one-dimensional crystal with l being a period of the lattice. This straightforwardly brings about the following continuum dynamical equation of motion

$$m\frac{\partial^2 u}{\partial t^2} - k\_u l^2 \frac{\partial^2 u}{\partial x^2} - QE - Au + Bu^3 + \gamma \frac{\partial u}{\partial t} = 0 \tag{3}$$

This is PDE that cannot be easily solved. Hopefully, this equation can be transformed into an ordinary differential equation (ODE). It is well known that, for a given wave equation, a traveling wave uð Þ ξ is a solution which depends upon x and t only through a unified variable ξ as ξ ¼ κx � ωt, where κ and ω are constants. If we substitute the variables x and t by ξ we straightforwardly transform Eq. (3) into the following ODE

$$
\alpha\_u \Psi'' - \rho\_u \Psi' - \Psi + \Psi^3 - \sigma = 0 \tag{4}
$$

where u<sup>0</sup> � du=dξ and

Hu <sup>¼</sup> <sup>X</sup> n

fourth degree term in the W-potential.

also on the used mathematical method.

sion of the terms un�1, that is, un�<sup>1</sup> ! <sup>u</sup> � <sup>∂</sup><sup>u</sup>

m ∂2 u ∂t <sup>2</sup> � ku l

W-potential.

m <sup>2</sup> <sup>u</sup>\_ <sup>n</sup> 2 þ ku

6 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>2</sup> ð Þ unþ<sup>1</sup> � un <sup>2</sup> � <sup>1</sup>

where dot means the first derivative with respect to time while the integer n determines the position of the considered dimer in PF. The first term obviously represents a kinetic energy of the dimer of mass m. The second one is interaction between the neighboring dimers belonging to the same PF in the nearest neighboring approximation and k is an intradimer stiffness parameter. The next two terms represent the W-potential energy mentioned earlier, where the parameters A and B should be determined or, at least, estimated and are assumed to be positive. We should point out that the double-well potential is rather common in physics [27, 29, 30]. The very last term is coming from the fact that the dimer is the electric dipole existing in the field of all other dimers, where Q > 0 represents the excess charge within the dipole and E > 0 is internal electric field. The last three terms together can be regarded as unsymmetrical

Our final goal is the function unð Þt , describing nonlinear dynamics of MT. This function is a solution of so-called dynamical equation of motion, which can be obtained from Eq. (1). To derive it, we introduce generalized coordinates qn and pn defined as qn ¼ un and pn ¼ mdun=dt. Using well-known Hamilton's equations of motion dpn=dt ¼ �dH=dq<sup>n</sup> and dqn=dt ¼ dH=dpn,

The last term is a viscosity force with γ being a viscosity coefficient [23]. Therefore, nonlinear dynamics of MTs has been described by Eq. (2). Obviously, nonlinearity is coming from the

It was explained earlier that we used some approximations to derive Eq. (2). However, we need one more to solve it. We now explain two mathematical methods for solving this equation. Practically, these two approaches are two approximations. They are continuum and semidiscrete approximations. We will see that the different mathematical procedures yield to different solutions. Therefore, the function unð Þt depends not only on the physical system but

Let us explain the continuum approximation first. A question if MTs are discrete or continuum systems was studied in Ref. [31], where it was shown that the continuum approximation is valid. The continuum approximation means a transition unðÞ!t u xð Þ ; t , which allows a series expan-

> <sup>∂</sup><sup>x</sup> <sup>l</sup> <sup>þ</sup> <sup>1</sup> 2 ∂2u <sup>∂</sup>x<sup>2</sup> l 2

straightforwardly brings about the following continuum dynamical equation of motion

<sup>2</sup> ∂<sup>2</sup> u

earlier. In fact, PF can be seen as one-dimensional crystal with l being a period of the lattice. This

This is PDE that cannot be easily solved. Hopefully, this equation can be transformed into an ordinary differential equation (ODE). It is well known that, for a given wave equation, a

<sup>∂</sup>x<sup>2</sup> � QE � Au <sup>þ</sup> Bu<sup>3</sup> <sup>þ</sup> <sup>γ</sup>

we obtain the following discrete differential equation that should be solved

mu€<sup>n</sup> ¼ kuðunþ<sup>1</sup> þ un�<sup>1</sup> � 2unÞ þ Aun � Bun

2 Au<sup>2</sup> <sup>n</sup> þ 1 4 Bu<sup>4</sup>

� �

<sup>n</sup> � QEun

<sup>3</sup> <sup>þ</sup> QE � <sup>γ</sup>u\_ <sup>n</sup> (2)

, where l is the dimer's length explained

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup> (3)

∂u

(1)

$$u = \sqrt{\frac{A}{B}} \Psi, \quad \alpha\_u = \frac{m\omega^2 - k\_u l^2 \kappa^2}{A}, \quad \sigma = \frac{qE}{A\sqrt{A/B}}, \quad \rho\_u = \frac{\gamma\omega}{A} \tag{5}$$

Eq. (4) becomes the appropriate one in Ref. [23] for α<sup>u</sup> ¼ �1. Therefore, the U-model is more general than its predecessor introduced in Ref. [23]. It is crucial that the parameter α<sup>u</sup> can be determined together with the function Ψ for known or estimated σ and ru. This is because α<sup>u</sup> has very important physical meaning. The first term in Eq. (3) is the inertial term and it is coming from the kinetic energy in Hamiltonian (1), while the second one is the elastic one. Therefore, positive α<sup>u</sup> means that the inertial term is bigger than the elastic one and vice versa.

Eq. (4) has already been solved using different mathematical procedures like standard procedure [23, 27, 29, 30] and method of factorization [31, 32]. There exists a group of procedures where the function Ψ is represented as a serious expansion over other known function like <sup>Ψ</sup> <sup>¼</sup> <sup>P</sup> N k¼0 AkΦ<sup>k</sup> . The function Φ is usually known and we plug Ψ into Eq. (4) and determine the coefficients Ak. A common example for Φ is a solution of Riccati equation, which is either tangent or tangent hyperbolic. As only the latter function may have physical meaning, we call the method as tangent hyperbolic function method (THFM) [25, 33–35] and extended or modified extended THFM [36]. The function Φ can also be one of Jacobian elliptic functions [37] and, even, unknown [38].

It is very likely that the most general procedure is the simplest equation method (SEM) [39–41] and its simplified version called as modified simplest equation method (MSEM) [42]. According to SEM, the series expansion is [39–41].

$$\Psi = A\_0 + \sum\_{k=1}^{N} \left( A\_k \Phi^k + B\_k \left( \frac{\Phi'}{\Phi} \right)^k \right) \tag{6}$$

where A0, Ak and Bk are coefficients that should be determined and Φ<sup>0</sup> represents the first derivative. In general, the function Φ ¼ Φð Þ ξ is known and represents a solution of a certain ODE of lower order than the equation that should be solved. A commonly used example is the Riccati equation [40]

$$
\Phi' + \Phi^2 - 2a\Phi - b = 0, \qquad a, b = \text{const} \tag{7}
$$

To determine the positive integer <sup>N</sup> in Eq. (6), we should plug <sup>Ψ</sup> <sup>¼</sup> <sup>c</sup>=ξ<sup>p</sup> , c ¼ const, into Eq. (4) and concentrate our attention on the leading terms [42]. One can easily show that N ¼ 1 for Eq. (4) as the leading terms are proportional to ξ�ð Þ <sup>p</sup>þ<sup>2</sup> and ξ�3<sup>p</sup> .

The well-known general solution of Eq. (7) is [39, 40]

$$\Phi = a + \sqrt{a^2 + b} \tanh\left[\sqrt{a^2 + b}(\xi - \xi\_0)\right] \tag{8}$$

In what follows, we assume ξ<sup>0</sup> ¼ 0.

Our next step is determination of the parameters A0, A1, B1, a, b and αu. According to Eqs. (6) for N ¼ 1 and (7), we obtain the expressions for Ψ<sup>0</sup> , Ψ<sup>00</sup> and Ψ<sup>3</sup> as required by Eq. (4), which yields to the following expression:

$$K\_3 \Phi^3 + K\_3^{'} \Phi^{-3} + K\_2 \Phi^2 + K\_2^{'} \Phi^{-2} + K\_1 \Phi + K\_1^{'} \Phi^{-1} + K\_0 = 0 \tag{9}$$

Obviously, this is satisfied if all the coefficients are simultaneously equal to zero. This brings about a system of seven equations, which can be obtained using Mathematica or similar software [39]. One of them can be written as

$$K\_3 \equiv (A\_1 - B\_1) \times \left[ 2\alpha\_u + (A\_1 - B\_1)^2 \right] = 0 \tag{10}$$

It was shown [42] that the second case is equal to the first one indicating that the value of a is irrelevant if B<sup>1</sup> ¼ 0. In other words, we could have assumed a simpler version of the Riccati

The third case is more interesting. It turns out that, instead of the three lines in Figure 2, that is, the three solutions, we obtain infinitely many lines corresponding to each of them [39]. However, they represent three groups of parallel lines, which means that all these solutions are only shifted functions and, consequently, have equal physical meaning. Therefore, this case does not

Case 4 is suggested by Eq. (10). The system of seven equations, mentioned earlier, gives the

2

1 sinh y � �, y <sup>¼</sup> <sup>3</sup>A<sup>0</sup>

<sup>2</sup> , B<sup>1</sup> ¼ � <sup>r</sup><sup>u</sup>

3A<sup>0</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � 3A<sup>0</sup> 2

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 9

2r<sup>u</sup>

(15)

ξ (16)

<sup>a</sup> <sup>¼</sup> <sup>0</sup>, A<sup>1</sup> <sup>¼</sup> <sup>2</sup>B1, <sup>α</sup><sup>u</sup> ¼ � <sup>B</sup><sup>1</sup>

tanh y þ

This case yields to a new solution, which was not obtained using less general mathematical methods. However, it may be interesting from mathematical point of view only as Ψ diverges

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � 3A<sup>0</sup> 2

q

equation neglecting the term 2aΦ.

Figure 2. The functions Ψ ξð Þ for r<sup>u</sup> ¼ 1 and σ ¼ 0:34.

bring about any physically new result.

The final expression for Ψ is

for ξ ¼ 0.

first and the last term in Eq. (11) as well as

Ψ ξð Þ¼ A<sup>0</sup> �

indicating two possible relationships between the parameters A<sup>1</sup> and B1. Hence, there are a few cases to be studied. They are as follows [39]: (1) B<sup>1</sup> ¼ 0, a ¼ 0; (2) B<sup>1</sup> ¼ 0, a 6¼ 0; (3) A<sup>1</sup> ¼ B1; (4) 2α<sup>u</sup> ¼ �ð Þ A<sup>1</sup> � B<sup>1</sup> 2 , A1B<sup>1</sup> 6¼ 0; (5) A<sup>1</sup> ¼ 0, a 6¼ 0 and (6) A<sup>1</sup> ¼ 0, a ¼ 0.

It is obvious that the first case represents nothing but a simpler method called extended tanhfunction method. The system mentioned earlier brings about [39]

$$8A\_0^{\;\;3} - 2A\_0 + \sigma = 0, \qquad a\_{\mu} = -\frac{A\_1^{\;\;2}}{2}, \qquad A\_1 = -\frac{\rho\_u}{3A\_0}, \qquad b = \frac{1 - 3A\_0^{\;2}}{A\_1^{\;\;2}} \tag{11}$$

The final result is [39]

$$
\Psi\_i = A\_{0i} + A\_{1i} \Phi\_{i\nu} \qquad \Phi\_i = \sqrt{b\_i} \tanh\left(\sqrt{b\_i}\,\xi\right) \tag{12}
$$

where A0<sup>i</sup> is the following three real solutions of the first of Eqs. (11)

$$A\_{01} = \frac{1}{2\sqrt{3}} \left( \cos F + \sqrt{3} \sin F \right), \quad A\_{02} = \frac{1}{2\sqrt{3}} \left( \cos F - \sqrt{3} \sin F \right) \tag{13}$$

$$A\_{03} = -\frac{1}{\sqrt{3}} \cos F, \quad F = \frac{1}{3} \arccos \left(\frac{\sigma}{\sigma\_0}\right), \quad \sigma\_0 = \frac{2}{3\sqrt{3}} \tag{14}$$

Of course, these three solutions exist for σ < σ0. The case σ > σ<sup>0</sup> was discussed in Ref. [25].

All the three solutions are shown in Figure 2 for σ ≈ 0:9σ<sup>0</sup> and r<sup>u</sup> ¼ 1. Of course, these solutions reproduce previously known results [25]. Figure 2 shows that the solutions of Eq. (4) are kink and antikink solitons. More detailed analysis of their physical meaning is given in Ref. [25].

Figure 2. The functions Ψ ξð Þ for r<sup>u</sup> ¼ 1 and σ ¼ 0:34.

The well-known general solution of Eq. (7) is [39, 40]

8 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

for N ¼ 1 and (7), we obtain the expressions for Ψ<sup>0</sup>

0

function method. The system mentioned earlier brings about [39]

<sup>3</sup> � <sup>2</sup>A<sup>0</sup> <sup>þ</sup> <sup>σ</sup> <sup>¼</sup> <sup>0</sup>, <sup>α</sup><sup>u</sup> ¼ � <sup>A</sup><sup>1</sup>

where A0<sup>i</sup> is the following three real solutions of the first of Eqs. (11)

ffiffiffi 3

<sup>p</sup> cos <sup>F</sup> <sup>þ</sup> ffiffiffi

<sup>A</sup><sup>03</sup> ¼ � <sup>1</sup>

<sup>K</sup>3Φ<sup>3</sup> <sup>þ</sup> <sup>K</sup><sup>3</sup>

software [39]. One of them can be written as

2

<sup>A</sup><sup>01</sup> <sup>¼</sup> <sup>1</sup> 2 ffiffiffi 3

In what follows, we assume ξ<sup>0</sup> ¼ 0.

yields to the following expression:

(4) 2α<sup>u</sup> ¼ �ð Þ A<sup>1</sup> � B<sup>1</sup>

The final result is [39]

8A<sup>0</sup>

<sup>Φ</sup> <sup>¼</sup> <sup>a</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>Φ</sup>�<sup>3</sup> <sup>þ</sup> <sup>K</sup>2Φ<sup>2</sup> <sup>þ</sup> <sup>K</sup><sup>2</sup>

<sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup> <sup>p</sup> tanh ffiffiffiffiffiffiffiffiffiffiffiffi

Our next step is determination of the parameters A0, A1, B1, a, b and αu. According to Eqs. (6)

0

Obviously, this is satisfied if all the coefficients are simultaneously equal to zero. This brings about a system of seven equations, which can be obtained using Mathematica or similar

indicating two possible relationships between the parameters A<sup>1</sup> and B1. Hence, there are a few cases to be studied. They are as follows [39]: (1) B<sup>1</sup> ¼ 0, a ¼ 0; (2) B<sup>1</sup> ¼ 0, a 6¼ 0; (3) A<sup>1</sup> ¼ B1;

, A1B<sup>1</sup> 6¼ 0; (5) A<sup>1</sup> ¼ 0, a 6¼ 0 and (6) A<sup>1</sup> ¼ 0, a ¼ 0. It is obvious that the first case represents nothing but a simpler method called extended tanh-

2

<sup>Ψ</sup><sup>i</sup> <sup>¼</sup> <sup>A</sup>0<sup>i</sup> <sup>þ</sup> <sup>A</sup>1iΦi, <sup>Φ</sup><sup>i</sup> <sup>¼</sup> ffiffiffiffi

3 <sup>p</sup> sin <sup>F</sup> � �

<sup>p</sup> cos F, F <sup>¼</sup> <sup>1</sup>

K<sup>3</sup> � ð Þ� A<sup>1</sup> � B<sup>1</sup> 2α<sup>u</sup> þ ð Þ A<sup>1</sup> � B<sup>1</sup>

<sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup> <sup>p</sup> ð Þ <sup>ξ</sup> � <sup>ξ</sup><sup>0</sup> h i

<sup>Φ</sup>�<sup>2</sup> <sup>þ</sup> <sup>K</sup>1<sup>Φ</sup> <sup>þ</sup> <sup>K</sup><sup>1</sup>

<sup>2</sup> h i

<sup>2</sup> , A<sup>1</sup> ¼ � <sup>r</sup><sup>u</sup>

bi p tanh ffiffiffiffi

> 2 ffiffiffi 3

σ σ0 � �

, A<sup>02</sup> <sup>¼</sup> <sup>1</sup>

3 arccos

All the three solutions are shown in Figure 2 for σ ≈ 0:9σ<sup>0</sup> and r<sup>u</sup> ¼ 1. Of course, these solutions reproduce previously known results [25]. Figure 2 shows that the solutions of Eq. (4) are kink and antikink solitons. More detailed analysis of their physical meaning is given in Ref. [25].

Of course, these three solutions exist for σ < σ0. The case σ > σ<sup>0</sup> was discussed in Ref. [25].

3A<sup>0</sup>

bi p ξ � �

<sup>p</sup> cos <sup>F</sup> � ffiffiffi

, <sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>

, b <sup>¼</sup> <sup>1</sup> � <sup>3</sup>A<sup>0</sup>

3 <sup>p</sup> sin <sup>F</sup> � �

> 3 ffiffiffi 3

A1

0

, Ψ<sup>00</sup> and Ψ<sup>3</sup> as required by Eq. (4), which

<sup>Φ</sup>�<sup>1</sup> <sup>þ</sup> <sup>K</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup> (9)

¼ 0 (10)

2

p (14)

<sup>2</sup> (11)

(12)

(13)

(8)

It was shown [42] that the second case is equal to the first one indicating that the value of a is irrelevant if B<sup>1</sup> ¼ 0. In other words, we could have assumed a simpler version of the Riccati equation neglecting the term 2aΦ.

The third case is more interesting. It turns out that, instead of the three lines in Figure 2, that is, the three solutions, we obtain infinitely many lines corresponding to each of them [39]. However, they represent three groups of parallel lines, which means that all these solutions are only shifted functions and, consequently, have equal physical meaning. Therefore, this case does not bring about any physically new result.

Case 4 is suggested by Eq. (10). The system of seven equations, mentioned earlier, gives the first and the last term in Eq. (11) as well as

$$a = 0, \qquad A\_1 = 2B\_1, \qquad \alpha\_u = -\frac{B\_1}{2}^2, \qquad B\_1 = -\frac{\rho\_u}{3A\_0} \tag{15}$$

The final expression for Ψ is

$$\Psi(\xi) = A\_0 - \sqrt{1 - 3A\_0^2} \left( \tanh y + \frac{1}{\sinh y} \right), \quad y = \frac{3A\_0}{2\rho\_u} \sqrt{1 - 3A\_0^2} \xi \tag{16}$$

This case yields to a new solution, which was not obtained using less general mathematical methods. However, it may be interesting from mathematical point of view only as Ψ diverges for ξ ¼ 0.

Case 5 is a simplified version of SEM, explained in Ref. [42]. The mentioned system brings about r<sup>u</sup> ¼ 0 as well as

$$B\_1 = -\frac{\mathbf{A}\_0}{a}, \quad \alpha\_\iota = -\frac{B\_1}{2}^2, \quad \mathbf{A}\_0^{\ast 3} - \mathbf{A}\_0 - \sigma = \mathbf{0}, \quad b = \frac{a^2 \left(\mathbf{A}\_0^{\ast 2} - 1\right)}{2\mathbf{A}\_0^{\ast 2}}\tag{17}$$

where a notation A0 has been introduced to distinguish this parameter from A0, used in the previous cases. It is interesting to compare the polynomials for A<sup>0</sup> and A0, existing in Eqs. (11) and (17). We can see that.

$$\mathbf{A}\_{0i} = -2\mathbf{A}\_{0i} \qquad \mathbf{i} = \mathbf{1}, \ \mathbf{2}, \ \mathbf{3} \tag{18}$$

which means that the values for A0<sup>i</sup> are given by Eqs. (13), (14) and (18). We can easily show that the final solution for Ψ is [39]

$$\Psi = \mathbf{A}\_0 - \frac{\mathbf{A}\_0 \mathbf{K}^2}{\cosh^2(aK\xi) \ [1 + K \tanh(aK\xi)]}, \qquad \mathbf{K} = \frac{\sqrt{3\mathbf{A}\_0^2 - 1}}{\mathbf{A}\_0 \sqrt{2}} \tag{19}$$

Obviously, this function cannot diverge for any value of aξ but only for �1 < K < 1. Also, K should be real and these two requirements eliminate Ψ<sup>2</sup> and Ψ<sup>3</sup> [39], which means that Ψ and A0 in Eq. (19) are Ψ<sup>1</sup> and A01.

The function Ψ1ð Þ ξ is shown in Figure 3 for a ¼ 0:1 and for two values of the parameter σ. We notice very interesting result that is a bell-type soliton! This certainly demonstrates the advantage of SEM method over the less general ones.

It is important to study the physical meanings of the parameters a and σ. Eq. (19) indicates that solitonic width is inversely proportional to j j a and that a does not affect maximum of the wave. Figure 3 shows that the amplitude of Ψ<sup>1</sup> is a decreasing function with respect to σ.

Finally, the last case gives the solution

$$\Psi' = \pm \frac{\sqrt{2}}{\sin \left( 2\sqrt{-b}\xi \right)}, \qquad b < 0 \tag{20}$$

the method is a multiple-scale method or a derivative-expansion method [16, 44]. We assume

According to the semi-discrete approximation, we look for wave solution which is a modu-

<sup>i</sup>θ<sup>n</sup> <sup>þ</sup> <sup>ε</sup> <sup>F</sup>0ð Þþ <sup>ξ</sup> <sup>ε</sup>F2ð Þ <sup>ξ</sup> <sup>e</sup>

where ω is the optical frequency of the linear approximation, q = 2π/λ > 0 is the wave number, cc represents complex conjugate terms and the function F<sup>0</sup> is real. Of course, l is the dimer's length, as mentioned earlier. The function F is continuous and represents an envelope, while exp(iθn), including discreteness, is a carrier component. Notice that the parameter ε exists in

<sup>ε</sup>mΦ€ <sup>n</sup> <sup>¼</sup> <sup>ε</sup> <sup>k</sup>ðΦ<sup>n</sup>þ<sup>1</sup> <sup>þ</sup> <sup>Φ</sup><sup>n</sup>�<sup>1</sup> � <sup>2</sup>ΦnÞ þ <sup>ε</sup>AΦ<sup>n</sup> � <sup>ε</sup><sup>3</sup>

unðÞ ¼ t ε Φnð Þt , ε << 1 (21)

BΦ<sup>n</sup>

ξ ¼ ð Þ εnl; εt , θ<sup>n</sup> ¼ nql � ωt (24)

<sup>3</sup> <sup>þ</sup> qE <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>4</sup> (22)

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 11

<sup>i</sup>2θ<sup>n</sup> <sup>þ</sup> cc <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup> (23)

small oscillations

lated wave, that is [28, 45]

which straightforwardly transforms Eq. (2) into

Figure 3. A bell-type soliton for a = 0.1 and σ = 0.34 (a) and σ = 0.1 (b).

ΦnðÞ¼ t Fð Þ ξ e

which is obviously divergent for ffiffiffiffiffiffi �<sup>b</sup> <sup>p</sup> <sup>ξ</sup> <sup>¼</sup> <sup>k</sup>π, <sup>k</sup> <sup>¼</sup> <sup>0</sup>, � <sup>1</sup>, � <sup>2</sup>, ….

Therefore, all the cases are explained and we can see that the continuum approximation yields to both kink solitons and bell-type solitons. The latter may exist only if viscosity is neglected.

It was stated earlier that the coordinate u was the projection of the top of the dimer on the direction of MT. A patient reader may ask how u can be negative when this is the projection. This question is answered in Ref. [28].

It was mentioned earlier that there are two approximations that can be used to solve Eq. (2). Now we get back to Eq. (2) and study semi-discrete one [15, 28, 43]. A mathematical basis for

Figure 3. A bell-type soliton for a = 0.1 and σ = 0.34 (a) and σ = 0.1 (b).

Case 5 is a simplified version of SEM, explained in Ref. [42]. The mentioned system brings

where a notation A0 has been introduced to distinguish this parameter from A0, used in the previous cases. It is interesting to compare the polynomials for A<sup>0</sup> and A0, existing in Eqs. (11)

which means that the values for A0<sup>i</sup> are given by Eqs. (13), (14) and (18). We can easily show

cosh <sup>2</sup>ð Þ aK<sup>ξ</sup> ½ � <sup>1</sup> <sup>þ</sup> <sup>K</sup>tanhð Þ aK<sup>ξ</sup> , K <sup>¼</sup>

Obviously, this function cannot diverge for any value of aξ but only for �1 < K < 1. Also, K should be real and these two requirements eliminate Ψ<sup>2</sup> and Ψ<sup>3</sup> [39], which means that Ψ and

The function Ψ1ð Þ ξ is shown in Figure 3 for a ¼ 0:1 and for two values of the parameter σ. We notice very interesting result that is a bell-type soliton! This certainly demonstrates the advan-

It is important to study the physical meanings of the parameters a and σ. Eq. (19) indicates that solitonic width is inversely proportional to j j a and that a does not affect maximum of the wave.

> ffiffiffi 2 p sin 2 ffiffiffiffiffiffi

�<sup>b</sup> <sup>p</sup> <sup>ξ</sup> <sup>¼</sup> <sup>k</sup>π, <sup>k</sup> <sup>¼</sup> <sup>0</sup>, � <sup>1</sup>, � <sup>2</sup>, ….

Therefore, all the cases are explained and we can see that the continuum approximation yields to both kink solitons and bell-type solitons. The latter may exist only if viscosity is neglected. It was stated earlier that the coordinate u was the projection of the top of the dimer on the direction of MT. A patient reader may ask how u can be negative when this is the projection.

It was mentioned earlier that there are two approximations that can be used to solve Eq. (2). Now we get back to Eq. (2) and study semi-discrete one [15, 28, 43]. A mathematical basis for

Figure 3 shows that the amplitude of Ψ<sup>1</sup> is a decreasing function with respect to σ.

Ψ ¼ �

<sup>3</sup> � A0 � <sup>σ</sup> <sup>¼</sup> <sup>0</sup>, b <sup>¼</sup> <sup>a</sup><sup>2</sup> A0

A0<sup>i</sup> ¼ �2A0i, i ¼ 1, 2, 3 (18)

<sup>2</sup> � <sup>1</sup> � � 2A0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3A0 <sup>2</sup> � <sup>1</sup> <sup>p</sup> A0 ffiffiffi 2

�<sup>b</sup> <sup>p</sup> <sup>ξ</sup> � � , b <sup>&</sup>lt; <sup>0</sup> (20)

<sup>2</sup> (17)

p (19)

about r<sup>u</sup> ¼ 0 as well as

and (17). We can see that.

that the final solution for Ψ is [39]

A0 in Eq. (19) are Ψ<sup>1</sup> and A01.

tage of SEM method over the less general ones.

Finally, the last case gives the solution

which is obviously divergent for ffiffiffiffiffiffi

This question is answered in Ref. [28].

<sup>B</sup><sup>1</sup> ¼ � A0

<sup>a</sup> , <sup>α</sup><sup>u</sup> ¼ � <sup>B</sup><sup>1</sup>

10 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>Ψ</sup> <sup>¼</sup> A0 � A0K<sup>2</sup>

2 <sup>2</sup> , A0

> the method is a multiple-scale method or a derivative-expansion method [16, 44]. We assume small oscillations

$$
\mu\_n(t) \; := \varepsilon \, \Phi\_n(t), \qquad \varepsilon << 1 \tag{21}
$$

which straightforwardly transforms Eq. (2) into

$$
\varepsilon \,\varepsilon m \ddot{\Phi}\_n = \varepsilon \, k (\Phi\_{n+1} + \Phi\_{n-1} - 2\Phi\_n) + \varepsilon \, A \Phi\_n - \varepsilon^3 B \Phi\_n^{,3} + qE + \mathcal{O}(\varepsilon^4) \tag{22}
$$

According to the semi-discrete approximation, we look for wave solution which is a modulated wave, that is [28, 45]

$$\Phi\_n(t) = F(\xi)e^{i\theta\_n} + \varepsilon \,\, F\_0(\xi) + \varepsilon \, F\_2(\xi)e^{i2\theta\_n} + \infty + \mathcal{O}(\varepsilon^2) \tag{23}$$

$$
\xi = (\varepsilon nl, \varepsilon t), \qquad \Theta\_n = nql - \omega t \tag{24}
$$

where ω is the optical frequency of the linear approximation, q = 2π/λ > 0 is the wave number, cc represents complex conjugate terms and the function F<sup>0</sup> is real. Of course, l is the dimer's length, as mentioned earlier. The function F is continuous and represents an envelope, while exp(iθn), including discreteness, is a carrier component. Notice that the parameter ε exists in the function F, but does not in exp(iθn). This is because the frequency of the carrier wave is much higher than the frequency of the envelope and we need two time scales, t and εt, for those two functions. The same holds for the coordinate scales.

To simplify the problem, a continuum limit nl ! z should be introduced as well as new transformations Z ¼ εz and T ¼ εt. This allows a series expansion of Fð Þ ξ , that is

$$F(\varepsilon(n \pm 1)l, \varepsilon l) \to F(Z, T) \pm F\varepsilon(Z, T)\varepsilon l + \frac{1}{2}F\varepsilon\mathbf{z}(Z, T)\,\varepsilon^2l^2\tag{25}$$

absent in the space scaling is given in Refs. [45, 46]. If we consider the terms for e<sup>i</sup><sup>θ</sup> again we

2 � �, Q ¼ � <sup>3</sup><sup>B</sup>

Even though Eq. (31) is PDE, its solution exists. This well-known solution, existing for PQ > 0,

where parameters ue and uc represent envelope and carrier component velocities, while the

It is very difficult to deal with the parameters ue and uc as ue > 2uc is completely unprecise statement. However, uc=ue < 0:5 seems to be more practical. Hence, new parameters Ue and η have been introduced as Ue ¼ εue, η ¼ uc=ue and 0 ≤ η < 0:5 [45]. Finally, we can easily obtain

� � cos ð Þ� <sup>Θ</sup>nl � <sup>Ω</sup><sup>t</sup>

Ue

One more parameter can be eliminated using the idea of coherent mode [49]. This mode means that the envelope and the carrier wave velocities are equal. It follows from Eq. (35) that Ve ¼ Ω=Θ, which yields to the function Ueð Þ η . This means that the wave unð Þt is the one phase

To plot the function unð Þt or, equivalently, Unð Þt we should know or estimate the values of a couple of the parameters. Of course, if 2D plot is chosen, Unð Þt can be presented as either a

, L � Le

<sup>2</sup><sup>P</sup> , <sup>Ω</sup> <sup>¼</sup> <sup>ω</sup> <sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � 2η <sup>2</sup>PQ <sup>s</sup>

iueð Þ S � ucτ

, Le <sup>¼</sup> <sup>2</sup><sup>P</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ue

<sup>2</sup> � 2ue uc

C

<sup>ε</sup> <sup>¼</sup> <sup>2</sup><sup>P</sup>

<sup>A</sup> � Unð Þ� <sup>t</sup>

Ue j j ffiffiffiffiffiffiffiffiffiffiffiffiffi

V<sup>g</sup> þ ηUe � �Ue

F ¼ 0 (31)

<sup>2</sup><sup>P</sup> , ue <sup>&</sup>gt; <sup>2</sup>uc (33)

<sup>p</sup> (34)

C

<sup>A</sup> (35)

<sup>1</sup> � <sup>2</sup><sup>η</sup> <sup>p</sup> (36)

<sup>2</sup><sup>P</sup> (37)

<sup>2</sup>m<sup>ω</sup> (32)

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 13

iF<sup>τ</sup> <sup>þ</sup> P FSS <sup>þ</sup> Q Fj j<sup>2</sup>

obtain the well-known nonlinear Schrödinger equation (NLSE) for the function F

cos ð Þ� ql Vg

where the dispersion coefficient P and the coefficient of nonlinearity Q are

Le � �exp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ue

<sup>2</sup> � 2ueuc <sup>2</sup>PQ <sup>s</sup>

the expression for the longitudinal displacement of the dimer at the position n

L

kul 2 m

F Sð Þ¼ ; <sup>τ</sup> Ae sech <sup>S</sup> � ue<sup>τ</sup>

amplitude Ae and the soliton width Le have the forms

Ae ¼

unðÞ¼ <sup>t</sup> <sup>A</sup>0sech nl � Ve <sup>t</sup>

A<sup>0</sup> � 2εAe ¼ 2 Ue j j

function, preserving its shape in time.

Ve ¼ Vg þ Ue, Θ ¼ q þ

<sup>P</sup> <sup>¼</sup> <sup>1</sup> 2ω

is [15, 47, 48]

where

where indexes Z and ZZ denote the first and the second derivative with respect to Z. Hence, the function Φnð Þt becomes

$$\Phi\_n(t) = F e^{i\theta} + F^\* e^{-i\theta} + \varepsilon \, F\_0 + F\_2 e^{i2\theta} + F\_2^\* \, e^{-i2\theta} \tag{26}$$

where <sup>∗</sup> stands for complex conjugate and <sup>F</sup> � F Zð Þ ; <sup>T</sup> . All this allows us to obtain the expressions existing in Eq. (22), such as Φ<sup>n</sup>þ<sup>1</sup> þ Φ<sup>n</sup>�<sup>1</sup> � 2Φn, Φ<sup>n</sup> and Φ<sup>n</sup> 3 , and Eq. (22) becomes [28]

$$\begin{aligned} & \left(\varepsilon^3 F\_{T7} - 2i\varepsilon^2 \omega F\_T - \varepsilon \omega^2 F\right) e^{i\theta} - \left(4i\varepsilon^3 \omega F\_{2T} + 4\varepsilon^2 \omega^2 F\_2\right) e^{i2\theta} + \csc \varepsilon = \\ & = \varepsilon \frac{k}{m} \left\{ 2F\_1 \left[ \cos\left(ql\right) - 1\right] + 2i\varepsilon l F\_Z \sin\left(ql\right) + \varepsilon^2 l^2 F\_{ZZ} \cos\left(ql\right) \right\} e^{i\theta} \\ & + \varepsilon \frac{k}{m} \left\{ 2\varepsilon F\_2 \left[ \cos\left(2ql\right) - 1\right] + 2i\varepsilon^2 l F\_{2Z} \sin\left(2ql\right) \right\} e^{i2\theta} + \frac{\mathsf{C}}{m} \\ & + \varepsilon \frac{A}{m} \left( F e^{i\theta} + \varepsilon F\_0 + \varepsilon F\_2 e^{i2\theta} \right) - \varepsilon^3 \frac{B}{m} \left( 3|F|^2 F e^{i\theta} + F^3 e^{i3\theta} \right) + \csc + \mathsf{O}\left(\varepsilon^4\right) \end{aligned} \tag{27}$$

This crucial expression represents a starting point for a series of important expressions. They can be obtained equating the coefficients for the various harmonics. For example, equating the coefficients for e<sup>i</sup><sup>θ</sup> and neglecting all the terms with ε<sup>2</sup> and ε<sup>3</sup> one obtains the following expressions for the dispersion relation ω ¼ ωð Þq and the group velocity dω=dq:

$$
\omega^2 = \frac{2k\_u}{m} \left[ 1 - \cos\left(ql\right) \right] - \frac{A}{m'} \qquad V\_{\mathcal{g}} = \frac{lk\_u}{m\omega} \sin\left(ql\right) \tag{28}
$$

Also, the coefficients for <sup>e</sup><sup>i</sup><sup>0</sup> <sup>¼</sup> 1 and <sup>e</sup><sup>i</sup>2<sup>θ</sup> , respectively give [28]

$$
\varepsilon^2 F\_0 = -\frac{\mathbf{C}}{A'} \qquad F\_2 = \mathbf{0} \tag{29}
$$

which yields to

$$
\mu\_{\rm tr} = \varepsilon F e^{i\theta\_{\rm tr}} - \frac{\mathbf{C}}{A} + \mathbf{cc} \tag{30}
$$

Eqs. (28) and (29) and new coordinates S and τ, defined as S ¼ Z � Vg T and τ ¼ εT, allows us to simplify Eq. (27). An explanation for why the parameter ε exists in the time scaling but is absent in the space scaling is given in Refs. [45, 46]. If we consider the terms for e<sup>i</sup><sup>θ</sup> again we obtain the well-known nonlinear Schrödinger equation (NLSE) for the function F

$$\left| iF\_{\pi} + PF\_{SS} + Q \right| F \big|^{2} F = 0 \tag{31}$$

where the dispersion coefficient P and the coefficient of nonlinearity Q are

$$P = \frac{1}{2\omega} \left[ \frac{k\_\mu l^2}{m} \cos\left(ql\right) - V\_\mathcal{g}^{\prime} \right], \qquad \qquad Q = -\frac{3B}{2m\omega} \tag{32}$$

Even though Eq. (31) is PDE, its solution exists. This well-known solution, existing for PQ > 0, is [15, 47, 48]

$$F(\mathcal{S}, \tau) = A\_{\varepsilon} \operatorname{sech} \left( \frac{\mathcal{S} - u\_{\varepsilon}\tau}{L\_{\varepsilon}} \right) \exp \frac{\operatorname{i} u\_{\varepsilon} (\mathcal{S} - u\_{\varepsilon}\tau)}{2P}, \qquad \qquad u\_{\varepsilon} > 2u\_{\varepsilon} \tag{33}$$

where parameters ue and uc represent envelope and carrier component velocities, while the amplitude Ae and the soliton width Le have the forms

$$A\_{\varepsilon} = \sqrt{\frac{\mu\_{\varepsilon}^{2} - 2\mu\_{\varepsilon}\mu\_{\varepsilon}}{2PQ}}, \qquad \qquad L\_{\varepsilon} = \frac{2P}{\sqrt{\mu\_{\varepsilon}^{2} - 2\mu\_{\varepsilon}\mu\_{\varepsilon}}} \tag{34}$$

It is very difficult to deal with the parameters ue and uc as ue > 2uc is completely unprecise statement. However, uc=ue < 0:5 seems to be more practical. Hence, new parameters Ue and η have been introduced as Ue ¼ εue, η ¼ uc=ue and 0 ≤ η < 0:5 [45]. Finally, we can easily obtain the expression for the longitudinal displacement of the dimer at the position n

$$u\_n(t) = A\_0 \text{sech}\left(\frac{nl - V\_\varepsilon t}{L}\right) \cdot \cos\left(\Theta nl - \Omega t\right) - \frac{\mathbb{C}}{A} \equiv U\_n(t) - \frac{\mathbb{C}}{A} \tag{35}$$

where

the function F, but does not in exp(iθn). This is because the frequency of the carrier wave is much higher than the frequency of the envelope and we need two time scales, t and εt, for

To simplify the problem, a continuum limit nl ! z should be introduced as well as new trans-

where indexes Z and ZZ denote the first and the second derivative with respect to Z. Hence,

where <sup>∗</sup> stands for complex conjugate and <sup>F</sup> � F Zð Þ ; <sup>T</sup> . All this allows us to obtain the expres-

FZZ cos ð Þ ql <sup>e</sup>

m

This crucial expression represents a starting point for a series of important expressions. They can be obtained equating the coefficients for the various harmonics. For example, equating the coefficients for e<sup>i</sup><sup>θ</sup> and neglecting all the terms with ε<sup>2</sup> and ε<sup>3</sup> one obtains the following

m

, respectively give [28]

<sup>3</sup>j j <sup>F</sup> <sup>2</sup>

�i<sup>θ</sup> <sup>þ</sup> <sup>ε</sup>F<sup>0</sup> <sup>þ</sup> <sup>F</sup><sup>2</sup> <sup>e</sup>

1 2

<sup>i</sup>2<sup>θ</sup> <sup>þ</sup> <sup>F</sup><sup>2</sup> ∗ e

<sup>e</sup><sup>i</sup>2<sup>θ</sup> <sup>þ</sup> cc <sup>¼</sup>

<sup>i</sup>2<sup>θ</sup> <sup>þ</sup> C m

l 2

F e<sup>i</sup><sup>θ</sup> <sup>þ</sup> <sup>F</sup><sup>3</sup> <sup>e</sup> <sup>i</sup>3<sup>θ</sup>

, Vg <sup>¼</sup> l ku

mω

<sup>A</sup> , F<sup>2</sup> <sup>¼</sup> <sup>0</sup> (29)

<sup>A</sup> <sup>þ</sup> cc (30)

3

FZZð Þ <sup>Z</sup>; <sup>T</sup> <sup>ε</sup><sup>2</sup>

l

<sup>2</sup> (25)

(27)

�i2<sup>θ</sup> (26)

, and Eq. (22) becomes [28]

iθ

<sup>þ</sup> cc <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>4</sup>

sin ð Þ ql (28)

those two functions. The same holds for the coordinate scales.

12 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>Φ</sup>nðÞ¼ <sup>t</sup> F e<sup>i</sup><sup>θ</sup> <sup>þ</sup> <sup>F</sup><sup>∗</sup> <sup>e</sup>

sions existing in Eq. (22), such as Φ<sup>n</sup>þ<sup>1</sup> þ Φ<sup>n</sup>�<sup>1</sup> � 2Φn, Φ<sup>n</sup> and Φ<sup>n</sup>

F ei<sup>θ</sup> <sup>þ</sup> <sup>ε</sup>F<sup>0</sup> <sup>þ</sup> <sup>ε</sup>F<sup>2</sup> <sup>e</sup> <sup>i</sup>2<sup>θ</sup> � <sup>ε</sup><sup>3</sup> <sup>B</sup>

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>ku

Also, the coefficients for <sup>e</sup><sup>i</sup><sup>0</sup> <sup>¼</sup> 1 and <sup>e</sup><sup>i</sup>2<sup>θ</sup>

which yields to

the function Φnð Þt becomes

¼ ε k m

þε k m

þε A m

formations Z ¼ εz and T ¼ εt. This allows a series expansion of Fð Þ ξ , that is

Fð Þ! εð Þ n � 1 l; εt F Zð Þ� ; T FZð Þ Z; T εl þ

<sup>ε</sup><sup>3</sup>FTT � <sup>2</sup>iε<sup>2</sup>ωFT � εω<sup>2</sup><sup>F</sup> ei<sup>θ</sup> � <sup>4</sup>iε<sup>3</sup>ωF2<sup>T</sup> <sup>þ</sup> <sup>4</sup>ε<sup>2</sup>ω<sup>2</sup>F<sup>2</sup>

<sup>2</sup>F½ �þ cos ð Þ� ql <sup>1</sup> <sup>2</sup>iεlFZ sin ð Þþ ql <sup>ε</sup><sup>2</sup>

<sup>2</sup>εF<sup>2</sup> <sup>½</sup> cos 2ð Þ� ql <sup>1</sup>� þ <sup>2</sup>iε<sup>2</sup> lF2<sup>Z</sup> sin 2ð Þ ql <sup>e</sup>

expressions for the dispersion relation ω ¼ ωð Þq and the group velocity dω=dq:

<sup>m</sup> ½ �� <sup>1</sup> � cos ð Þ ql <sup>A</sup>

<sup>F</sup><sup>0</sup> ¼ � <sup>C</sup>

un <sup>¼</sup> <sup>ε</sup>Fei<sup>θ</sup><sup>n</sup> � <sup>C</sup>

Eqs. (28) and (29) and new coordinates S and τ, defined as S ¼ Z � Vg T and τ ¼ εT, allows us to simplify Eq. (27). An explanation for why the parameter ε exists in the time scaling but is

ε2

$$A\_0 \equiv 2\varepsilon A\_\varepsilon = 2|\mathcal{U}\_\varepsilon|\sqrt{\frac{1-2\eta}{2PQ}}, \qquad \qquad L \equiv \frac{L\_\varepsilon}{\varepsilon} = \frac{2P}{|\mathcal{U}\_\varepsilon|\sqrt{1-2\eta}}\tag{36}$$

$$V\_{\varepsilon} = V\_{\mathcal{S}} + \mathcal{U}\_{\varepsilon} \qquad \Theta = q + \frac{\mathcal{U}\_{\varepsilon}}{2P} \qquad \Omega = \omega + \frac{\left(\mathcal{V}\_{\mathcal{S}} + \eta \mathcal{U}\_{\varepsilon}\right) \mathcal{U}\_{\varepsilon}}{2P} \tag{37}$$

One more parameter can be eliminated using the idea of coherent mode [49]. This mode means that the envelope and the carrier wave velocities are equal. It follows from Eq. (35) that Ve ¼ Ω=Θ, which yields to the function Ueð Þ η . This means that the wave unð Þt is the one phase function, preserving its shape in time.

To plot the function unð Þt or, equivalently, Unð Þt we should know or estimate the values of a couple of the parameters. Of course, if 2D plot is chosen, Unð Þt can be presented as either a

<sup>H</sup><sup>φ</sup> <sup>¼</sup> <sup>X</sup> n

not exist in Eq. (38) even though the terms including φ<sup>n</sup>

Following the procedure explained earlier, we obtain

<sup>φ</sup> <sup>¼</sup> <sup>Ψ</sup> ffiffiffi 6

<sup>α</sup><sup>u</sup> <sup>¼</sup> <sup>m</sup>κ<sup>2</sup>

αφ <sup>¼</sup> <sup>I</sup>κ<sup>2</sup>

class of supersonic solitons. We will return to this issue in the next section.

<sup>ε</sup>IΦ€<sup>n</sup> <sup>¼</sup> <sup>ε</sup> <sup>k</sup>φðΦ<sup>n</sup>þ<sup>1</sup> <sup>þ</sup> <sup>Φ</sup><sup>n</sup>�<sup>1</sup> � <sup>2</sup>ΦnÞ � <sup>ε</sup>pEΦ<sup>n</sup> <sup>þ</sup> <sup>ε</sup><sup>3</sup>

<sup>I</sup> sin <sup>2</sup>ð Þ ql=<sup>2</sup> , <sup>ω</sup><sup>0</sup> <sup>¼</sup>

Like above, the solutions are kink solitons [50].

where

and

can be written as

other hand, αφ ¼ 2r<sup>φ</sup>

equation of motion, which is [51]

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

<sup>0</sup> þ 4k

I 2 φ\_ n 2 þ kφ

<sup>2</sup> <sup>φ</sup><sup>n</sup>þ<sup>1</sup> � <sup>φ</sup><sup>n</sup>

where I is a moment of inertia of the dimer at the position n. Notice that the W-potential does

expansion of the cosine function. Instead of viscosity force introduced in the previous section, we introduce viscosity momentum Mv ¼ �Γφ\_ , where Γ is the viscosity coefficient [51–53].

It is very interesting to compare the expressions for α<sup>u</sup> and αφ, given by Eqs. (5) and (40). They

<sup>2</sup> � �, cu

<sup>2</sup> � �, c<sup>φ</sup>

where v ¼ ω=κ is the soliton velocity, while cu and c<sup>φ</sup> are corresponding sound velocities. According to Eq. (11), we can see that the U-model predicts cu > v as A is positive. On the

Now, we switch to the semi-discrete approximation within the φ-model to solve the dynamical

where φ ¼ εΦ has been used. Of course, Eq. (43) is analog to Eq. (22). Following the procedure explained in the previous section, we straightforwardly obtain F<sup>0</sup> ¼ 0 and F2ð Þ¼ ξ 0, as well as

where ω<sup>0</sup> is the lowest frequency of the oscillations [59]. Also, we easily obtain NLSE (31), where

ffiffiffiffiffiffiffiffiffiffi pE=I q

2 κ2 pE , <sup>r</sup><sup>φ</sup> <sup>¼</sup> <sup>ω</sup><sup>Γ</sup>

<sup>p</sup> , αφ <sup>¼</sup> <sup>I</sup>ω<sup>2</sup> � <sup>k</sup>φ<sup>l</sup>

<sup>A</sup> <sup>v</sup><sup>2</sup> � cu

pE <sup>v</sup><sup>2</sup> � <sup>c</sup><sup>φ</sup>

� �<sup>2</sup> � pE cosφ<sup>n</sup>

<sup>2</sup> and φ<sup>n</sup>

<sup>2</sup> <sup>¼</sup> kul 2

<sup>2</sup> <sup>¼</sup> <sup>k</sup>φ<sup>l</sup> 2

<sup>2</sup>=9 > 0 [50] means that, according to the φ-model, the kink belongs to the

pEΦ<sup>n</sup>

, Vg <sup>¼</sup> l k

Iω

� � (38)

αφΨ<sup>00</sup> � <sup>r</sup>φΨ<sup>0</sup> <sup>þ</sup> <sup>Ψ</sup> � <sup>Ψ</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup> (39)

<sup>4</sup> appear as a result of a series

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 15

pE (40)

<sup>m</sup> (41)

<sup>I</sup> (42)

<sup>3</sup> <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>4</sup> � � (43)

sin ð Þ ql (44)

Figure 4. Function U nð Þ for <sup>t</sup> <sup>¼</sup> 50ns, <sup>A</sup> <sup>¼</sup> <sup>2</sup>:<sup>9</sup> � <sup>10</sup>�3N=m, <sup>B</sup> <sup>¼</sup> <sup>1</sup>:<sup>7</sup> � <sup>10</sup>14N=m3, ku <sup>¼</sup> <sup>150</sup>A, <sup>N</sup> <sup>¼</sup> 15 and <sup>η</sup> <sup>¼</sup> <sup>0</sup>:43.

function of t at a certain position n or as a function of n for chosen t. Very detailed analysis of the parameter selection was done in Ref. [28]. One example for q ¼ 2π=Nl is shown in Figure 4. Obviously, this is a localized modulated wave usually called as breather. We can see that its width is about 200 nm, which means that it covers about 25 dimers.

As a conclusion, we can state that the two mathematical procedures bring about even three results, that is, three different solitons. These are kinks, bell-type solitons and breathers. They may be signals for the motor proteins to start moving, as explained in Introduction.

Obviously, viscosity has been neglected. This will be explained in the following section, within the φ-model.

#### 4. φ-model

A weak point of the U-model is the last term in Eq. (1). A scalar product � p ! � <sup>E</sup> ! ¼ �QdE cosφ<sup>n</sup> would be better choice for the potential energy, where d is the distance between the centers of positive and negative charges within the dipole. This potential indicates the angle as a coordinate instead of the projection u and the Hamiltonian for the radial model, which we call as φ-model, is [50, 51]

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 15

$$H\_{\varphi} = \sum\_{n} \left[ \frac{I}{2} \dot{\boldsymbol{\phi}}\_{n}^{2} + \frac{k\_{\varphi}}{2} \left( \boldsymbol{\varphi}\_{n+1} - \boldsymbol{\varphi}\_{n} \right)^{2} - pE \cos \boldsymbol{\varphi}\_{n} \right] \tag{38}$$

where I is a moment of inertia of the dimer at the position n. Notice that the W-potential does not exist in Eq. (38) even though the terms including φ<sup>n</sup> <sup>2</sup> and φ<sup>n</sup> <sup>4</sup> appear as a result of a series expansion of the cosine function. Instead of viscosity force introduced in the previous section, we introduce viscosity momentum Mv ¼ �Γφ\_ , where Γ is the viscosity coefficient [51–53]. Following the procedure explained earlier, we obtain

$$
\alpha\_{\psi} \Psi'' - \rho\_{\psi} \Psi' + \Psi - \Psi^3 = 0 \tag{39}
$$

where

$$
\varphi = \Psi \sqrt{\mathbf{\hat{6}}}, \qquad \alpha\_{\varphi} = \frac{\mathbf{I} \omega^2 - k\_{\varphi} l^2 \kappa^2}{pE}, \qquad \rho\_{\varphi} = \frac{\omega \Gamma}{pE} \tag{40}
$$

Like above, the solutions are kink solitons [50].

It is very interesting to compare the expressions for α<sup>u</sup> and αφ, given by Eqs. (5) and (40). They can be written as

$$\alpha\_{\rm u} = \frac{m\kappa^2}{A} \left(v^2 - c\_{\rm u}{}^2\right), \quad c\_{\rm u}{}^2 = \frac{k\_{\rm u}l^2}{m} \tag{41}$$

and

function of t at a certain position n or as a function of n for chosen t. Very detailed analysis of the parameter selection was done in Ref. [28]. One example for q ¼ 2π=Nl is shown in Figure 4. Obviously, this is a localized modulated wave usually called as breather. We can see that its

Figure 4. Function U nð Þ for <sup>t</sup> <sup>¼</sup> 50ns, <sup>A</sup> <sup>¼</sup> <sup>2</sup>:<sup>9</sup> � <sup>10</sup>�3N=m, <sup>B</sup> <sup>¼</sup> <sup>1</sup>:<sup>7</sup> � <sup>10</sup>14N=m3, ku <sup>¼</sup> <sup>150</sup>A, <sup>N</sup> <sup>¼</sup> 15 and <sup>η</sup> <sup>¼</sup> <sup>0</sup>:43.

As a conclusion, we can state that the two mathematical procedures bring about even three results, that is, three different solitons. These are kinks, bell-type solitons and breathers. They

Obviously, viscosity has been neglected. This will be explained in the following section, within

would be better choice for the potential energy, where d is the distance between the centers of positive and negative charges within the dipole. This potential indicates the angle as a coordinate instead of the projection u and the Hamiltonian for the radial model, which we call as

! � <sup>E</sup> !

¼ �QdE cosφ<sup>n</sup>

may be signals for the motor proteins to start moving, as explained in Introduction.

A weak point of the U-model is the last term in Eq. (1). A scalar product � p

width is about 200 nm, which means that it covers about 25 dimers.

14 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

the φ-model.

4. φ-model

φ-model, is [50, 51]

$$\alpha\_{\varphi} = \frac{I\kappa^2}{pE} \left(\upsilon^2 - c\_{\varphi}{}^2\right), \quad c\_{\varphi}{}^2 = \frac{k\_{\varphi}l^2}{I} \tag{42}$$

where v ¼ ω=κ is the soliton velocity, while cu and c<sup>φ</sup> are corresponding sound velocities. According to Eq. (11), we can see that the U-model predicts cu > v as A is positive. On the other hand, αφ ¼ 2r<sup>φ</sup> <sup>2</sup>=9 > 0 [50] means that, according to the φ-model, the kink belongs to the class of supersonic solitons. We will return to this issue in the next section.

Now, we switch to the semi-discrete approximation within the φ-model to solve the dynamical equation of motion, which is [51]

$$
\varepsilon I \ddot{\Phi}\_n = \varepsilon \left( k\_\psi (\Phi\_{n+1} + \Phi\_{n-1} - 2\Phi\_n) - \varepsilon p E \Phi\_n + \varepsilon^3 p E \Phi\_n^{-3} + \mathcal{O} \left( \varepsilon^4 \right) \right) \tag{43}
$$

where φ ¼ εΦ has been used. Of course, Eq. (43) is analog to Eq. (22). Following the procedure explained in the previous section, we straightforwardly obtain F<sup>0</sup> ¼ 0 and F2ð Þ¼ ξ 0, as well as

$$
\omega^2 = a\_0^2 + \frac{4k}{I} \sin^2(ql/2), \qquad \omega\_0 = \sqrt{pE/I}, \qquad V\_{\mathcal{S}} = \frac{lk}{I\omega} \sin(ql) \tag{44}
$$

where ω<sup>0</sup> is the lowest frequency of the oscillations [59]. Also, we easily obtain NLSE (31), where

$$P = \frac{1}{2\omega} \left[ \frac{l^2 k}{I} \cos\left(ql\right) - \left. V\_{\mathcal{S}} \right|^2 \right] \qquad Q = \frac{3pE}{2I\omega}.\tag{45}$$

All of them are studied in Ref. [54] and they will be explained here briefly.

<sup>Ψ</sup> � <sup>K</sup>Ψ, <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>I</sup>ω<sup>2</sup> � kl<sup>2</sup>

4r<sup>1</sup> ξ

Eq. (50) holds for both positive and negative r1. Therefore, φ ξð Þ represents kink soliton if

One of the advantages of the GM over the φ-model is the value of amplitude. Namely, the

<sup>1</sup> <sup>þ</sup> tanh <sup>3</sup>

<sup>α</sup>1Ψ<sup>00</sup> � <sup>r</sup>1Ψ<sup>0</sup> <sup>þ</sup> <sup>Ψ</sup> � <sup>Ψ</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup> (48)

pE � <sup>A</sup> , <sup>r</sup><sup>1</sup> <sup>¼</sup> <sup>Γ</sup><sup>ω</sup>

� � � � , <sup>α</sup><sup>1</sup> <sup>&</sup>gt; <sup>0</sup> (50)

6

<sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>K</sup>tanhð Þ <sup>ξ</sup>=<sup>a</sup> , a<sup>2</sup> <sup>¼</sup> <sup>2</sup>α<sup>1</sup> (51)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pE � A B � pE=6

� � � � , <sup>r</sup><sup>2</sup> 6¼ <sup>0</sup> (53)

tan ð Þ <sup>ξ</sup>=<sup>a</sup> , a<sup>2</sup> ¼ �2α, <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> (54)

Ψ � K<sup>0</sup>

s

<sup>p</sup> , the appropriate factor, existing in the GM, is <sup>K</sup>, given

pE � <sup>A</sup> (49)

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 17

<sup>p</sup> , coming from Eq. (40). This is

Ψ (52)

κ2

Case 1 straightforwardly yields to

φ ¼ �

and the final solution is [54]

s

unrealistic, too big value. Instead of ffiffiffi

Case 2 straightforwardly yields to

and to the final results

and

diverge.

Case 3 brings about

If viscosity is neglected, the GM brings about

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A � pE B � pE=6

> φ ξð Þ¼ <sup>K</sup> 2

r<sup>1</sup> > 0 and antikink one for negative r1, which is shown in Figure 5.

6

<sup>α</sup>2Ψ<sup>00</sup> � <sup>r</sup>2Ψ<sup>0</sup> <sup>þ</sup> <sup>Ψ</sup> <sup>þ</sup> <sup>Ψ</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>φ</sup> ¼ �

K0

<sup>2</sup> <sup>1</sup> <sup>þ</sup> tanh <sup>3</sup>

4r<sup>2</sup> ξ

It is obvious that these results do not have physical meaning as φ<sup>2</sup> is complex, while φ<sup>20</sup> may

φ<sup>2</sup> ¼ i

φ<sup>20</sup> ¼ K<sup>0</sup>

amplitude of the kink soliton, according to the φ-model, is ffiffiffi

where

by Eq. (49).

The final solution φnð Þt is the same as Unð Þt except that P and Q are different. Therefore, both the U- and the φ-models predict the breather waves moving through MT.

Finally, viscosity should be introduced in the semi-discrete approximation [51]. Due to viscosity momentum Mv ¼ �ΓΦ\_ <sup>n</sup>, the final result <sup>φ</sup>nð Þ<sup>t</sup> includes the expected exponential term <sup>e</sup>�<sup>β</sup> <sup>t</sup> , where β ¼ Γ=2I [51].

#### 5. General model of MTs

It was mentioned earlier that the weak point of the U-model is the last term in Eq. (1). Also, it is better to use the radial coordinate φ than the longitudinal one as we assume angular oscillations of the dimers. The scalar product � p ! � <sup>E</sup> ! ¼ �QdE cosφn, existing in the φ-model, solved these problems but the W-potential has been missing. In fact, a series expansion of cosφ<sup>n</sup> gives φn <sup>2</sup> and φ<sup>n</sup> <sup>4</sup> terms but with opposite signs from those in the U-model. These two terms are, practically, a potential that looks like W in a mirror having only one minimum surrounded by two maxima and, due to its shape, can be called as M-potential [54]. This potential brings about αφ > 0, which is disputable result.

Therefore, we want to solve the mentioned problem regarding the U-model but to keep the Wpotential, the coordinate φ and, probably, Iω<sup>2</sup> < kl<sup>2</sup> κ2, that is, α < 0. This suggests the following Hamiltonian

$$H = \sum\_{n} \left[ \frac{I}{2} \dot{\boldsymbol{\varphi}}\_{n}^{2} + \frac{k}{2} (\boldsymbol{\varphi}\_{n+1} - \boldsymbol{\varphi}\_{n})^{2} - \frac{A}{2} \boldsymbol{\varphi}\_{n}^{2} + \frac{B}{4} \boldsymbol{\varphi}\_{n}^{4} - pE \cos \boldsymbol{\varphi}\_{n} \right] \tag{46}$$

where A > 0, B > 0 and φ<sup>n</sup> has the same meaning as in the φ-model. Let us call the model as general one (GM). The procedure mentioned earlier brings about

$$\left(\text{I}\omega^2 - \text{kl}^2\text{\textdegree k}^2\right)\varphi'' - \Gamma\omega\varphi' - (A - pE)\varphi + \left(B - \frac{pE}{6}\right)\varphi^3 = 0\tag{47}$$

where, of course, φ � φ ξð Þ. If we consider Eqs. (3) and (47), we can see that the last two terms in Eq. (47) may be the first derivatives of either W- or M-potential, depending on the sign of the terms in the brackets. However, these brackets may have different signs or can be zero. Therefore, the possible cases are:

$$\begin{array}{ll} \textbf{Case 1:} (A - pE) \left( B - \frac{pE}{6} \right) > 0, & \textbf{Case 2:} (A - pE) \left( B - \frac{pE}{6} \right) < 0, \\\\ \textbf{Case 3:} A = pE, B \neq \frac{pE}{6}, & \textbf{Case 4:} A \neq pE, B = pE/6. \end{array}$$

All of them are studied in Ref. [54] and they will be explained here briefly.

Case 1 straightforwardly yields to

$$
\alpha\_1 \Psi'' - \rho\_1 \Psi' + \Psi - \Psi^3 = 0 \tag{48}
$$

where

<sup>P</sup> <sup>¼</sup> <sup>1</sup> 2ω l 2 k

16 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

where β ¼ Γ=2I [51].

φn

<sup>2</sup> and φ<sup>n</sup>

ing Hamiltonian

5. General model of MTs

tions of the dimers. The scalar product � p

about αφ > 0, which is disputable result.

<sup>H</sup> <sup>¼</sup> <sup>X</sup> n

Therefore, the possible cases are:

Case 1: ð Þ <sup>A</sup> � pE <sup>B</sup> � pE

Case 3: <sup>A</sup> <sup>¼</sup> pE, <sup>B</sup> 6¼ pE

potential, the coordinate φ and, probably, Iω<sup>2</sup> < kl<sup>2</sup>

I 2 φ\_ n 2 þ k

<sup>I</sup>ω<sup>2</sup> � kl<sup>2</sup>

6 � �

general one (GM). The procedure mentioned earlier brings about

<sup>I</sup> cos ð Þ� ql Vg

the U- and the φ-models predict the breather waves moving through MT.

The final solution φnð Þt is the same as Unð Þt except that P and Q are different. Therefore, both

Finally, viscosity should be introduced in the semi-discrete approximation [51]. Due to viscosity momentum Mv ¼ �ΓΦ\_ <sup>n</sup>, the final result <sup>φ</sup>nð Þ<sup>t</sup> includes the expected exponential term <sup>e</sup>�<sup>β</sup> <sup>t</sup>

It was mentioned earlier that the weak point of the U-model is the last term in Eq. (1). Also, it is better to use the radial coordinate φ than the longitudinal one as we assume angular oscilla-

these problems but the W-potential has been missing. In fact, a series expansion of cosφ<sup>n</sup> gives

practically, a potential that looks like W in a mirror having only one minimum surrounded by two maxima and, due to its shape, can be called as M-potential [54]. This potential brings

Therefore, we want to solve the mentioned problem regarding the U-model but to keep the W-

where A > 0, B > 0 and φ<sup>n</sup> has the same meaning as in the φ-model. Let us call the model as

where, of course, φ � φ ξð Þ. If we consider Eqs. (3) and (47), we can see that the last two terms in Eq. (47) may be the first derivatives of either W- or M-potential, depending on the sign of the terms in the brackets. However, these brackets may have different signs or can be zero.

<sup>κ</sup><sup>2</sup> � �φ<sup>00</sup> � <sup>Γ</sup>ωφ<sup>0</sup> � ð Þ <sup>A</sup> � pE <sup>φ</sup> <sup>þ</sup> <sup>B</sup> � pE

<sup>&</sup>gt; 0, Case 2: ð Þ <sup>A</sup> � pE <sup>B</sup> � pE

<sup>6</sup> , Case 4: A 6¼ pE, B ¼ pE=6.

<sup>2</sup> <sup>φ</sup><sup>n</sup>þ<sup>1</sup> � <sup>φ</sup><sup>n</sup> � �<sup>2</sup> � <sup>A</sup>

<sup>4</sup> terms but with opposite signs from those in the U-model. These two terms are,

<sup>2</sup> <sup>φ</sup><sup>n</sup> 2 þ B <sup>4</sup> <sup>φ</sup><sup>n</sup>

6 � �

< 0,

� �

! � <sup>E</sup> !

� �

2

, Q <sup>¼</sup> <sup>3</sup>pE

<sup>2</sup>I<sup>ω</sup> : (45)

¼ �QdE cosφn, existing in the φ-model, solved

κ2, that is, α < 0. This suggests the follow-

<sup>φ</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup> (47)

<sup>4</sup> � pE cosφ<sup>n</sup>

6 � � ,

(46)

$$\varphi = \pm \sqrt{\frac{A - pE}{B - pE/6}} \Psi \equiv K \Psi, \qquad a\_1 = \frac{I\omega^2 - kl^2 \chi^2}{pE - A}, \qquad \rho\_1 = \frac{\Gamma \omega}{pE - A} \tag{49}$$

and the final solution is [54]

$$\varphi(\xi) = \frac{K}{2} \left[ 1 + \tanh\left(\frac{3}{4\rho\_1}\xi\right) \right], \qquad \qquad a\_1 > 0 \tag{50}$$

Eq. (50) holds for both positive and negative r1. Therefore, φ ξð Þ represents kink soliton if r<sup>1</sup> > 0 and antikink one for negative r1, which is shown in Figure 5.

One of the advantages of the GM over the φ-model is the value of amplitude. Namely, the amplitude of the kink soliton, according to the φ-model, is ffiffiffi 6 <sup>p</sup> , coming from Eq. (40). This is unrealistic, too big value. Instead of ffiffiffi 6 <sup>p</sup> , the appropriate factor, existing in the GM, is <sup>K</sup>, given by Eq. (49).

If viscosity is neglected, the GM brings about

$$
\varphi\_0 = \text{Ktanh}(\xi/a), \qquad \qquad a^2 = 2\alpha\_1 \tag{51}
$$

Case 2 straightforwardly yields to

$$
\rho\_2 \Psi'' - \rho\_2 \Psi' + \Psi + \Psi^3 = 0, \quad \wp = \pm \sqrt{\frac{pE - A}{B - pE/6}} \Psi \equiv \mathcal{K}' \Psi \tag{52}
$$

and to the final results

$$\varphi\_2 = i \frac{K'}{2} \left[ 1 + \tanh\left(\frac{\mathfrak{Z}}{4\rho\_2} \xi\right) \right], \quad \rho\_2 \neq 0 \tag{53}$$

and

$$
\varphi\_{20} = K' \tan\left(\xi/a\right), \quad a^2 = -2a, \quad \rho\_2 = 0 \tag{54}
$$

It is obvious that these results do not have physical meaning as φ<sup>2</sup> is complex, while φ<sup>20</sup> may diverge.

Case 3 brings about

$$
\alpha\_3 \Psi'' - \rho\_3 \Psi' + \Psi^3 = 0, \qquad \alpha\_3 = \frac{I\omega^2 - kl^2 \kappa^2}{B - pE/6}, \qquad \rho\_3 = \frac{\Gamma \omega}{B - pE/6} \tag{55}
$$

6. Conclusion and future research

cannot be without experimental results.

electro-acoustic wave excitations [58].

logical Development (grant No. III45010).

explained in this chapter [59–61].

Acknowledgements

a lot.

In this chapter, the three models describing nonlinear dynamics of MTs are shown. The first one, the U-model, is the improved version of the first nonlinear model and it predicts subsonic kink solitons moving along MT. The second one, the radial φ-model, predicts the supersonic kinks. Finally, the GM is explained. This is the radial model which yields both possibilities regarding the kink's speed. If we assume that the kink soliton is subsonic wave then we know

Mechanical Models of Microtubules http://dx.doi.org/10.5772/intechopen.71181 19

Two mathematical procedures are explained, continuum and semi-discrete approximations. It is very interesting that the final result depends not only on the physical system but on the mathematical methods as well. These solutions are the kink soliton and the breather. The question is which one, if any, really moves along MTs. This is not known in the moment and

It was demonstrated that the GM is better than the previous two models. However, this does not mean that it should not be improved. For example, there has been an attempt to improve the model introducing Morse potential instead of the harmonic one [55]. The harmonic potential energy assumes that attractive and repulsive forces are equal. Morse potential is not

In this chapter, the dimers are considered as elementary units. However, their structure is more complicated and they include tubulin tales (TTs). Consequently, nonlinear dynamics of TTs

The W-potential has two minima which means that it assumes existence of the two angles between the dimer and the direction of PF around which the dimer oscillates. One of the future tasks should be measuring these angles. First of all, such experiment would check if the Wpotential is correct or not. If it is, then our knowledge of their values would improve the theory

One of the future research goals should be two-component model. This may mean that we should construct the model assuming two degrees of freedom. However, one of these degrees can be an internal one, which means that oscillations if monomers within the dimer should be taken into consideration. Notice that the two-component model may be the one studying

Finally, we should bear in mind the cytological and medical applications of the research

This work was supported by funds from Serbian Ministry of Education, Sciences and Techno-

the minimum value of the parameter A, that is, A > pE, as explained earlier.

symmetric and is good for both strong and weak interactions.

should also be studied and some results already exist [56, 57].

as well as a<sup>0</sup> ¼ a ¼ α<sup>3</sup> ¼ r<sup>3</sup> ¼ 0, which certainly means that Eq. (55) does not have any solution having physical sense.

The remaining Case 4 linearizes Eq. (47) and will not be studied here.

Therefore, the GM yields to the kink solitons as the previous two models do. However, this is the radial model and the problems with both the last term in Eq. (1) and the huge amplitude in the case of the φ-model have been solved. We should study one more issue. It was mentioned earlier that the U-model predicts the subsonic kink soliton, while the φ-model predicts the supersonic wave. How about the GM? It was shown that Case 1 yields to the solutions having physical sense and that α<sup>1</sup> > 0. According to Eq. (49), we easily reach the final conclusion:


All this certainly suggests the advantages of the GM with respect to the previous two.

Figure 5. Kink soliton for r<sup>1</sup> ¼ �1 (a) and antikink soliton for r<sup>1</sup> ¼ 1 (b) for K ¼ 1.

## 6. Conclusion and future research

<sup>α</sup><sup>3</sup> <sup>Ψ</sup><sup>00</sup> � <sup>r</sup>3Ψ<sup>0</sup> <sup>þ</sup> <sup>Ψ</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>α</sup><sup>3</sup> <sup>¼</sup> <sup>I</sup>ω<sup>2</sup> � kl<sup>2</sup>

18 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

The remaining Case 4 linearizes Eq. (47) and will not be studied here.

the positive K in Eq. (50) and antikink otherwise.

for the positive K in Eq. (50) and kink otherwise.

Figure 5. Kink soliton for r<sup>1</sup> ¼ �1 (a) and antikink soliton for r<sup>1</sup> ¼ 1 (b) for K ¼ 1.

having physical sense.

as well as a<sup>0</sup> ¼ a ¼ α<sup>3</sup> ¼ r<sup>3</sup> ¼ 0, which certainly means that Eq. (55) does not have any solution

Therefore, the GM yields to the kink solitons as the previous two models do. However, this is the radial model and the problems with both the last term in Eq. (1) and the huge amplitude in the case of the φ-model have been solved. We should study one more issue. It was mentioned earlier that the U-model predicts the subsonic kink soliton, while the φ-model predicts the supersonic wave. How about the GM? It was shown that Case 1 yields to the solutions having physical sense and that α<sup>1</sup> > 0. According to Eq. (49), we easily reach the final conclusion:

a. If A > pE and B > pE=6, then r<sup>1</sup> < 0 and the function φð Þ x; t is subsonic soliton, kink for

b. If A < pE and B < pE=6, then r<sup>1</sup> > 0 and the function φð Þ x; t is supersonic soliton, antikink

All this certainly suggests the advantages of the GM with respect to the previous two.

κ2

<sup>B</sup> � pE=<sup>6</sup> , <sup>r</sup><sup>3</sup> <sup>¼</sup> <sup>Γ</sup><sup>ω</sup>

<sup>B</sup> � pE=<sup>6</sup> (55)

In this chapter, the three models describing nonlinear dynamics of MTs are shown. The first one, the U-model, is the improved version of the first nonlinear model and it predicts subsonic kink solitons moving along MT. The second one, the radial φ-model, predicts the supersonic kinks. Finally, the GM is explained. This is the radial model which yields both possibilities regarding the kink's speed. If we assume that the kink soliton is subsonic wave then we know the minimum value of the parameter A, that is, A > pE, as explained earlier.

Two mathematical procedures are explained, continuum and semi-discrete approximations. It is very interesting that the final result depends not only on the physical system but on the mathematical methods as well. These solutions are the kink soliton and the breather. The question is which one, if any, really moves along MTs. This is not known in the moment and cannot be without experimental results.

It was demonstrated that the GM is better than the previous two models. However, this does not mean that it should not be improved. For example, there has been an attempt to improve the model introducing Morse potential instead of the harmonic one [55]. The harmonic potential energy assumes that attractive and repulsive forces are equal. Morse potential is not symmetric and is good for both strong and weak interactions.

In this chapter, the dimers are considered as elementary units. However, their structure is more complicated and they include tubulin tales (TTs). Consequently, nonlinear dynamics of TTs should also be studied and some results already exist [56, 57].

The W-potential has two minima which means that it assumes existence of the two angles between the dimer and the direction of PF around which the dimer oscillates. One of the future tasks should be measuring these angles. First of all, such experiment would check if the Wpotential is correct or not. If it is, then our knowledge of their values would improve the theory a lot.

One of the future research goals should be two-component model. This may mean that we should construct the model assuming two degrees of freedom. However, one of these degrees can be an internal one, which means that oscillations if monomers within the dimer should be taken into consideration. Notice that the two-component model may be the one studying electro-acoustic wave excitations [58].

Finally, we should bear in mind the cytological and medical applications of the research explained in this chapter [59–61].

#### Acknowledgements

This work was supported by funds from Serbian Ministry of Education, Sciences and Technological Development (grant No. III45010).

#### Author details

Slobodan Zdravković

Address all correspondence to: szdjidji@vin.bg.ac.rs

Vinča Institute of Nuclear Sciences, University of Belgrade, Atomic Physics Laboratory, Serbia

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**Chapter 2**

Provisional chapter

**The Dynamics Analysis of Two Delayed Epidemic**

DOI: 10.5772/intechopen.71087

The Dynamics Analysis of Two Delayed Epidemic

Two novel delayed epidemic spreading models with latent period on scale-free network are presented. The formula of the basic reproductive number and the analysis of dynamical behaviors for the models are presented. Meanwhile, numerical simulations are given

Keywords: epidemic spreading, scale-free network, basic reproductive number, time

Following the seminal work on scale-free network, in which the probability of p(k) for any node with k links to other nodes is distributed according to the power law p(k) =Ck<sup>γ</sup> (2 < γ ≤ 3), suggested by Barabási and Albert [1], the researches of complex network have attracted more and more interests. It was found that many relevant networks, for instance, the internet, the World Wide Web (WWW), the patterns of human sexual contacts, biology network, transpor-

The dynamical behaviors of epidemic diseases have been studied for a long time. The epidemic spreading process on network is primarily dominated by two factors: one is the macroscopic topology of the underlying network, and the other is the microscopic infection scheme, which includes properties of disease, infection pattern, individual differences, infectivity of individuals, etc. The traditional epidemic dynamics is based on homogeneous network, and the infectivity rate is equally likely over all links [2]. However, the real disease transmission network exhibits scale-free properties, and the spreading of epidemic disease (e.g., computer

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

tation infrastructure, etc., exhibit power-law or "scale-free" degree distributions.

**Spreading Models with Latent Period on**

Spreading Models with Latent Period on

**Heterogeneous Network**

Heterogeneous Network

http://dx.doi.org/10.5772/intechopen.71087

to verify the main results.

Abstract

delay, stability

1. Introduction

Qiming Liu, Meici Sun and Shihua Zhang

Qiming Liu, Meici Sun and Shihua Zhang

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network** The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

DOI: 10.5772/intechopen.71087

Qiming Liu, Meici Sun and Shihua Zhang Qiming Liu, Meici Sun and Shihua Zhang

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71087

#### Abstract

Two novel delayed epidemic spreading models with latent period on scale-free network are presented. The formula of the basic reproductive number and the analysis of dynamical behaviors for the models are presented. Meanwhile, numerical simulations are given to verify the main results.

Keywords: epidemic spreading, scale-free network, basic reproductive number, time delay, stability

#### 1. Introduction

Following the seminal work on scale-free network, in which the probability of p(k) for any node with k links to other nodes is distributed according to the power law p(k) =Ck<sup>γ</sup> (2 < γ ≤ 3), suggested by Barabási and Albert [1], the researches of complex network have attracted more and more interests. It was found that many relevant networks, for instance, the internet, the World Wide Web (WWW), the patterns of human sexual contacts, biology network, transportation infrastructure, etc., exhibit power-law or "scale-free" degree distributions.

The dynamical behaviors of epidemic diseases have been studied for a long time. The epidemic spreading process on network is primarily dominated by two factors: one is the macroscopic topology of the underlying network, and the other is the microscopic infection scheme, which includes properties of disease, infection pattern, individual differences, infectivity of individuals, etc. The traditional epidemic dynamics is based on homogeneous network, and the infectivity rate is equally likely over all links [2]. However, the real disease transmission network exhibits scale-free properties, and the spreading of epidemic disease (e.g., computer

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

virus spreading, epidemic disease between human beings) on heterogeneous network, i.e., scale-free network, has been studied by many researchers [3–27].

<sup>n</sup> <sup>¼</sup> mN<sup>1</sup>=ð Þ <sup>γ</sup>�<sup>1</sup> : (1)

http://dx.doi.org/10.5772/intechopen.71087

SkðÞþ t EkðÞþ t IkðÞþ t RkðÞ¼ t 1 (3)

ϕð Þk p kð ÞIkð Þt (4)

Skð Þs Θð Þs ds (5)

Skð Þs Θð Þs ds þ IkðÞþ t RkðÞ¼ t 1: (6)

), 0 < α < 1 in [7], and so on. Here,

(2)

27

Let τ be the latent period of the disease, i.e., each exposed node becomes an infected node after τ. The relative density Sk(t), Ek(t), Ik(t) and Rk(t), at the mean-field level, satisfy the following set

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

<sup>k</sup>ðÞ ¼ t �λð Þk Skð Þt Θð Þt ,

<sup>k</sup>ðÞ ¼ t λð Þk Skð Þt ΘðÞ� t λð Þk Skð Þ t � τ Θð Þ t � τ ,

I <sup>k</sup>ðÞ ¼ t λð Þk Skð Þ t � τ Θð Þ� t � τ μIkð Þt , <sup>R</sup>\_ <sup>k</sup>ðÞ ¼ <sup>t</sup> <sup>μ</sup>Ikð Þ<sup>t</sup>

holds due to the fact that the number of total nodes with degree k is a constant p(k)N during the period of epidemic spreading. Where λ(k) is the degree-dependent infection rate such as λ(k) = λk [3] and λC(k) [4], μ is the recovery rate of the infected nodes; Θ(t) represents the probability that any given link points to an infected node. Assuming that the network has no

ΘðÞ¼ t

1 h ik Xn k¼m

in which 〈k〉 = ∑kp(k)k stands for the average node degree and ϕ(k) means the occupied edges which can transmit the disease (i.e., represents the infectivity of infected nodes) [22]; they have

we point out that the delay τ in the model (2) in this paper is different from one in the model (2)–(4) in [15]. The incubation period τ in the model in [15] is another kind of time period, during which the infectious agents develop in the vector and the infected vector becomes

> ðt t�τ

/(1 + bk<sup>α</sup>

Skð Þ¼ θ φkð Þ θ , Ikð Þ¼ θ Ψkð Þ θ , RkðÞ¼ t ζkð Þ θ , θ∈½ � �τ; 0 (7)

of coupled different equations when t > 0 [15, 16]: S\_

E\_

8 >>>><

>>>>:

with the normalization condition

degree correlations [7, 16, 22], we have

infectious after that time.

many different forms, such as φ(k) = A in [5], φ(k) = ak<sup>α</sup>

Note that we obtain from the third equation of system (2) that

Skð Þþ t λð Þk

The initial conditions of system (2) are

EkðÞ¼ t λð Þk

and the normalization condition becomes the following mathematical form

ðt t�τ

\_

A handful of existing works address the complex behavior of epidemic spreading using compartmental differential equations [2, 15]. Comparing with the ordinary differential equation models, more realistic models should be retarded functional differential equation models which can include some of the past states of these systems. Time delay plays an important role in the process of the epidemic spreading, for instance, the latent period of the infectious diseases or computer virus, the infection period of infective members, and the immunity period of the recovered individuals can be represented by time delays [2]. Recently, some researchers discussed the epidemic spreading model with time delays [7, 15–17]. Susceptible-Infected-Removed (SIR) model is a basic and important epidemic model, Zou and Wu discussed a delayed SIR model without birth and death [15], and Wang and Wang et al. discussed a delayed SIR model with birth rate and death rate [16]. However, it is suitable to divide the nodes being considered into disjoint classes of susceptible, exposed, infective, and recovered nodes in modeling disease transmission [2, 10], i.e., a susceptive node first through an incubation period (and it is said to become exposed) after infection and before becoming infectious. For example, the latent period of epidemic cholera is about 1–3 days, hepatitis B virus 100 days, measles 10–11 days, chincough 7–10 days, diphtheria 2–4 days, scarlatina 2–5 days, poliomyelitis 7–14 days, and so on [25]; the resulting model is Susceptible-Exposed-Infected-Removed (SEIR) model. In addition, some diseases confer temporary immunity, and the recovered nodes cycle back into the susceptive class after an immune period; the resulting model is Susceptible-Exposed-Infected-Removed-susceptible (SEIRS) model.

In this paper, we will present a suitable SEIR model with time delay and a suitable SEIRS model with time delay on heterogeneous network by using functional differential equation to investigate the dynamical behaviors of epidemic spreading.

The rest of this paper is organized as follows: In Section 2, the SEIRS model with time delay on scale-free network is discussed. The SEIRS model with time delay on scale-free network is discussed in Section 3. Finally, the main conclusions of this work are summarized in Section 4.

#### 2. Analysis of the SEIR model with time delay

#### 2.1. The SEIR model

Suppose that the size of the network is a constant N and the degree of each node is time invariant during the period of epidemic spreading, p(k) denotes the degree distribution of the network. We classify all the nodes in the network into n groups such that the nodes in the same group have the same degree. That is, each node in the kth group has the same connectivity (k = m, m + 1, ⋯, n). Let Sk(t), Ek(t), Ik(t) and Rk(t) be the relative density of susceptible nodes, exposed nodes, infected nodes, and recovered nodes of connectivity k at time t, respectively, where k = m, m + 1, ⋯, n (m and n are the minimum and maximum degree in network topology) and n is related to the network age, measured as the number of nodes N [3]:

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network http://dx.doi.org/10.5772/intechopen.71087 27

$$m = mN^{1/(\gamma - 1)}.\tag{1}$$

Let τ be the latent period of the disease, i.e., each exposed node becomes an infected node after τ. The relative density Sk(t), Ek(t), Ik(t) and Rk(t), at the mean-field level, satisfy the following set of coupled different equations when t > 0 [15, 16]:

$$\begin{cases} \dot{\mathcal{S}}\_k(t) &= 0 \\ \dot{\mathcal{E}}\_k(t) &= \lambda(k)\mathcal{S}\_k(t)\Theta(t) - \lambda(k)\mathcal{S}\_k(t-\tau)\Theta(t-\tau), \\ \dot{I}\_k(t) &= 0 \\ \dot{\mathcal{R}}\_k(t) &= 0 \end{cases} \tag{2}$$

with the normalization condition

virus spreading, epidemic disease between human beings) on heterogeneous network, i.e.,

A handful of existing works address the complex behavior of epidemic spreading using compartmental differential equations [2, 15]. Comparing with the ordinary differential equation models, more realistic models should be retarded functional differential equation models which can include some of the past states of these systems. Time delay plays an important role in the process of the epidemic spreading, for instance, the latent period of the infectious diseases or computer virus, the infection period of infective members, and the immunity period of the recovered individuals can be represented by time delays [2]. Recently, some researchers discussed the epidemic spreading model with time delays [7, 15–17]. Susceptible-Infected-Removed (SIR) model is a basic and important epidemic model, Zou and Wu discussed a delayed SIR model without birth and death [15], and Wang and Wang et al. discussed a delayed SIR model with birth rate and death rate [16]. However, it is suitable to divide the nodes being considered into disjoint classes of susceptible, exposed, infective, and recovered nodes in modeling disease transmission [2, 10], i.e., a susceptive node first through an incubation period (and it is said to become exposed) after infection and before becoming infectious. For example, the latent period of epidemic cholera is about 1–3 days, hepatitis B virus 100 days, measles 10–11 days, chincough 7–10 days, diphtheria 2–4 days, scarlatina 2–5 days, poliomyelitis 7–14 days, and so on [25]; the resulting model is Susceptible-Exposed-Infected-Removed (SEIR) model. In addition, some diseases confer temporary immunity, and the recovered nodes cycle back into the susceptive class after an immune period; the resulting

scale-free network, has been studied by many researchers [3–27].

26 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

model is Susceptible-Exposed-Infected-Removed-susceptible (SEIRS) model.

and n is related to the network age, measured as the number of nodes N [3]:

investigate the dynamical behaviors of epidemic spreading.

2. Analysis of the SEIR model with time delay

2.1. The SEIR model

In this paper, we will present a suitable SEIR model with time delay and a suitable SEIRS model with time delay on heterogeneous network by using functional differential equation to

The rest of this paper is organized as follows: In Section 2, the SEIRS model with time delay on scale-free network is discussed. The SEIRS model with time delay on scale-free network is discussed in Section 3. Finally, the main conclusions of this work are summarized in Section 4.

Suppose that the size of the network is a constant N and the degree of each node is time invariant during the period of epidemic spreading, p(k) denotes the degree distribution of the network. We classify all the nodes in the network into n groups such that the nodes in the same group have the same degree. That is, each node in the kth group has the same connectivity (k = m, m + 1, ⋯, n). Let Sk(t), Ek(t), Ik(t) and Rk(t) be the relative density of susceptible nodes, exposed nodes, infected nodes, and recovered nodes of connectivity k at time t, respectively, where k = m, m + 1, ⋯, n (m and n are the minimum and maximum degree in network topology)

$$S\_k(t) + E\_k(t) + I\_k(t) + R\_k(t) = 1\tag{3}$$

holds due to the fact that the number of total nodes with degree k is a constant p(k)N during the period of epidemic spreading. Where λ(k) is the degree-dependent infection rate such as λ(k) = λk [3] and λC(k) [4], μ is the recovery rate of the infected nodes; Θ(t) represents the probability that any given link points to an infected node. Assuming that the network has no degree correlations [7, 16, 22], we have

$$\Theta(t) = \frac{1}{\langle k \rangle} \sum\_{k=m}^{n} \phi(k) p(k) I\_k(t) \tag{4}$$

in which 〈k〉 = ∑kp(k)k stands for the average node degree and ϕ(k) means the occupied edges which can transmit the disease (i.e., represents the infectivity of infected nodes) [22]; they have many different forms, such as φ(k) = A in [5], φ(k) = ak<sup>α</sup> /(1 + bk<sup>α</sup> ), 0 < α < 1 in [7], and so on. Here, we point out that the delay τ in the model (2) in this paper is different from one in the model (2)–(4) in [15]. The incubation period τ in the model in [15] is another kind of time period, during which the infectious agents develop in the vector and the infected vector becomes infectious after that time.

Note that we obtain from the third equation of system (2) that

$$E\_k(t) = \lambda(k) \int\_{t-\tau}^t S\_k(s)\Theta(s)ds\tag{5}$$

and the normalization condition becomes the following mathematical form

$$\mathcal{S}\_k(t) + \lambda(k) \int\_{t-\tau}^t \mathcal{S}\_k(s)\Theta(s)ds + I\_k(t) + R\_k(t) = 1. \tag{6}$$

The initial conditions of system (2) are

$$S\_k(\theta) = \varphi\_k(\theta), \newline I\_k(\theta) = \mathcal{V}\_k(\theta), \newline R\_k(t) = \mathsf{I}\_k(\theta), \newline \theta \in [-\pi, 0] \tag{7}$$

and satisfy Skð Þþ <sup>0</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>Ð</sup> <sup>0</sup> �<sup>τ</sup> Skð Þ<sup>s</sup> <sup>Θ</sup>ð Þ<sup>s</sup> ds <sup>þ</sup> Ikð Þþ <sup>0</sup> Rkð Þ¼ <sup>0</sup> 1 which guarantees the normalization condition holds. And Φ<sup>k</sup> = (ϕk(θ), Ψk(θ), ζ(θ), k = m, m + 1, ⋯, n � m + 1) ∈ C are nonnegative continuous on [�τ, 0], ϕk(0) > 0, Ψk(0) > 0, and ζ(θ) = 0 for θ = 0. C denotes the Banach space <sup>C</sup>([�τ, 0], <sup>R</sup>3(<sup>n</sup> � <sup>m</sup> + 1)) with the norm, where <sup>∣</sup>f(θ)∣<sup>τ</sup> = sup�<sup>τ</sup> <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> <sup>0</sup> <sup>∣</sup> <sup>f</sup>(θ)∣. k k <sup>ω</sup> <sup>¼</sup> <sup>P</sup><sup>n</sup> <sup>i</sup>¼<sup>m</sup> j j <sup>Ψ</sup>ið Þ <sup>θ</sup> <sup>2</sup> <sup>τ</sup> þ ϕ<sup>i</sup> ð Þ <sup>θ</sup> � � � � 2 <sup>τ</sup> <sup>þ</sup> j j <sup>ζ</sup>ið Þ <sup>θ</sup> <sup>2</sup> τ � �<sup>1</sup>=<sup>2</sup> : �

#### 2.2. The main results for the model

In this section, we first discuss the final size relation of solutions for system (2).

It is easy to know that system (2) only has a disease-free equilibrium set

$$M\_0 = \left\{ \left( \widehat{\mathcal{S}}, \widehat{E}, \widehat{I}, \widehat{R} \right) \middle| E\_k = I\_k = 0, \mathcal{S}\_k + R\_k = 1, R\_k, \mathcal{S}\_k \ge 0, k = m, m+1, \dots, n \right\} \tag{8}$$

in which <sup>b</sup><sup>S</sup> <sup>¼</sup> ð Þ Sm; Smþ1;⋯; Sn ,Eb ¼ ð Þ Em; Emþ<sup>1</sup>; ⋯; En ,bI ¼ ð Þ Im; Imþ<sup>1</sup>; ⋯; In , Rb ¼ ð Þ Rm;Rmþ<sup>1</sup>; ⋯;Rn :

Supposing f(t) is an arbitrary nonnegative continuous function f(t), we adopt the following convention:

$$f(+\infty) = \lim\_{t \to +\infty} f(t) \tag{9}$$

Integrating (13) from 0 to +∞, we obtain that

Substituting (15) into (16), we obtain that

Sk(0) ≈ 1 and obtain from (17) that

Hence, we have the following result.

by formulas (18) and (19).

Consequently,

from (16) that

literature [19].

ln Skð Þ<sup>0</sup>

Skð Þ <sup>þ</sup><sup>∞</sup> <sup>¼</sup> <sup>λ</sup>ð Þ<sup>k</sup>

h ik

ln Skð Þ¼ <sup>þ</sup><sup>∞</sup> <sup>λ</sup>ð Þ<sup>k</sup>

ðþ<sup>∞</sup> 0

ln Skð Þ<sup>0</sup>

have from (14) that

Skð Þ� 0 Skð Þþ þ∞ Ekð Þ� 0 Ekð Þþ þ∞ Ikð Þ� 0 Ikð Þ¼ þ∞ μ

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

Noting that Ik(+ <sup>∞</sup>) = 0, Ek(0) = Ek(+ <sup>∞</sup>) = 0, and Sk(+ <sup>∞</sup>) exists due to existence of <sup>Ð</sup> <sup>þ</sup><sup>∞</sup>

Ikð Þ<sup>s</sup> ds <sup>¼</sup> <sup>1</sup>

Skð Þ <sup>∞</sup> <sup>¼</sup> <sup>λ</sup>ð Þ<sup>k</sup> h ik

> X k

> > μh ik

Additionally, integrating the first equation of system (2) from 0 to + ∞, we have

X k

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>1</sup>

X k

Theorem 2.1. The equilibrium set M<sup>0</sup> <sup>¼</sup> <sup>b</sup>S; <sup>E</sup>b;bI; <sup>R</sup><sup>b</sup> � �jEk <sup>¼</sup> Ik <sup>¼</sup> <sup>0</sup>; Sk <sup>þ</sup> Rk <sup>¼</sup> <sup>1</sup>; <sup>k</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>⋯</sup>; <sup>n</sup>

system (2) is globally attractive, i.e., limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0, limt!<sup>∞</sup>Ek(t)=0. And Rk(+∞), Sk(+∞) are given

Note that it is impossible for every susceptible to be infected. Supposing Sk(+∞) = 0, we know

X k

Obviously, Eq. (20) does not hold, i.e., Sk(+∞) = 0. Similar results were obtained in the early

Secondly, we discuss the basic reproductive number of model (2). The basic reproductive number is an important conception; it represents the average number of secondary infections

<sup>þ</sup><sup>∞</sup> <sup>¼</sup> <sup>λ</sup>ð Þ<sup>k</sup> μh ik

Because there are only several infective nodes at the beginning of disease spreading, we take

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>ð</sup>þ<sup>∞</sup>

0

ðþ<sup>∞</sup> 0

http://dx.doi.org/10.5772/intechopen.71087

<sup>μ</sup> ð Þ Skð Þþ <sup>0</sup> Ikð Þ� <sup>0</sup> Skð Þ <sup>þ</sup><sup>∞</sup> : (15)

Ikð Þs ds: (16)

<sup>μ</sup> ð Þ Skð Þþ <sup>0</sup> Ikð Þ� <sup>0</sup> Skð Þ <sup>þ</sup><sup>∞</sup> : (17)

ϕð Þk p kð Þð Þ Skð Þ� þ∞ 1 : (18)

Rkð Þ¼ ∞ 1 � Skð Þ þ∞ : (19)

n o of

ϕð Þk p kð Þ (20)

Ikð Þs ds: (14)

<sup>0</sup> Ikð Þs ds, we

29

and we obtain from the last equation of system (2) that

$$R\_k(+\!\!\!\ast\!) - R\_k(0) = \mu \int\_0^{+\!\!\ast\!\ast} I\_k(\text{s})d\text{s}.\tag{10}$$

According to the last equation of system (2), Rk(t) is increasing and bounded above by 1, and it has a limit as t ! + ∞. Thus, the left-hand side of (10) is finite due to boundedness of Rk(+∞), and Rk(0) exits, i.e., 0 <sup>&</sup>lt; <sup>Ð</sup> <sup>þ</sup><sup>∞</sup> <sup>0</sup> Ikð Þ u du < þ∞. Since Ik(t) is smooth nonnegative function, we know Ik(+∞) = 0, i.e., limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0.

Furthermore, we have from (5) that

$$0 \le E\_k(t) = \lambda(k) \int\_{t-\tau}^t S\_k(s)\Theta(s)ds \le \lambda(k) \int\_{t-\tau}^t \Theta(s)ds,\tag{11}$$

In addition, by using mean value theorem for integrals, we have

$$\int\_{t-\tau}^{t} \Theta(s)ds = \Theta(\xi)\tau, \,(t-\tau \le \xi \le t). \tag{12}$$

We obtain from Ik(+∞) = 0 that limξ! <sup>+</sup> <sup>∞</sup>Θ(ξ)τ = 0 and then limt! <sup>+</sup> <sup>∞</sup>Ek(t) = 0. Hence, M<sup>0</sup> is globally attractive [27].

In addition, it follows from (2) that

$$
\dot{S}\_k(t) + \dot{E}\_k(t) + \dot{I}\_k(t) = -\mu I\_k(t). \tag{13}
$$

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network http://dx.doi.org/10.5772/intechopen.71087 29

Integrating (13) from 0 to +∞, we obtain that

and satisfy Skð Þþ <sup>0</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>Ð</sup> <sup>0</sup>

<sup>i</sup>¼<sup>m</sup> j j <sup>Ψ</sup>ið Þ <sup>θ</sup> <sup>2</sup>

2.2. The main results for the model

<sup>M</sup><sup>0</sup> <sup>¼</sup> <sup>b</sup>S; <sup>E</sup>b;bI;R<sup>b</sup> � �

<sup>τ</sup> þ ϕ<sup>i</sup> ð Þ <sup>θ</sup> � � � � 2

and we obtain from the last equation of system (2) that

0 ≤ EkðÞ¼ t λð Þk

In addition, by using mean value theorem for integrals, we have ðt t�τ

> S\_ <sup>k</sup>ð Þþ <sup>t</sup> <sup>E</sup>\_

� �<sup>1</sup>=<sup>2</sup>

28 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

k k <sup>ω</sup> <sup>¼</sup> <sup>P</sup><sup>n</sup>

convention:

and Rk(0) exits, i.e., 0 <sup>&</sup>lt; <sup>Ð</sup> <sup>þ</sup><sup>∞</sup>

globally attractive [27].

In addition, it follows from (2) that

Ik(+∞) = 0, i.e., limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0.

Furthermore, we have from (5) that

�

�<sup>τ</sup> Skð Þ<sup>s</sup> <sup>Θ</sup>ð Þ<sup>s</sup> ds <sup>þ</sup> Ikð Þþ <sup>0</sup> Rkð Þ¼ <sup>0</sup> 1 which guarantees the normali-

jEk ¼ Ik ¼ 0; Sk þ Rk ¼ 1;Rk; Sk ≥ 0; k ¼ m; m þ 1; ⋯; n

ðþ<sup>∞</sup> 0

<sup>0</sup> Ikð Þ u du < þ∞. Since Ik(t) is smooth nonnegative function, we know

ðt t�τ

Θð Þs ds ¼ Θð Þ ξ τ, tð Þ � τ ≤ ξ ≤ t : (12)

I <sup>k</sup>ðÞ¼� t μIkð Þt : (13)

<sup>f</sup>ð Þ¼ <sup>þ</sup><sup>∞</sup> lim<sup>t</sup>!þ<sup>∞</sup> f tð Þ (9)

Ikð Þs ds: (10)

Θð Þs ds, (11)

(8)

n o

in which <sup>b</sup><sup>S</sup> <sup>¼</sup> ð Þ Sm; Smþ1;⋯; Sn ,Eb ¼ ð Þ Em; Emþ<sup>1</sup>; ⋯; En ,bI ¼ ð Þ Im; Imþ<sup>1</sup>; ⋯; In , Rb ¼ ð Þ Rm;Rmþ<sup>1</sup>; ⋯;Rn : Supposing f(t) is an arbitrary nonnegative continuous function f(t), we adopt the following

According to the last equation of system (2), Rk(t) is increasing and bounded above by 1, and it has a limit as t ! + ∞. Thus, the left-hand side of (10) is finite due to boundedness of Rk(+∞),

Skð Þs Θð Þs ds ≤ λð Þk

We obtain from Ik(+∞) = 0 that limξ! <sup>+</sup> <sup>∞</sup>Θ(ξ)τ = 0 and then limt! <sup>+</sup> <sup>∞</sup>Ek(t) = 0. Hence, M<sup>0</sup> is

<sup>k</sup>ð Þþ <sup>t</sup> \_

zation condition holds. And Φ<sup>k</sup> = (ϕk(θ), Ψk(θ), ζ(θ), k = m, m + 1, ⋯, n � m + 1) ∈ C are nonnegative continuous on [�τ, 0], ϕk(0) > 0, Ψk(0) > 0, and ζ(θ) = 0 for θ = 0. C denotes the Banach space <sup>C</sup>([�τ, 0], <sup>R</sup>3(<sup>n</sup> � <sup>m</sup> + 1)) with the norm, where <sup>∣</sup>f(θ)∣<sup>τ</sup> = sup�<sup>τ</sup> <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> <sup>0</sup> <sup>∣</sup> <sup>f</sup>(θ)∣.

:

<sup>τ</sup> <sup>þ</sup> j j <sup>ζ</sup>ið Þ <sup>θ</sup> <sup>2</sup>

In this section, we first discuss the final size relation of solutions for system (2).

Rkð Þ� þ∞ Rkð Þ¼ 0 μ

ðt t�τ

It is easy to know that system (2) only has a disease-free equilibrium set

τ

$$\mathcal{S}\_k(0) - \mathcal{S}\_k(+\infty) + E\_k(0) - E\_k(+\infty) + I\_k(0) - I\_k(+\infty) = \mu \int\_0^{+\infty} I\_k(s)ds.\tag{14}$$

Noting that Ik(+ <sup>∞</sup>) = 0, Ek(0) = Ek(+ <sup>∞</sup>) = 0, and Sk(+ <sup>∞</sup>) exists due to existence of <sup>Ð</sup> <sup>þ</sup><sup>∞</sup> <sup>0</sup> Ikð Þs ds, we have from (14) that

$$\int\_0^{+\infty} I\_k(s)ds = \frac{1}{\mu} \left( \mathcal{S}\_k(0) + I\_k(0) - \mathcal{S}\_k(+\infty) \right). \tag{15}$$

Additionally, integrating the first equation of system (2) from 0 to + ∞, we have

$$\ln \frac{S\_k(0)}{S\_k(\circ \circ)} = \frac{\lambda(k)}{\langle k \rangle} \sum\_k \phi(k) p(k) \int\_0^{+\circ} I\_k(s) ds. \tag{16}$$

Substituting (15) into (16), we obtain that

$$\ln \frac{S\_k(0)}{S\_k(+\infty)} = \frac{\lambda(k)}{\langle k \rangle} \sum\_k \phi(k) p(k) \frac{1}{\mu} (S\_k(0) + I\_k(0) - S\_k(+\infty)). \tag{17}$$

Because there are only several infective nodes at the beginning of disease spreading, we take Sk(0) ≈ 1 and obtain from (17) that

$$\ln S\_k(+\!\!\!\/) = \frac{\lambda(k)}{\mu(k)} \sum\_{k} \phi(k) p(k) (S\_k(+\!\!\/) - 1). \tag{18}$$

Consequently,

$$R\_k(\\*\*suit) = 1 - S\_k(+\\*\*suit).\tag{19}$$

Hence, we have the following result.

Theorem 2.1. The equilibrium set M<sup>0</sup> <sup>¼</sup> <sup>b</sup>S; <sup>E</sup>b;bI; <sup>R</sup><sup>b</sup> � �jEk <sup>¼</sup> Ik <sup>¼</sup> <sup>0</sup>; Sk <sup>þ</sup> Rk <sup>¼</sup> <sup>1</sup>; <sup>k</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>⋯</sup>; <sup>n</sup> n o of system (2) is globally attractive, i.e., limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0, limt!<sup>∞</sup>Ek(t)=0. And Rk(+∞), Sk(+∞) are given by formulas (18) and (19).

Note that it is impossible for every susceptible to be infected. Supposing Sk(+∞) = 0, we know from (16) that

$$+\!\!\!\!\/+\!\!\/+\!\frac{\lambda(k)}{\mu\langle k\rangle}\sum\_{k}\phi(k)p(k)\tag{20}$$

Obviously, Eq. (20) does not hold, i.e., Sk(+∞) = 0. Similar results were obtained in the early literature [19].

Secondly, we discuss the basic reproductive number of model (2). The basic reproductive number is an important conception; it represents the average number of secondary infections infected by an individual of infective during the whole course of disease in the case that all the members of the population are susceptible [2].

Theorem 2.2. For system (2),

$$R\_0 = \frac{\langle \lambda(k)\phi(k)\rangle}{\mu \langle k\rangle} \tag{21}$$

By way of contradiction, supposing that T < + ∞, then we have <sup>d</sup>

Θ(t2) < Θ(t1). Note that Θ(t<sup>2</sup> � τ) ≥ Θ(t<sup>1</sup> � τ). It follows from (25) that

R<sup>0</sup> ≤ 1. Hence, R<sup>0</sup> is the basic reproductive number for system (2).

Furthermore, the basic reproductive number for system (2) is R<sup>0</sup> = (λ〈k

and analyze the effect of time delay on behaviors of disease spreading.

The degree distribution of scale-free network is p(k) =Ck�<sup>γ</sup>

t

epidemic, limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0 due to global attractivity of M0.

cal with the results that the epidemic threshold λ<sup>c</sup> = (λ〈k〉)/(〈k

2.3. Numerical simulation for the model

/(1 + b α 0 < d dt <sup>Θ</sup>ð Þ <sup>t</sup><sup>1</sup> <sup>≤</sup>

dt Θð Þ t<sup>1</sup> > 0. It follows that there is a t2∈[t

d

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

which is a contradiction. Hence, Θ(t) decreases on [0, +∞), and there is no one outbreak when

It follows from Theorems 2.1 and 2.2 that R<sup>0</sup> is the basic reproductive number for system (2), which is irrelative to τ. There exists at least one outbreak for the spreading of epidemic if R<sup>0</sup> > 1, and there is no outbreak if R<sup>0</sup> ≤ 1. Whether or not there exists one outbreak for the spreading of

Besides, if we let τ = 0, φ(k) = k, λ(k) = λk, μ = 1, the model (2) reduces to the model in [9].

than unity when N is large enough [3, 7], and it means the lack of any basic reproductive number. This result is consistent with the results in epidemic dynamics on heterogeneous

Now, we present numerical simulations to support the results obtained in previous sections

Here, we set the maximum degree n = 100 and the minimum degree m = 1. Consider system

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

(a) (b)

Figure 1. (a) The time evolutions for system (2) with λ = 0.5, μ = 0.4, τ = 5 and Ik(0) = 0.2, k = 2, 3, 4, 5,Ik(0) = 0 for the other k and R<sup>0</sup> = 0.4006. (b) The time evolutions for system (2) with different initial values, λ = 0.4, μ = 0.4, τ = 5 and R<sup>0</sup> = 0.4006.

Density I(t)

show the dynamic behaviors of system (2) with the initial functions satisfying condition (7).

t1∈(t ∗ , t

network [3, 10].

(2), let φ(k) = ak<sup>α</sup>

Density

<sup>∗</sup> + τ] such that <sup>d</sup>

dt Θ t

, t1) such that <sup>d</sup>

http://dx.doi.org/10.5772/intechopen.71087

dt <sup>Θ</sup>ð Þ¼ <sup>t</sup><sup>2</sup> <sup>0</sup>, (27)

2

, and C satisfies P<sup>n</sup>

I(t) with Ik

I(t) with Ik

I(t) with Ik

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>0</sup>

t

2

) in which a = 0.5, α= 0.75, b = 0.02 and λ(k) =λk, and let γ= 2.5. Figures 1–4

∗

<sup>∗</sup> ð Þ¼ 0, and there exists a

〉)/(〈k〉), which is identi-

<sup>k</sup>¼<sup>1</sup> p kð Þ¼ 1.

(0)=0.2, k=2,3,4,5.

(0)=0.3, k=2,3,4,5.

(0)=0.4, k=2,3,4,5.

〉) in [9]. And, R<sup>0</sup> is always more

dt Θð Þ¼ t<sup>2</sup> 0 and

31

is the basic reproductive number for system (2).

Proof. Note that P<sup>n</sup> <sup>k</sup>¼<sup>m</sup> <sup>φ</sup>ð Þ<sup>k</sup> p kð ÞIkð Þ<sup>t</sup> may be considered as the force of infection [15] and <sup>Θ</sup>(t) may be considered as the average force of infection. Letting Θ(t) be an auxiliary function and computing its time derivative along the solution of (2), we get

$$\begin{split} \frac{d\Theta(t)}{dt} &= \frac{1}{\langle k \rangle} \sum\_{k} \phi(k) p(k) \dot{I}\_{k}(t) \\ &= \frac{1}{\langle k \rangle} \sum\_{k} \phi(k) p(k) \Big(\lambda(k) \mathbf{S}\_{k}(t-\tau) \Theta(t-\tau) - \mu I\_{k}(t)\Big) \\ &= \Theta(t-\tau) \frac{1}{\langle k \rangle} \sum\_{k} \lambda(k) \phi(k) p(k) \mathbf{S}\_{k}(t-\tau) - \mu \Theta(t). \end{split} \tag{22}$$

We have

$$\left. \frac{d\Theta(t)}{dt} \right|\_{t=0} = \Theta(-\tau) \frac{1}{\langle k \rangle} \sum\_{k} \lambda(k) \phi(k) p(k) S\_k(-\tau) - \mu \Theta(0). \tag{23}$$

Since each exposed node becomes infected node after τ, Ik(�τ) = Ik(0). It follows that Θ(�τ) =Θ(0). Meanwhile, Sk(�τ) ≈ 1. Hence, we have from (23) that

$$\left. \frac{d\Theta(t)}{dt} \right|\_{t=0} = \mu \left( \frac{1}{\mu \langle k \rangle} \sum\_{k} \lambda(k) \phi(k) p(k) - 1 \right) \Theta(0) = \mu (R\_0 - 1) \Theta(0). \tag{24}$$

If R<sup>0</sup> > 1, <sup>d</sup>Θð Þ<sup>t</sup> dt � � � t¼0 > 0, which means that Θ(t) increases at the beginning of the epidemic and there exists at least one outbreak.

Meanwhile, if R<sup>0</sup> ≤ 1, we obtain from (24) that <sup>d</sup>Θð Þ<sup>t</sup> dt � � � t¼0 ≤ 0. Let t <sup>∗</sup> = sup {T ≥ 0 : Θ(t) decreases on [0, T]}. Then, it follows from the above discussion that T ≥ 0. We will prove that T = + ∞ . Note that we obtain from the first equation of system (2) that

$$\mathcal{S}\_k(t) = \mathcal{S}\_k(0)\mathbf{e}^{-\lambda(k)\Psi(t)} \tag{25}$$

in which <sup>Ψ</sup>ðÞ¼ <sup>t</sup> <sup>1</sup> h ik P k Ðt <sup>0</sup> φð Þk p kð ÞIkð Þ u du: Hence, it follows from Eqs. (22) and (25) that

$$\frac{d\Theta(t)}{dt} = \Theta(t-\tau)\frac{1}{\langle k\rangle}\sum\_{k}\lambda(k)\phi(k)p(k)\mathbf{S}\_{k}(0)\mathbf{e}^{-\lambda(k)\Psi(t-\tau)} - \mu\Theta(t).\tag{26}$$

By way of contradiction, supposing that T < + ∞, then we have <sup>d</sup> dt Θ t <sup>∗</sup> ð Þ¼ 0, and there exists a t1∈(t ∗ , t <sup>∗</sup> + τ] such that <sup>d</sup> dt Θð Þ t<sup>1</sup> > 0. It follows that there is a t2∈[t ∗ , t1) such that <sup>d</sup> dt Θð Þ¼ t<sup>2</sup> 0 and Θ(t2) < Θ(t1). Note that Θ(t<sup>2</sup> � τ) ≥ Θ(t<sup>1</sup> � τ). It follows from (25) that

$$0 < \frac{d}{dt}\Theta(t\_1) \le \frac{d}{dt}\Theta(t\_2) = 0,\tag{27}$$

which is a contradiction. Hence, Θ(t) decreases on [0, +∞), and there is no one outbreak when R<sup>0</sup> ≤ 1. Hence, R<sup>0</sup> is the basic reproductive number for system (2).

It follows from Theorems 2.1 and 2.2 that R<sup>0</sup> is the basic reproductive number for system (2), which is irrelative to τ. There exists at least one outbreak for the spreading of epidemic if R<sup>0</sup> > 1, and there is no outbreak if R<sup>0</sup> ≤ 1. Whether or not there exists one outbreak for the spreading of epidemic, limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0 due to global attractivity of M0.

Besides, if we let τ = 0, φ(k) = k, λ(k) = λk, μ = 1, the model (2) reduces to the model in [9]. Furthermore, the basic reproductive number for system (2) is R<sup>0</sup> = (λ〈k 2 〉)/(〈k〉), which is identical with the results that the epidemic threshold λ<sup>c</sup> = (λ〈k〉)/(〈k 2 〉) in [9]. And, R<sup>0</sup> is always more than unity when N is large enough [3, 7], and it means the lack of any basic reproductive number. This result is consistent with the results in epidemic dynamics on heterogeneous network [3, 10].

#### 2.3. Numerical simulation for the model

infected by an individual of infective during the whole course of disease in the case that all the

<sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � �

may be considered as the average force of infection. Letting Θ(t) be an auxiliary function and

I <sup>k</sup>ð Þt

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ\_

1 h ik X k

> 1 h ik X k

Since each exposed node becomes infected node after τ, Ik(�τ) = Ik(0). It follows that Θ(�τ) =Θ(0).

λð Þk ϕð Þk p kð Þ� 1

dt � � � t¼0

[0, T]}. Then, it follows from the above discussion that T ≥ 0. We will prove that T = + ∞ . Note

> 0, which means that Θ(t) increases at the beginning of the epidemic and

<sup>0</sup> φð Þk p kð ÞIkð Þ u du: Hence, it follows from Eqs. (22) and (25) that

≤ 0. Let t

SkðÞ¼ <sup>t</sup> Skð Þ<sup>0</sup> <sup>e</sup>�λð Þ<sup>k</sup> <sup>Ψ</sup>ð Þ<sup>t</sup> (25)

<sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> p kð ÞSkð Þ<sup>0</sup> <sup>e</sup>�λð Þ<sup>k</sup> <sup>Ψ</sup>ð Þ <sup>t</sup>�<sup>τ</sup> � μΘð Þ<sup>t</sup> : (26)

!

<sup>k</sup>¼<sup>m</sup> <sup>φ</sup>ð Þ<sup>k</sup> p kð ÞIkð Þ<sup>t</sup> may be considered as the force of infection [15] and <sup>Θ</sup>(t)

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>λ</sup>ð Þ<sup>k</sup> Skð Þ <sup>t</sup> � <sup>τ</sup> <sup>Θ</sup>ð Þ� <sup>t</sup> � <sup>τ</sup> <sup>μ</sup>Ikð Þ<sup>t</sup> � �

λð Þk ϕð Þk p kð ÞSkð Þ� t � τ μΘð Þt :

<sup>μ</sup>h i<sup>k</sup> (21)

λð Þk ϕð Þk p kð ÞSkð Þ� �τ μΘð ÞÞ 0 : (23)

Θð Þ¼ 0 μð Þ R<sup>0</sup> � 1 Θð Þ0 : (24)

<sup>∗</sup> = sup {T ≥ 0 : Θ(t) decreases on

(22)

members of the population are susceptible [2].

30 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

is the basic reproductive number for system (2).

dΘð Þt dt <sup>¼</sup> <sup>1</sup> h ik X k

dΘð Þt dt

dΘð Þt dt

there exists at least one outbreak.

h ik P k Ðt

dΘð Þt

� � � � t¼0 ¼ μ

� � � � t¼0

Meanwhile, Sk(�τ) ≈ 1. Hence, we have from (23) that

Meanwhile, if R<sup>0</sup> ≤ 1, we obtain from (24) that <sup>d</sup>Θð Þ<sup>t</sup>

that we obtain from the first equation of system (2) that

dt <sup>¼</sup> <sup>Θ</sup>ð Þ <sup>t</sup> � <sup>τ</sup>

1 h ik X k

computing its time derivative along the solution of (2), we get

¼ Θð Þ t � τ

¼ Θð Þ �τ

1 μh ik

X k

¼ 1 h ik X k

Theorem 2.2. For system (2),

Proof. Note that P<sup>n</sup>

We have

If R<sup>0</sup> > 1, <sup>d</sup>Θð Þ<sup>t</sup> dt � � � t¼0

in which <sup>Ψ</sup>ðÞ¼ <sup>t</sup> <sup>1</sup>

Now, we present numerical simulations to support the results obtained in previous sections and analyze the effect of time delay on behaviors of disease spreading.

The degree distribution of scale-free network is p(k) =Ck�<sup>γ</sup> , and C satisfies P<sup>n</sup> <sup>k</sup>¼<sup>1</sup> p kð Þ¼ 1. Here, we set the maximum degree n = 100 and the minimum degree m = 1. Consider system (2), let φ(k) = ak<sup>α</sup> /(1 + b α ) in which a = 0.5, α= 0.75, b = 0.02 and λ(k) =λk, and let γ= 2.5. Figures 1–4 show the dynamic behaviors of system (2) with the initial functions satisfying condition (7).

Figure 1. (a) The time evolutions for system (2) with λ = 0.5, μ = 0.4, τ = 5 and Ik(0) = 0.2, k = 2, 3, 4, 5,Ik(0) = 0 for the other k and R<sup>0</sup> = 0.4006. (b) The time evolutions for system (2) with different initial values, λ = 0.4, μ = 0.4, τ = 5 and R<sup>0</sup> = 0.4006.

Figure 2. (a) The time evolutions for system (2) with λ = 1, μ = 0.1, τ = 5 and Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k and R<sup>0</sup> = 3.2051. (b) The time evolutions for system (2) with different initial values, λ = 1, μ = 0.1, τ = 5 and R<sup>0</sup> = 3.2051.

Figure 3. (a) Phase trajectories on SR-plane of system (2) with different initial values, λ = 0.5, μ = 0.4, τ = 5 and R<sup>0</sup> = 0.4006. (b) Phase trajectories on SR-plane of system (2) with different initial values, λ = 1, μ = 0.1, τ = 5 and R<sup>0</sup> = 3.2051.

Denote that

$$S(t) = \sum\_{k=m}^{n} p(k) S\_k(t), \\ I(t) = \sum\_{k=m}^{n} p(k) I\_k(t), \\ R(t) = \sum\_{k=m}^{n} p(k) R\_k(t). \tag{28}$$

Second, time delay τ has no effects on the basic reproductive number R<sup>0</sup> according to (21), but it has much impact on the of process of the disease; the slower the relative density of infected nodes converges to zero, the larger τ gets, i.e., time delay may slow down the speed of disappearing the disease spreading on network. Meanwhile, time delay may effectively reduce the peak value of the relative density of infected nodes when R<sup>0</sup> > 1. Thus, the delay

Figure 4. (a) The time evolutions of the average relative density I(t) for system (2) with different τ as well as Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k, λ = 0.5, μ = 0.4 and R<sup>0</sup> = 0.4006. (b) The time evolutions of the average relative I(t) for system (2) with different τ as well as Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k, λ = 1, μ = 0.1 and

0.02

(a) (b)

0.04

0.06

Desity I(t)

0.08

0.1

0.12

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>

http://dx.doi.org/10.5772/intechopen.71087

τ=2 τ=6 τ=10 τ=18 33

τ=2 τ=6 τ=10 τ=18

t

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>

t

τ=2 τ=6 τ=10 τ=18

At last, we know from Figure 5 that time evolutions of the average force of infection for system (2) is consistent with time evolutions of the average relative density I(t). However,

0.02

0.04

0.06

Θ(t)

(a) (b)

Figure 5. (a) The time evolutions of the average force of infection Θ(t) for system (2) with different τ as well as with Ik(0) = 0.2, k = 2, 3, 4, 5,Ik(0) = 0 for the other k, λ = 0.5, μ = 0.4 and R<sup>0</sup> = 0.4006. (b) The time evolutions of the average force of infection Θ(t) for system (2) with different τ as well as Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k, λ = 1, μ = 0.1 and

0.08

0.1

0.12

there is only one outbreak, which is different from the phenomenon in [15].

τ=2 τ=6 τ=10 τ=18

cannot be ignored.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

R<sup>0</sup> = 3.2051.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

R<sup>0</sup> = 3.2051.

Θ(t)

Density I(t)

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>0</sup>

t

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>0</sup>

t

They are the relative average density of susceptible nodes, exposed nodes, infected nodes, and recovered nodes at time t, respectively.

First, Figures 1 and 2 show that the infection eventually disappears, whatever R<sup>0</sup> < 1 or not, and the outbreak of disease spreading appears when R<sup>0</sup> > 1 and the outbreak of disease spreading does not appear when R<sup>0</sup> ≤ 1. Meanwhile, Figure 3 shows that phase trajectories on SR-plane of system (2) with different initial values tend to be S(t) +R(t) = 1, i.e., ∑kp(k)(Sk(t) +Rk(t)) = 1, which is consistent with the fact that the equilibrium M<sup>0</sup> is globally attractive. The numerical simulation results are identical with Theorems 2.1–2.2.

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network http://dx.doi.org/10.5772/intechopen.71087 33

Figure 4. (a) The time evolutions of the average relative density I(t) for system (2) with different τ as well as Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k, λ = 0.5, μ = 0.4 and R<sup>0</sup> = 0.4006. (b) The time evolutions of the average relative I(t) for system (2) with different τ as well as Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k, λ = 1, μ = 0.1 and R<sup>0</sup> = 3.2051.

Second, time delay τ has no effects on the basic reproductive number R<sup>0</sup> according to (21), but it has much impact on the of process of the disease; the slower the relative density of infected nodes converges to zero, the larger τ gets, i.e., time delay may slow down the speed of disappearing the disease spreading on network. Meanwhile, time delay may effectively reduce the peak value of the relative density of infected nodes when R<sup>0</sup> > 1. Thus, the delay cannot be ignored.

At last, we know from Figure 5 that time evolutions of the average force of infection for system (2) is consistent with time evolutions of the average relative density I(t). However, there is only one outbreak, which is different from the phenomenon in [15].

Denote that

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Density

S tðÞ¼ <sup>X</sup><sup>n</sup>

recovered nodes at time t, respectively.

k¼m

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>

S+R=1

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>0</sup>

t

R

p kð ÞSkð Þ<sup>t</sup> ,I tðÞ¼ <sup>X</sup><sup>n</sup>

numerical simulation results are identical with Theorems 2.1–2.2.

k¼m

They are the relative average density of susceptible nodes, exposed nodes, infected nodes, and

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) (b)

Figure 3. (a) Phase trajectories on SR-plane of system (2) with different initial values, λ = 0.5, μ = 0.4, τ = 5 and R<sup>0</sup> = 0.4006.

(b) Phase trajectories on SR-plane of system (2) with different initial values, λ = 1, μ = 0.1, τ = 5 and R<sup>0</sup> = 3.2051.

S

0.02

(a) (b)

Figure 2. (a) The time evolutions for system (2) with λ = 1, μ = 0.1, τ = 5 and Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k and R<sup>0</sup> = 3.2051. (b) The time evolutions for system (2) with different initial values, λ = 1, μ = 0.1, τ = 5 and R<sup>0</sup> = 3.2051.

0.04

0.06

Density I(t)

0.08

0.1

0.12

S50(t) E50(t) I 50(t) R50(t)

32 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

First, Figures 1 and 2 show that the infection eventually disappears, whatever R<sup>0</sup> < 1 or not, and the outbreak of disease spreading appears when R<sup>0</sup> > 1 and the outbreak of disease spreading does not appear when R<sup>0</sup> ≤ 1. Meanwhile, Figure 3 shows that phase trajectories on SR-plane of system (2) with different initial values tend to be S(t) +R(t) = 1, i.e., ∑kp(k)(Sk(t) +Rk(t)) = 1, which is consistent with the fact that the equilibrium M<sup>0</sup> is globally attractive. The

p kð ÞIkð Þ<sup>t</sup> ,R tðÞ¼ <sup>X</sup><sup>n</sup>

k¼m

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>

S+R=1

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>0</sup>

I(t) with Ik

I(t) with Ik

I(t) with Ik

(0)=0.2, k=2,3,4,5

(0)=0.3, k=2,3,4,5

(0)=0.4, k=2,3,4,5

t

I

p kð ÞRkð Þt : (28)

Figure 5. (a) The time evolutions of the average force of infection Θ(t) for system (2) with different τ as well as with Ik(0) = 0.2, k = 2, 3, 4, 5,Ik(0) = 0 for the other k, λ = 0.5, μ = 0.4 and R<sup>0</sup> = 0.4006. (b) The time evolutions of the average force of infection Θ(t) for system (2) with different τ as well as Ik(0) = 0.2, k = 2, 3, 4, 5, Ik(0) = 0 for the other k, λ = 1, μ = 0.1 and R<sup>0</sup> = 3.2051.

#### 3. Analysis of the SEIRS model with time delays

#### 3.1. The SEIRS model

Since some diseases confer temporary immunity, the recovered nodes cycle back into the susceptive class after an immune period. Let ω be furthermore the immune period of the recovered node, and the recovered node cycles back into the susceptive class after an immune period ω. Denote that σ = max {τ, ω}. Based on the model (2), the relative densities Sk(t), Ek(t), Ik(t) and Rk(t), at the mean-field level, satisfy the following set of coupled different equations when t > 0:

$$\begin{cases} \dot{\mathcal{S}}\_k(t) = -\lambda(k)\mathsf{S}\_k(t)\Theta(t) + \mu I\_k(t-\omega), \\ \dot{\mathcal{E}}\_k(t) = \lambda(k)\mathsf{S}\_k(t)\Theta(t) - \lambda(k)\mathsf{S}\_k(t-\tau)\Theta(t-\tau), \\ \dot{I}\_k(t) = \lambda(k)\mathsf{S}\_k(t-\tau)\Theta(t-\tau) - \mu I\_k(t), \\ \dot{\mathcal{R}}\_k(t) = \mu I\_k(t) - \mu I\_k(t-\omega) \end{cases} \tag{29}$$

3.2. The main results for the model

unique endemic equilibrium E<sup>∗</sup> <sup>¼</sup> <sup>S</sup><sup>∗</sup>

<sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> μ

<sup>m</sup>þ<sup>1</sup>; <sup>⋯</sup>; <sup>S</sup><sup>∗</sup>

<sup>k</sup>Θ<sup>∗</sup> <sup>þ</sup> <sup>μ</sup><sup>I</sup>

<sup>k</sup> <sup>þ</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>τ</sup>S<sup>∗</sup>

<sup>Θ</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup> h ik X k0

<sup>k</sup> <sup>¼</sup> <sup>λ</sup>kΘ<sup>∗</sup>

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>λ</sup>kΘ<sup>∗</sup>

<sup>μ</sup> <sup>þ</sup> <sup>λ</sup><sup>k</sup> <sup>1</sup> <sup>þ</sup> μτ <sup>þ</sup> ωμ � �Θ<sup>∗</sup> � �<sup>2</sup>

k 2

Hence, if R<sup>0</sup> > 1, Eq. (38) has a unique positive solution. Consequently, system (32) has a unique

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>λ</sup>k<sup>μ</sup>

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>λ</sup><sup>2</sup>

<sup>k</sup>¼<sup>m</sup> f kð Þp kð Þ in which <sup>f</sup>(k) is a function.

<sup>m</sup>; S<sup>∗</sup>

�λkS<sup>∗</sup>

S∗

I ∗

Substituting it into Eq. (4), we obtain the self-consistency equality:

Xn k¼m

<sup>Θ</sup><sup>∗</sup> <sup>¼</sup> <sup>λ</sup> h ik

> Xn k¼m

<sup>00</sup> <sup>Θ</sup><sup>∗</sup> ð Þ¼ �2<sup>λ</sup>

h ik

<sup>1</sup>; S<sup>∗</sup> <sup>2</sup>; ⋯; S<sup>∗</sup> n; I ∗ 1; I ∗ <sup>2</sup>; ⋯; I ∗ n � � since (35) and (37) hold.

Xn k¼m

<sup>Θ</sup>∗¼<sup>0</sup> <sup>¼</sup> <sup>λ</sup> h ik

f

endemic equilibrium E<sup>∗</sup> S<sup>∗</sup>

h i λð Þk φð Þk

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

Theorem 3.1. System (32) always has a disease-free equilibrium E<sup>0</sup> (1, 1, ⋯, 1, 0, 0, ⋯, 0), and it has a

n; I ∗ <sup>m</sup>; I ∗ <sup>m</sup>þ<sup>1</sup>; <sup>⋯</sup>; <sup>I</sup>

Proof. Obviously, the disease-free equilibrium E<sup>0</sup> of system (32) always exists. Now, we discuss the existence of the endemic equilibrium of system (32). Note that the equilibrium E<sup>∗</sup> should

> ∗ <sup>k</sup> ¼ 0,

<sup>k</sup>Θ<sup>∗</sup> <sup>þ</sup> <sup>I</sup> ∗ <sup>k</sup> þ ωμI ∗

� � when R<sup>0</sup> > 1.

ϕ k<sup>0</sup> ð Þp k<sup>0</sup> ð ÞI

∗

<sup>μ</sup> <sup>þ</sup> <sup>λ</sup><sup>k</sup> <sup>1</sup> <sup>þ</sup> μτ <sup>þ</sup> ωμ � �Θ<sup>∗</sup> : (37)

<sup>μ</sup> <sup>þ</sup> <sup>λ</sup><sup>k</sup> <sup>1</sup> <sup>þ</sup> μτ <sup>þ</sup> ωμ � �Θ<sup>∗</sup> <sup>¼</sup> <sup>f</sup> <sup>Θ</sup><sup>∗</sup> ð Þ: (38)

<sup>μ</sup> <sup>þ</sup> <sup>λ</sup><sup>k</sup> <sup>1</sup> <sup>þ</sup> μτ <sup>þ</sup> ωμ � �Θ<sup>∗</sup> � �<sup>3</sup> <sup>&</sup>lt; <sup>0</sup>: (40)

� � � � � Θ∗¼0

<sup>μ</sup> <sup>1</sup> <sup>þ</sup> μτ <sup>þ</sup> ωμ � �Θ<sup>∗</sup>

h i<sup>k</sup> , (34)

http://dx.doi.org/10.5772/intechopen.71087

35

<sup>k</sup> <sup>¼</sup> <sup>1</sup>, (35)

<sup>k</sup> : (36)

<sup>¼</sup> <sup>λ</sup> <sup>k</sup>ϕð Þ<sup>k</sup> � �

<sup>μ</sup>h i<sup>k</sup> <sup>¼</sup> <sup>R</sup><sup>0</sup> (39)

∗ n

Denote that

satisfy:

where

Note that

and

f <sup>0</sup> <sup>Θ</sup><sup>∗</sup> ð Þ� �

We obtain from (35) that

where h i f kð Þ <sup>¼</sup> <sup>P</sup><sup>n</sup>

with the normalization condition (3).

Furthermore, we obtain from the third equation and the fourth equation of system (29) that

$$E\_k(t) = \lambda(k) \int\_{t-\tau}^t \mathcal{S}\_k(s)\Theta(s)ds,\\ \mathcal{R}\_k(t) = \mu \int\_{t-\omega}^t I\_k(s)ds. \tag{30}$$

Hence, the normalization condition becomes the following mathematical form:

$$S\_k(t) + \lambda(k) \int\_{t-\tau}^t S\_k(s)\Theta(s)ds + I\_k(t) + \mu \int\_{t-\omega}^t I\_k(s)ds = 1. \tag{31}$$

Obviously, if we discuss the dynamical behaviors of system (29), we just need to discuss the following system:

$$\begin{cases} \dot{S}\_k(t) = -\lambda(k)\mathbb{S}\_k(t)\Theta(t) + \mu I\_k(t-\omega), \\ \dot{I}\_k(t) = \lambda(k)\mathbb{S}\_k(t-\tau)\Theta(t-\tau) - \mu I\_k(t) \end{cases} \tag{32}$$

with the normalization condition (31).

The initial conditions of system (32) are

$$S\_k(\theta) = \varphi\_k(\theta), I\_k(\theta) = \Psi\_k(\theta), \theta \in [-\sigma, 0]. \tag{33}$$

which satisfy Skð Þþ <sup>0</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>Ð</sup> <sup>0</sup> �<sup>τ</sup> Skð Þ<sup>s</sup> <sup>Θ</sup>ð Þ<sup>s</sup> ds <sup>þ</sup> Ikð Þþ <sup>0</sup> <sup>μ</sup> Ð 0 �<sup>ω</sup> Ikð Þ<sup>s</sup> ds <sup>¼</sup> 1. Hence, the normalization condition (31) holds. And, Φ<sup>k</sup> = (ϕk(θ), Ψk(θ), k =m, m+ 1, ⋯, n� m+ 1)∈C are nonnegative continuous on [�σ, 0], ϕk(0) > 0, Ψk(0) > 0, and ζ(θ) = 0 for θ = 0. C denotes the Banach space C([�σ, 0], <sup>R</sup>2(n� <sup>m</sup> + 1)) with the norm k k <sup>ω</sup> <sup>¼</sup> <sup>P</sup><sup>n</sup> <sup>i</sup>¼<sup>m</sup> j j <sup>Ψ</sup>ið Þ <sup>θ</sup> <sup>2</sup> <sup>σ</sup> þ φ<sup>i</sup> ð Þ <sup>θ</sup> � � � � 2 σ � �<sup>1</sup>=<sup>2</sup> � , where |f(θ)|<sup>σ</sup> = sup�τ≤θ≤<sup>0</sup> ∣f(θ)∣. The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network http://dx.doi.org/10.5772/intechopen.71087 35

#### 3.2. The main results for the model

Denote that

3. Analysis of the SEIRS model with time delays

34 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

S\_

8 >>>><

>>>>:

with the normalization condition (3).

with the normalization condition (31). The initial conditions of system (32) are

<sup>R</sup>2(n� <sup>m</sup> + 1)) with the norm k k <sup>ω</sup> <sup>¼</sup> <sup>P</sup><sup>n</sup>

which satisfy Skð Þþ <sup>0</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>Ð</sup> <sup>0</sup>

E\_

\_

EkðÞ¼ t λð Þk

SkðÞþ t λð Þk

Since some diseases confer temporary immunity, the recovered nodes cycle back into the susceptive class after an immune period. Let ω be furthermore the immune period of the recovered node, and the recovered node cycles back into the susceptive class after an immune period ω. Denote that σ = max {τ, ω}. Based on the model (2), the relative densities Sk(t), Ek(t), Ik(t) and Rk(t), at the mean-field level, satisfy the following set of coupled different equations

<sup>k</sup>ðÞ¼� t λð Þk Skð Þt Θð Þþ t μIkð Þ t � ω ,

I <sup>k</sup>ðÞ¼ t λð Þk Skð Þ t � τ Θð Þ� t � τ μIkð Þt ,

Furthermore, we obtain from the third equation and the fourth equation of system (29) that

Skð Þs Θð Þs ds, RkðÞ¼ t μ

Skð Þs Θð Þs ds þ Ikð Þþ t μ

Obviously, if we discuss the dynamical behaviors of system (29), we just need to discuss the

<sup>k</sup>ðÞ¼� t λð Þk Skð Þt Θð Þþ t μIkð Þ t � ω ,

I <sup>k</sup>ðÞ¼ t λð Þk Skð Þ t � τ Θð Þ� t � τ μIkð Þt

condition (31) holds. And, Φ<sup>k</sup> = (ϕk(θ), Ψk(θ), k =m, m+ 1, ⋯, n� m+ 1)∈C are nonnegative continuous on [�σ, 0], ϕk(0) > 0, Ψk(0) > 0, and ζ(θ) = 0 for θ = 0. C denotes the Banach space C([�σ, 0],

� �<sup>1</sup>=<sup>2</sup> �

<sup>σ</sup> þ φ<sup>i</sup> ð Þ <sup>θ</sup> � � � � 2 σ ðt t�ω

ðt t�ω

Skð Þ¼ θ φkð Þ θ , Ikð Þ¼ θ Ψkð Þ θ , θ∈½ � �σ; 0 , (33)

Ð 0

<sup>R</sup>\_ <sup>k</sup>ðÞ¼ <sup>t</sup> <sup>μ</sup>IkðÞ� <sup>t</sup> <sup>μ</sup>Ikð Þ <sup>t</sup> � <sup>ω</sup>

ðt t�τ

ðt t�τ

S\_

(

\_

Hence, the normalization condition becomes the following mathematical form:

�<sup>τ</sup> Skð Þ<sup>s</sup> <sup>Θ</sup>ð Þ<sup>s</sup> ds <sup>þ</sup> Ikð Þþ <sup>0</sup> <sup>μ</sup>

<sup>i</sup>¼<sup>m</sup> j j <sup>Ψ</sup>ið Þ <sup>θ</sup> <sup>2</sup>

<sup>k</sup>ðÞ¼ t λð Þk Skð Þt Θð Þ� t λð Þk Skð Þ t � τ Θð Þ t � τ ,

3.1. The SEIRS model

when t > 0:

following system:

$$R\_0 = \frac{1}{\mu} \frac{\langle \lambda(k)\varphi(k)\rangle}{\langle k\rangle},\tag{34}$$

where h i f kð Þ <sup>¼</sup> <sup>P</sup><sup>n</sup> <sup>k</sup>¼<sup>m</sup> f kð Þp kð Þ in which <sup>f</sup>(k) is a function.

Theorem 3.1. System (32) always has a disease-free equilibrium E<sup>0</sup> (1, 1, ⋯, 1, 0, 0, ⋯, 0), and it has a unique endemic equilibrium E<sup>∗</sup> <sup>¼</sup> <sup>S</sup><sup>∗</sup> <sup>m</sup>; S<sup>∗</sup> <sup>m</sup>þ<sup>1</sup>; <sup>⋯</sup>; <sup>S</sup><sup>∗</sup> n; I ∗ <sup>m</sup>; I ∗ <sup>m</sup>þ<sup>1</sup>; <sup>⋯</sup>; <sup>I</sup> ∗ n � � when R<sup>0</sup> > 1.

Proof. Obviously, the disease-free equilibrium E<sup>0</sup> of system (32) always exists. Now, we discuss the existence of the endemic equilibrium of system (32). Note that the equilibrium E<sup>∗</sup> should satisfy:

$$\begin{aligned} -\lambda k S\_k^\* \Theta^\* + \mu I\_k^\* &= 0, \\ S\_k^\* + \lambda (k) \tau S\_k^\* \Theta^\* + I\_k^\* + \omega \mu I\_k^\* &= 1, \end{aligned} \tag{35}$$

where

(29)

(32)

Ikð Þs ds: (30)

Ikð Þs ds ¼ 1: (31)

�<sup>ω</sup> Ikð Þ<sup>s</sup> ds <sup>¼</sup> 1. Hence, the normalization

, where |f(θ)|<sup>σ</sup> = sup�τ≤θ≤<sup>0</sup> ∣f(θ)∣.

$$
\Theta^\* = \frac{1}{\langle k \rangle} \sum\_{k'} \phi(k') p(k') I\_k^\*. \tag{36}
$$

We obtain from (35) that

$$I\_k^\* = \frac{\lambda k \Theta^\*}{\mu + \lambda k (1 + \mu \tau + \omega \mu) \Theta^\*} \,. \tag{37}$$

Substituting it into Eq. (4), we obtain the self-consistency equality:

$$\Theta^\* = \frac{\lambda}{\langle k \rangle} \sum\_{k=m}^n \phi(k) p(k) \frac{\lambda k \Theta^\*}{\mu + \lambda k (1 + \mu \tau + \omega \mu) \Theta^\*} = f(\Theta^\*). \tag{38}$$

Note that

$$\left.f'(\Theta^\*)\right|\_{\Theta^\*=0} = \frac{\lambda}{\langle k \rangle} \sum\_{k=m}^n \phi(k)p(k) \frac{\lambda k \mu}{\left(\mu + \lambda k \left(1 + \mu \tau + \omega \mu\right) \Theta^\*\right)^2} \bigg|\_{\Theta^\*=0} = \frac{\lambda \left}{\mu \left} = R\_0 \tag{39}$$

and

$$f''(\Theta^\*) = \frac{-2\lambda}{\langle k \rangle} \sum\_{k=m}^n \phi(k) p(k) \frac{\lambda^2 k^2 \mu \left(1 + \mu \tau + \omega \mu \right) \Theta^\*}{\left(\mu + \lambda k \left(1 + \mu \tau + \omega \mu \right) \Theta^\*\right)^3} < 0. \tag{40}$$

Hence, if R<sup>0</sup> > 1, Eq. (38) has a unique positive solution. Consequently, system (32) has a unique endemic equilibrium E<sup>∗</sup> S<sup>∗</sup> <sup>1</sup>; S<sup>∗</sup> <sup>2</sup>; ⋯; S<sup>∗</sup> n; I ∗ 1; I ∗ <sup>2</sup>; ⋯; I ∗ n � � since (35) and (37) hold.

Theorem 3.2. If R<sup>0</sup> ≤ 1, the disease-free equilibrium E<sup>0</sup> of system (32) is globally attractive.

Proof. We define a Lyapunov function V(t) as

$$V(t) = \frac{1}{2}\Theta^2(t) + \gamma \int\_{t-\tau}^t \Theta^2(\mu)d\mu,\tag{41}$$

Then, T(t) is uniformly persistent if and only if, for each Mi∈ M,

constant <sup>ε</sup> such that limt! <sup>+</sup> <sup>∞</sup> inf <sup>I</sup>(t) > <sup>ε</sup>, where I tðÞ¼ <sup>P</sup><sup>n</sup>

where <sup>S</sup>; <sup>Ψ</sup> � � <sup>¼</sup> ð Þ Sm; Smþ<sup>1</sup>; <sup>⋯</sup>; Sn; <sup>Ψ</sup> <sup>m</sup>; <sup>Ψ</sup> <sup>m</sup>þ<sup>1</sup>; <sup>⋯</sup>; <sup>Ψ</sup><sup>n</sup> .

initial function ω = (ζm(θ), Ψm(θ), ⋯, Ψn(θ), ϕn(θ)) and

Finally, the proof will be done if we prove Ws

When t > t1, we obtain from (32) that

ðt t�τ

On the other hand, for t > t<sup>1</sup> we have from (4) and (50) that

SkðÞ¼ t 1 � Ikð Þþ t λð Þk

where <sup>A</sup><sup>~</sup> <sup>∂</sup> <sup>¼</sup> <sup>⋃</sup>

Proof. Denote that

and, consequently,

rium of system (32) on ∂X<sup>0</sup>

and acyclic.

x∈ A<sup>∂</sup>

T∂(t) in ∂X<sup>0</sup> in which T∂(t) = T(t)|∂X0.

W<sup>s</sup>

ð Þ Mi <sup>∩</sup> <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>∅</sup>, (45)

http://dx.doi.org/10.5772/intechopen.71087

37

, <sup>A</sup><sup>~</sup> <sup>∂</sup> <sup>¼</sup> f g <sup>E</sup><sup>0</sup> , while <sup>E</sup><sup>0</sup> is isolated

(E0) is the stable manifold

(50)

ωð Þx , ω(x) is the omega limit set of T(x) through x, and A<sup>∂</sup> is global attractor of

<sup>X</sup> <sup>¼</sup> <sup>S</sup>; <sup>Ψ</sup> � � : <sup>Ψ</sup>kð Þ <sup>θ</sup> <sup>≥</sup> <sup>0</sup>; for all <sup>θ</sup><sup>∈</sup> ½ � �ζ; <sup>0</sup> ; <sup>k</sup> <sup>¼</sup> <sup>m</sup>; <sup>m</sup> <sup>þ</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup> � �, (46)

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>S</sup>; <sup>Ψ</sup> � � : <sup>Ψ</sup>kð Þ <sup>θ</sup> <sup>&</sup>gt; <sup>0</sup>; for some <sup>θ</sup><sup>∈</sup> ½ � �ζ; <sup>0</sup> ; <sup>k</sup> <sup>¼</sup> <sup>m</sup>; <sup>m</sup> <sup>þ</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup> � �, (47)

<sup>∂</sup>X<sup>0</sup> <sup>¼</sup> <sup>X</sup>=X<sup>0</sup> <sup>¼</sup> <sup>S</sup>; <sup>Ψ</sup> � � : <sup>Ψ</sup>ið Þ¼ <sup>θ</sup> <sup>0</sup>; for all <sup>θ</sup>∈½ � �σ; <sup>0</sup> ; <sup>i</sup>∈f g <sup>m</sup>; <sup>m</sup> <sup>þ</sup> <sup>1</sup>; <sup>⋯</sup>; <sup>n</sup> � �, (48)

(E0) ∩ X<sup>0</sup> = ∅, where Ws

lim<sup>t</sup>!þ<sup>∞</sup> inf SkðÞ¼ <sup>t</sup> <sup>1</sup>, lim<sup>t</sup>!þ<sup>∞</sup> inf IkðÞ¼ <sup>t</sup> <sup>0</sup>, k <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n: (49)

Let (Sm(t), Im(t), ⋯, Sn, In(t)) = (Sm(t, ω), Im(t, ω), ⋯, Sn(t, ω), In(t, ω)) be the solution of (32) with

T(t)(ω)(θ)=(Sm(t + θ, ω), Im(t + θ, ω), ⋯, Sn(t + θ, ω), In(t + θ, ω)), θ ∈ [�σ, 0]. Obviously, X and X<sup>0</sup> are positively invariant sets for T(t). T(t) is completely continuous for t > 0. Also, it follows from 0 < Sk(t), Ik(t) ≤ 1 for t > 0 that T(t) is point dissipative. E<sup>0</sup> is the unique equilib-

Since R<sup>0</sup> > 1, we may choose 0 < η < 1 such that α = η(λ(k)τ〈φ(k)〉 +1+ μω) satisfies (1 � α)R<sup>0</sup> > 1. At the same time, there exists a t<sup>1</sup> > τ such that Ik(t) < η for t > t<sup>1</sup> due to limt! <sup>+</sup> <sup>∞</sup> inf Ik = 0.

Ikð Þ<sup>s</sup> ds � � <sup>≥</sup> <sup>1</sup> � <sup>η</sup> <sup>þ</sup> <sup>λ</sup>ð Þ<sup>k</sup> τη ϕð Þ<sup>k</sup> � � <sup>þ</sup> μωη � � <sup>¼</sup> <sup>1</sup> � <sup>α</sup>:

ðt t�ω

, and it is globally stable on ∂X<sup>0</sup>

of E0. Suppose it is not true, then there exists a solution S; I � � in X<sup>0</sup> such that

Skð Þs Θð Þs ds þ μ

<sup>k</sup>¼<sup>m</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> p kð ÞIkð Þ<sup>t</sup> .

Theorem 3.3. For system (32), if R<sup>0</sup> > 1, the disease is uniformly persistent, i.e., there exists a positive

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

where <sup>γ</sup> is a constant to be determined. Let <sup>G</sup> <sup>¼</sup> <sup>ϕ</sup> : <sup>V</sup>\_ <sup>ϕ</sup> � � <sup>¼</sup> <sup>0</sup> � �, and <sup>M</sup> is the largest set in <sup>G</sup> which is invariant with respect to system (32). Clearly, M is not empty since E0∈ M. Calculating the derivative of V(t) along the solution of (32), we get

$$\begin{split} \left. \dot{V}(t) \right|\_{\langle 3,3 \rangle} &= \Theta(t) \left[ \frac{1}{\lambda(k)} \sum\_{k} q(k) p(k) \left( -\lambda k \mathbb{S}\_{k}(t-\tau) \Theta(t-\tau) - \mu \Theta(t) \right) \right] + \gamma \Theta^{2}(t) - \gamma \Theta^{2}(t-\tau) \\ &\leq \Theta(t) \left[ \frac{1}{\langle k \rangle} \langle \lambda(k) \phi(k) \rangle \Theta(t-\tau) - \mu \Theta(t) \right] + \gamma \Theta^{2}(t) - \gamma \Theta^{2}(t-\tau) \\ &\leq \frac{1}{2\langle k \rangle} \left\langle \lambda(k) \phi(k) \right\rangle \Theta^{2}(t) + \frac{1}{2\langle k \rangle} \left\langle \lambda(k) \phi(k) \right\rangle \Theta^{2}(t-\tau) - \mu \Theta^{2}(t) + \gamma \Theta^{2}(t) - \gamma \Theta^{2}(t-\tau) \\ &= \left( \frac{1}{2\langle k \rangle} \langle \lambda(k) \phi(k) \rangle - \mu + \gamma \right) \Theta^{2}(t) + \left( \frac{1}{2\langle k \rangle} \langle \lambda(k) \phi(k) \rangle - \gamma \right) \Theta^{2}(t-\tau). \end{split} \tag{42}$$

Note that R<sup>0</sup> ≤ 1 implies <sup>1</sup> h i<sup>k</sup> h i <sup>λ</sup>ð Þ<sup>k</sup> <sup>φ</sup>ð Þ<sup>k</sup> <sup>≤</sup> <sup>μ</sup> <sup>&</sup>lt; 0; if we let <sup>γ</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup>h i<sup>k</sup> h i <sup>λ</sup>ð Þ<sup>k</sup> <sup>φ</sup>ð Þ<sup>k</sup> , we have from (42) that

$$\left. \dot{V}(t) \right|\_{(3,3)} \leq \left( \frac{1}{\langle k \rangle} \left\langle \lambda(k)\phi(k) \right\rangle - \mu \right) \Theta^2(t) \leq 0. \tag{43}$$

It follows from Sk(t) + Ek(t) + Ik(t) + Rk(t) = 1 that M = E0. Therefore, by the LaSalle invariance principle [24], the disease-free equilibrium E<sup>0</sup> is globally attractive.

Lemma 3.1. [28] Consider the following equation:

$$
\dot{\mathbf{x}}(t) = a\_1 \mathbf{x}(t - \tau) - a\_2 \mathbf{x}(t), \tag{44}
$$

where a1, a2, τ > 0; x(t) > 0 for �τ ≤ t ≤ 0. We have


Lemma 3.2. ([29], p 273–280) Let X be a complete metric space, X = X<sup>0</sup> ∪ ∂X<sup>0</sup> , where ∂X<sup>0</sup> , assumed to be nonempty, is the boundary of X<sup>0</sup> . Assume the C<sup>0</sup> – semigroup T(t) on X satisfies T(x) : <sup>X</sup>0!X<sup>0</sup> , <sup>T</sup>(x) : <sup>∂</sup>X0!∂X<sup>0</sup> and


Then, T(t) is uniformly persistent if and only if, for each Mi∈ M,

$$\mathcal{W}^s(M\_i) \cap X^0 = \mathcal{Q}\_{\prime} \tag{45}$$

where <sup>A</sup><sup>~</sup> <sup>∂</sup> <sup>¼</sup> <sup>⋃</sup> x∈ A<sup>∂</sup> ωð Þx , ω(x) is the omega limit set of T(x) through x, and A<sup>∂</sup> is global attractor of T∂(t) in ∂X<sup>0</sup> in which T∂(t) = T(t)|∂X0.

Theorem 3.3. For system (32), if R<sup>0</sup> > 1, the disease is uniformly persistent, i.e., there exists a positive constant <sup>ε</sup> such that limt! <sup>+</sup> <sup>∞</sup> inf <sup>I</sup>(t) > <sup>ε</sup>, where I tðÞ¼ <sup>P</sup><sup>n</sup> <sup>k</sup>¼<sup>m</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> p kð ÞIkð Þ<sup>t</sup> .

Proof. Denote that

Theorem 3.2. If R<sup>0</sup> ≤ 1, the disease-free equilibrium E<sup>0</sup> of system (32) is globally attractive.

ðt t�τ

which is invariant with respect to system (32). Clearly, M is not empty since E0∈ M. Calculat-

<sup>2</sup>h i<sup>k</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � �Θ<sup>2</sup>

h i<sup>k</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � � � <sup>μ</sup> � �

It follows from Sk(t) + Ek(t) + Ik(t) + Rk(t) = 1 that M = E0. Therefore, by the LaSalle invariance

1

<sup>Θ</sup><sup>2</sup>ð Þþ <sup>t</sup>

h i<sup>k</sup> h i <sup>λ</sup>ð Þ<sup>k</sup> <sup>φ</sup>ð Þ<sup>k</sup> <sup>≤</sup> <sup>μ</sup> <sup>&</sup>lt; 0; if we let <sup>γ</sup> <sup>¼</sup> <sup>1</sup>

1

<sup>φ</sup>ð Þ<sup>k</sup> p kðÞ�λkSkð Þ <sup>t</sup> � <sup>τ</sup> <sup>Θ</sup>ð Þ� <sup>t</sup> � <sup>τ</sup> μΘð Þ<sup>t</sup> � � " #

Θ<sup>2</sup> μ

<sup>þ</sup> <sup>γ</sup>Θ<sup>2</sup>ð Þ� <sup>t</sup> <sup>γ</sup>Θ<sup>2</sup>ð Þ <sup>t</sup> � <sup>τ</sup>

ð Þ� <sup>t</sup> � <sup>τ</sup> <sup>μ</sup>Θ<sup>2</sup>

<sup>2</sup>h i<sup>k</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � � � <sup>γ</sup> � �

Θ2

x t \_ðÞ¼ a1x tð Þ� � τ a2x tð Þ, (44)

. Assume the C<sup>0</sup> – semigroup T(t) on X satisfies T(x) : <sup>X</sup>0!X<sup>0</sup>

� �dμ, (41)

<sup>þ</sup> <sup>γ</sup>Θ<sup>2</sup>ð Þ� <sup>t</sup> <sup>γ</sup>Θ<sup>2</sup>ð Þ <sup>t</sup> � <sup>τ</sup>

ð Þ� <sup>t</sup> <sup>γ</sup>Θ<sup>2</sup>

ð Þ t � τ

(42)

� � <sup>¼</sup> <sup>0</sup> � �, and <sup>M</sup> is the largest set in <sup>G</sup>

ð Þþ <sup>t</sup> <sup>γ</sup>Θ<sup>2</sup>

<sup>Θ</sup><sup>2</sup>ð Þ <sup>t</sup> � <sup>τ</sup> :

<sup>2</sup>h i<sup>k</sup> h i <sup>λ</sup>ð Þ<sup>k</sup> <sup>φ</sup>ð Þ<sup>k</sup> , we have from (42) that

ð Þt ≤ 0: (43)

, where ∂X<sup>0</sup>

, assumed to

,

V tðÞ¼ <sup>1</sup> 2 Θ2 ðÞþ t γ

h i<sup>k</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � �Θð Þ� <sup>t</sup> � <sup>τ</sup> μΘð Þ<sup>t</sup> � �

ð Þþ t

1

where <sup>γ</sup> is a constant to be determined. Let <sup>G</sup> <sup>¼</sup> <sup>ϕ</sup> : <sup>V</sup>\_ <sup>ϕ</sup>

36 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

ing the derivative of V(t) along the solution of (32), we get

Proof. We define a Lyapunov function V(t) as

1 <sup>λ</sup>ð Þ<sup>k</sup> <sup>i</sup>

1

<sup>2</sup>h i<sup>k</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � �Θ<sup>2</sup>

Lemma 3.1. [28] Consider the following equation:

where a1, a2, τ > 0; x(t) > 0 for �τ ≤ t ≤ 0. We have

i. there is a t<sup>0</sup> such that T(t) is compact for t > t0.

iii. A~ <sup>∂</sup> is isolated and has an acyclic covering M.

i. if a<sup>1</sup> < a2, then lim<sup>t</sup>! <sup>+</sup> <sup>∞</sup>x(t) = 0, ii. if a<sup>1</sup> > a2, then lim<sup>t</sup>! <sup>+</sup> <sup>∞</sup>x(t)= + ∞.

be nonempty, is the boundary of X<sup>0</sup>

ii. T(t) is point dissipative in X.

<sup>T</sup>(x) : <sup>∂</sup>X0!∂X<sup>0</sup> and

<sup>2</sup>h i<sup>k</sup> <sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � � � <sup>μ</sup> <sup>þ</sup> <sup>γ</sup> � �

> V t \_ ð Þ� � ð Þ <sup>3</sup>:<sup>3</sup> <sup>≤</sup>

principle [24], the disease-free equilibrium E<sup>0</sup> is globally attractive.

Lemma 3.2. ([29], p 273–280) Let X be a complete metric space, X = X<sup>0</sup> ∪ ∂X<sup>0</sup>

X k

V t \_ ð Þ� �

ð Þ <sup>3</sup>:<sup>3</sup> <sup>¼</sup> <sup>Θ</sup>ð Þ<sup>t</sup>

≤ Θð Þt

<sup>¼</sup> <sup>1</sup>

≤ 1

Note that R<sup>0</sup> ≤ 1 implies <sup>1</sup>

$$X = \{ (\overline{\mathbb{S}}, \overline{\mathbb{V}}) : \mathbb{V}\_k(\theta) \ge 0, \text{for all } \theta \in [-\zeta, 0], k = m, m+1, \dots, n \}. \tag{46}$$

$$X^0 = \left\{ \left( \overline{\mathbb{S}}, \overline{\mathbb{V}} \right) : \mathbb{V}\_k(\theta) > 0, \text{for some } \theta \in [-\zeta, 0], k = m, m+1, \dots, n \right\},\tag{47}$$

and, consequently,

$$\partial X^0 = X/X^0 = \left\{ (\overline{\mathbb{S}}, \overline{\mathbb{V}}) : \mathbb{V}\_i(\theta) = 0, \text{for all } \theta \in [-\sigma, 0], i \in \{m, m+1, \dots, n\} \right\},\tag{48}$$

where <sup>S</sup>; <sup>Ψ</sup> � � <sup>¼</sup> ð Þ Sm; Smþ<sup>1</sup>; <sup>⋯</sup>; Sn; <sup>Ψ</sup> <sup>m</sup>; <sup>Ψ</sup> <sup>m</sup>þ<sup>1</sup>; <sup>⋯</sup>; <sup>Ψ</sup><sup>n</sup> .

Let (Sm(t), Im(t), ⋯, Sn, In(t)) = (Sm(t, ω), Im(t, ω), ⋯, Sn(t, ω), In(t, ω)) be the solution of (32) with initial function ω = (ζm(θ), Ψm(θ), ⋯, Ψn(θ), ϕn(θ)) and

T(t)(ω)(θ)=(Sm(t + θ, ω), Im(t + θ, ω), ⋯, Sn(t + θ, ω), In(t + θ, ω)), θ ∈ [�σ, 0]. Obviously, X and X<sup>0</sup> are positively invariant sets for T(t). T(t) is completely continuous for t > 0. Also, it follows from 0 < Sk(t), Ik(t) ≤ 1 for t > 0 that T(t) is point dissipative. E<sup>0</sup> is the unique equilibrium of system (32) on ∂X<sup>0</sup> , and it is globally stable on ∂X<sup>0</sup> , <sup>A</sup><sup>~</sup> <sup>∂</sup> <sup>¼</sup> f g <sup>E</sup><sup>0</sup> , while <sup>E</sup><sup>0</sup> is isolated and acyclic.

Finally, the proof will be done if we prove Ws (E0) ∩ X<sup>0</sup> = ∅, where Ws (E0) is the stable manifold of E0. Suppose it is not true, then there exists a solution S; I � � in X<sup>0</sup> such that

$$\lim\_{t \to +\infty} \inf \mathcal{S}\_k(t) = 1, \lim\_{t \to +\infty} \inf I\_k(t) = 0, k = 1, 2, \cdots, n. \tag{49}$$

Since R<sup>0</sup> > 1, we may choose 0 < η < 1 such that α = η(λ(k)τ〈φ(k)〉 +1+ μω) satisfies (1 � α)R<sup>0</sup> > 1. At the same time, there exists a t<sup>1</sup> > τ such that Ik(t) < η for t > t<sup>1</sup> due to limt! <sup>+</sup> <sup>∞</sup> inf Ik = 0.

When t > t1, we obtain from (32) that

$$S\_k(t) = 1 - \left(I\_k(t) + \lambda(k)\int\_{t-\tau}^t S\_k(s)\Theta(s)ds + \mu\int\_{t-\omega}^t I\_k(s)ds\right) \geq 1 - \left(\eta + \lambda(k)\tau\eta\langle\phi(k)\rangle + \mu\omega\eta\right) = 1 - \alpha. \tag{50}$$

On the other hand, for t > t<sup>1</sup> we have from (4) and (50) that

$$\begin{aligned} \Theta(t) &= \frac{1}{\langle k \rangle} \sum\_{k=m}^{n} \phi(k) p(k) \dot{I}\_k(t) \\\\ &= \frac{1}{\langle k \rangle} \sum\_{k=m}^{n} \phi(k) p(k) \left[ \lambda(k) S\_k(t-\tau) \Theta(t-\tau) - \mu I\_k(t) \right] \\\\ &\ge (1-\alpha) \frac{\langle \lambda(k) \phi(k) \rangle}{\langle k \rangle} \Theta(t-\tau) (-\mu \Theta(t) \end{aligned} \tag{51}$$

S\_

8 ><

>:

nomenon which means that a bifurcation may appear.

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> <sup>0</sup>

time t

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> <sup>0</sup>

time t

system (32) with ω = 6, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037.

system (32) with τ = 3, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

I

I

with the normalization condition

\_

<sup>k</sup>ðÞ¼� t λð Þk Skð Þt Θð Þþ t μIkð Þ t � ω ,

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

Skð Þþ t Ikð Þþ t RkðÞ¼ t 1: (53)

http://dx.doi.org/10.5772/intechopen.71087

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>0</sup>

time t

(52)

39

ω ω ω ω ω ω

I 15,τ=3 I 15,τ=6 I 15,τ=9 I 85,τ=3 I 85,τ=6 I 85,τ=9

I <sup>k</sup>ðÞ ¼ t λð Þk Skð Þt Θð Þ� t μIkð Þt , <sup>R</sup>\_ <sup>k</sup>ðÞ¼ <sup>t</sup> <sup>μ</sup>Ikð Þ� <sup>t</sup> <sup>μ</sup>Ikð Þ <sup>t</sup> � <sup>ω</sup> :

Figure 10 shows that when R<sup>0</sup> > 1, the quarantine delay ω can impact the density of infected nodes at the stationary state, and raising the quarantine period will suppress the viruses when ω is not large enough, which coincides with formula (37). Moreover, there exists periodic oscillation near the endemic equilibrium when ω is large enough. This is an interesting phe-

(a) (b)

Figure 6. (a) Evolutions of I(t) for system (32) with τ = 3, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for

0.1

(a) (b)

Figure 7. (a) Evolutions of I(t) for system (32) with ω = 6, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for

0.2

0.3

I15/I85

0.4

0.5

τ=3 τ=6 τ=9

ω=2 ω=6 ω=10

Note that (1 � <sup>α</sup>)R<sup>0</sup> > 1, and it follows with 1ð Þ � <sup>α</sup> h i <sup>λ</sup>ð Þ<sup>k</sup> <sup>φ</sup>ð Þ<sup>k</sup> h i<sup>k</sup> <sup>&</sup>gt; <sup>μ</sup>. Hence, we obtain from (51) that limt! <sup>+</sup> <sup>∞</sup>Θ(t)= + ∞ according to Lemma 3.1 contradicts lim<sup>t</sup> ! <sup>+</sup> <sup>∞</sup>Θ(t) = 0 due to limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0. Then, Ws (E0) ∩ X<sup>0</sup> = ∅.

Hence, the infection is uniformly persistent according to Lemma 3.2, i.e., there exists a positive constant <sup>ε</sup> such that limt! <sup>+</sup> <sup>∞</sup> inf Ik <sup>&</sup>gt; <sup>ε</sup> and, consequently, lim<sup>t</sup>!þ<sup>∞</sup>infI tð Þ <sup>&</sup>gt; <sup>P</sup><sup>n</sup> <sup>k</sup>¼<sup>m</sup> p kð Þ<sup>ε</sup> <sup>¼</sup> <sup>ε</sup>. This completes the proof.

In addition, Liu and Zhang discussed a simple SEIRS model without delay in [25], and the basic productive number for the model in [25] is λA/γ, which is consistent with R<sup>0</sup> for the model (32) in which ϕ(k) = A in this paper.

#### 3.3. Numerical simulations for the model

Now, we present the results of numerical simulations. The degree distribution of the scale-free network is p(k) =Ck�<sup>γ</sup> , and C satisfies P<sup>n</sup> <sup>k</sup>¼<sup>1</sup> p kð Þ¼ 1. Here, we set still the maximum degree n = 100 and the minimum degree m = 1.

Consider system (32). Let φ(k = ak<sup>α</sup> /(1 + b<sup>α</sup> ) in which a = 0.5, α = 0.75, b = 0.02 and λ(k) = λk, and let γ = 2.5 and μ = 0.06. Figures 1–4 show that the dynamic behaviors of system (32) with the initial functions satisfy condition (33) in which Ik(s) = 0.45, k = 2, 3, 4, 5 for s ∈[�σ, 0] and Ik = 0, k 6¼ 2, 3, 4, 5.

Although R<sup>0</sup> is irrelative to τ and ω. Figures 6 and 7 show that both the delay τ and ω have certain influence on the relative density of the infected nodes when R<sup>0</sup> < 1, for example, the faster the relative density of infected nodes converges to zero, the larger ω gets or the smaller τ gets. In addition, Figures 6 and 7 show that the average relative density of the infected nodes I(t) monotonically decreases to zero, whereas the relative density of infected nodes of connectivity k always breaks out first and then decreases to zero; the reason of the phenomenon appears that the spreading network is a scale-free one. Note that Figures 8 and 9 show that the delay τ and ω have much impact on the steady state of density of the infected nodes when R<sup>0</sup> > 1, the density of infected decreases as the delay τ and ω increase, which is consistent with the formula (37). We also know from (37) that I(t) !0 as ω ! + ∞ or τ ! + ∞.

Especially, system (29) reduces to the following SIRS model [26]:

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network http://dx.doi.org/10.5772/intechopen.71087 39

$$\begin{cases} \dot{\mathcal{S}}\_k(t) = -\lambda(k)\mathcal{S}\_k(t)\Theta(t) + \mu I\_k(t-\omega), \\ \dot{I}\_k(t) = \lambda(k)\mathcal{S}\_k(t)\Theta(t) - \mu I\_k(t), \\ \dot{R}\_k(t) = \mu I\_k(t) - \mu I\_k(t-\omega). \end{cases} \tag{52}$$

with the normalization condition

Θ ̇ ðÞ¼ t

1 h ik

38 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

¼ 1 h ik

Note that (1 � <sup>α</sup>)R<sup>0</sup> > 1, and it follows with 1ð Þ � <sup>α</sup> h i <sup>λ</sup>ð Þ<sup>k</sup> <sup>φ</sup>ð Þ<sup>k</sup>

≥ ð Þ 1 � α

Then, Ws

(E0) ∩ X<sup>0</sup> = ∅.

model (32) in which ϕ(k) = A in this paper.

3.3. Numerical simulations for the model

n = 100 and the minimum degree m = 1.

Consider system (32). Let φ(k = ak<sup>α</sup>

completes the proof.

network is p(k) =Ck�<sup>γ</sup>

k 6¼ 2, 3, 4, 5.

or τ ! + ∞.

Xn k¼m

Xn k¼m

<sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup> � �

constant <sup>ε</sup> such that limt! <sup>+</sup> <sup>∞</sup> inf Ik <sup>&</sup>gt; <sup>ε</sup> and, consequently, lim<sup>t</sup>!þ<sup>∞</sup>infI tð Þ <sup>&</sup>gt; <sup>P</sup><sup>n</sup>

, and C satisfies P<sup>n</sup>

Especially, system (29) reduces to the following SIRS model [26]:

/(1 + b<sup>α</sup>

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ\_

I <sup>k</sup>ð Þt

h i<sup>k</sup> <sup>Θ</sup> <sup>t</sup> � <sup>τ</sup>ÞÞ � μΘð Þ<sup>t</sup> �

limt! <sup>+</sup> <sup>∞</sup>Θ(t)= + ∞ according to Lemma 3.1 contradicts lim<sup>t</sup> ! <sup>+</sup> <sup>∞</sup>Θ(t) = 0 due to limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0.

Hence, the infection is uniformly persistent according to Lemma 3.2, i.e., there exists a positive

In addition, Liu and Zhang discussed a simple SEIRS model without delay in [25], and the basic productive number for the model in [25] is λA/γ, which is consistent with R<sup>0</sup> for the

Now, we present the results of numerical simulations. The degree distribution of the scale-free

γ = 2.5 and μ = 0.06. Figures 1–4 show that the dynamic behaviors of system (32) with the initial functions satisfy condition (33) in which Ik(s) = 0.45, k = 2, 3, 4, 5 for s ∈[�σ, 0] and Ik = 0,

Although R<sup>0</sup> is irrelative to τ and ω. Figures 6 and 7 show that both the delay τ and ω have certain influence on the relative density of the infected nodes when R<sup>0</sup> < 1, for example, the faster the relative density of infected nodes converges to zero, the larger ω gets or the smaller τ gets. In addition, Figures 6 and 7 show that the average relative density of the infected nodes I(t) monotonically decreases to zero, whereas the relative density of infected nodes of connectivity k always breaks out first and then decreases to zero; the reason of the phenomenon appears that the spreading network is a scale-free one. Note that Figures 8 and 9 show that the delay τ and ω have much impact on the steady state of density of the infected nodes when R<sup>0</sup> > 1, the density of infected decreases as the delay τ and ω increase, which is consistent with the formula (37). We also know from (37) that I(t) !0 as ω ! + ∞

<sup>ϕ</sup>ð Þ<sup>k</sup> p kð Þ <sup>λ</sup>ð Þ<sup>k</sup> Skð Þ <sup>t</sup> � <sup>τ</sup> <sup>Θ</sup>ð Þ� <sup>t</sup> � <sup>τ</sup> <sup>μ</sup>Ikð Þ<sup>t</sup> � �

(51)

<sup>k</sup>¼<sup>m</sup> p kð Þ<sup>ε</sup> <sup>¼</sup> <sup>ε</sup>. This

h i<sup>k</sup> <sup>&</sup>gt; <sup>μ</sup>. Hence, we obtain from (51) that

<sup>k</sup>¼<sup>1</sup> p kð Þ¼ 1. Here, we set still the maximum degree

) in which a = 0.5, α = 0.75, b = 0.02 and λ(k) = λk, and let

$$R\_k(t) + I\_k(t) + R\_k(t) = 1.\tag{53}$$

Figure 10 shows that when R<sup>0</sup> > 1, the quarantine delay ω can impact the density of infected nodes at the stationary state, and raising the quarantine period will suppress the viruses when ω is not large enough, which coincides with formula (37). Moreover, there exists periodic oscillation near the endemic equilibrium when ω is large enough. This is an interesting phenomenon which means that a bifurcation may appear.

Figure 6. (a) Evolutions of I(t) for system (32) with τ = 3, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for system (32) with τ = 3, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037.

Figure 7. (a) Evolutions of I(t) for system (32) with ω = 6, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for system (32) with ω = 6, λ(k) = 0.03k, and R<sup>0</sup> = 0.8037.

4. Conclusion and discussion

basic reproduction number is

will always exists.

rium E<sup>∗</sup> when R<sup>0</sup> > 1.

Acknowledgements

Author details

density of infected nodes at the stationary state.

i.e., stochastic differential equation models.

Qiming Liu\*, Meici Sun and Shihua Zhang

\*Address all correspondence to: lqmmath@163.com

Shijiazhuang Branch, Arm Engineering University, Shijiazhuang, China

An SEIR model with time delay on the scale-free network, which formulated a disease or computer virus transmission with constant latent period, is presented. For SEIR model, the

The Dynamics Analysis of Two Delayed Epidemic Spreading Models with Latent Period on Heterogeneous Network

When R<sup>0</sup> ≤ 1, there is no outbreak of the disease spreading, and the infection eventually disappears. When R<sup>0</sup> > 1, there exists at least one outbreak for the spreading of epidemic, and then limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0 due to global attractivity of M0. If the recovered nodes cycle back into the susceptive class after an immune period, we obtain a SEIRS model with two time delays on the scale-free network, which formulated a disease transmission with constant latent and immune periods. For SEIRS model, the basic reproduction number is still R<sup>0</sup> shown in (54). If R<sup>0</sup> ≤ 1, although the equilibrium E<sup>0</sup> is globally stable and the infection eventually disappears, the equilibrium E<sup>0</sup> may lose its stability when R<sup>0</sup> > 1 and the infection

Although R<sup>0</sup> is irrelevant to time delays, they influence the dynamical behaviors of the model such as slowing down the speed disappear of disease spreading on network, depressing the

In addition, for SEIRS model, numerical simulations show that the endemic equilibrium E<sup>∗</sup> may be globally asymptotically stable under some conditions when R<sup>0</sup> > 1 (shown in Figures 8 and 9). We would like to mention here that it is interesting but challenging to discuss the stability of equilib-

Furthermore, more and more researchers realize the fundamental role of the stochastic nature of diseases on their dynamics. In order to gain analytical insight into the behavior of the epidemic spreading, we also may extend the models (2) and (29) to the ones with random perturbations,

This research was supported by the Hebei Provincial Natural Science Foundation of China under Grant No. A2016506002. Qiming Liu would like to thank Professor Zhen Jin of Shanxi University in China for his helpful suggestions that led to truly significant improvements of this chapter.

<sup>λ</sup>ð Þ<sup>k</sup> <sup>ϕ</sup>ð Þ<sup>k</sup>

h i<sup>k</sup> : (54)

http://dx.doi.org/10.5772/intechopen.71087

41

<sup>R</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> μ

Figure 8. (a) Evolutions of I(t) for system (32) with τ = 3, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for system (32) with τ = 3, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505.

Figure 9. (a) Evolutions of I(t) for system (32) with ω = 6, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for system (32) with ω = 6, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505.

Figure 10. (a) Evolutions of I(t), for system (52) with ω = 3, 8, 14, 23, respectively, when R<sup>0</sup> = 1.3881 > 1. (b) Evolutions of I15, I45, I85, for system (1) with ω = 23, respectively, when R<sup>0</sup> = 1.3881 > 1.

### 4. Conclusion and discussion

An SEIR model with time delay on the scale-free network, which formulated a disease or computer virus transmission with constant latent period, is presented. For SEIR model, the basic reproduction number is

$$R\_0 = \frac{1}{\mu} \frac{\left<\lambda(k)\phi(k)\right>}{\langle k\rangle}.\tag{54}$$

When R<sup>0</sup> ≤ 1, there is no outbreak of the disease spreading, and the infection eventually disappears. When R<sup>0</sup> > 1, there exists at least one outbreak for the spreading of epidemic, and then limt! <sup>+</sup> <sup>∞</sup>Ik(t) = 0 due to global attractivity of M0. If the recovered nodes cycle back into the susceptive class after an immune period, we obtain a SEIRS model with two time delays on the scale-free network, which formulated a disease transmission with constant latent and immune periods. For SEIRS model, the basic reproduction number is still R<sup>0</sup> shown in (54). If R<sup>0</sup> ≤ 1, although the equilibrium E<sup>0</sup> is globally stable and the infection eventually disappears, the equilibrium E<sup>0</sup> may lose its stability when R<sup>0</sup> > 1 and the infection will always exists.

Although R<sup>0</sup> is irrelevant to time delays, they influence the dynamical behaviors of the model such as slowing down the speed disappear of disease spreading on network, depressing the density of infected nodes at the stationary state.

In addition, for SEIRS model, numerical simulations show that the endemic equilibrium E<sup>∗</sup> may be globally asymptotically stable under some conditions when R<sup>0</sup> > 1 (shown in Figures 8 and 9). We would like to mention here that it is interesting but challenging to discuss the stability of equilibrium E<sup>∗</sup> when R<sup>0</sup> > 1.

Furthermore, more and more researchers realize the fundamental role of the stochastic nature of diseases on their dynamics. In order to gain analytical insight into the behavior of the epidemic spreading, we also may extend the models (2) and (29) to the ones with random perturbations, i.e., stochastic differential equation models.

#### Acknowledgements

This research was supported by the Hebei Provincial Natural Science Foundation of China under Grant No. A2016506002. Qiming Liu would like to thank Professor Zhen Jin of Shanxi University in China for his helpful suggestions that led to truly significant improvements of this chapter.

#### Author details

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>0</sup>

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>0</sup>

40 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

ω=2 ω=6 ω=10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(a) (b)

Figure 8. (a) Evolutions of I(t) for system (32) with τ = 3, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for

I15/I85

time t

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>0</sup>

time t

system (32) with ω = 6, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505.

system (32) with τ = 3, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505.

τ=3 τ=8 τ=14 τ=23

τ=3 τ=6 τ=9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

I15/I85

(a) (b)

Figure 9. (a) Evolutions of I(t) for system (32) with ω = 6, λ(k) = 0.14k, and R<sup>0</sup> = 3.7505. (b) Evolutions of I<sup>15</sup> and I<sup>85</sup> for

0.02 0.04 0.06 0.08 0.1 0.12 0.14

(a) (b)

Figure 10. (a) Evolutions of I(t), for system (52) with ω = 3, 8, 14, 23, respectively, when R<sup>0</sup> = 1.3881 > 1. (b) Evolutions of I15,

<sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> <sup>500</sup> <sup>0</sup>

time t

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>0</sup>

I 15,ω=2 I 15,ω=6 I 15,ω=10 I 85,ω=2 I 85,ω=6 I 85,ω=10

time t

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>0</sup>

time t

I 15 I 45 I 85

I 15,τ=3 I 15,τ=6 I 15,τ=9 I 85,τ=3 I 85,τ=6 I 85,τ=9

time t

I45, I85, for system (1) with ω = 23, respectively, when R<sup>0</sup> = 1.3881 > 1.

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

I(t)

0.05

0.05

0.1

0.15

I

0.2

0.25

0.1

0.15

I

0.2

0.25

Qiming Liu\*, Meici Sun and Shihua Zhang

\*Address all correspondence to: lqmmath@163.com

Shijiazhuang Branch, Arm Engineering University, Shijiazhuang, China

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**Chapter 3**

Provisional chapter

**Stability and Hopf Bifurcation Analysis of a Simple**

DOI: 10.5772/intechopen.71624

Stability and Hopf Bifurcation Analysis of a Simple

A 3-dimensional nutrient-prey-predator model with intratrophic predation is proposed and studied. Some elementary properties such as invariance of nonnegativity, boundedness and dissipativity of the system are presented. The purpose of this chapter is to study the existence and stability of equilibria along with the effects of intratrophic predation towards the positions and stability of those equilibria of the system. We also investigate the occurrence of Hopf bifurcation. In the case when there is no presence of predator organisms, intratrophic predation may not give impact on the stability of equilibria of the system. We also analysed global stability of the equilibrium point. A suitable Lyapunov function is defined for global stability analysis and some results of persistence analysis are presented for the existence of positive interior equilibrium point. Besides that, Hopf

Keywords: chemostat, intratrophic predation, local and global stability, Hopf

Chemostat, a piece of laboratory apparatus is frequently used in mathematical ecology. This device carries an important role for ecological studies because the mathematics are tractable, and the relevant parameters are readily measurable. Chemostat also can be used for modelling microorganism systems such as lake and wastewater treatment. For detailed information

There are abundant of research journals and papers for analysing chemostat models. One of them is Li and Kuang [2], considered a simple food chain with distinct removal rate, which in this case,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

**Nutrient-Prey-Predator Model with Intratrophic**

Nutrient-Prey-Predator Model with Intratrophic

**Predation in Chemostat**

Predation in Chemostat

Zabidin Salleh and Liyana Abd Rahim

Zabidin Salleh and Liyana Abd Rahim

http://dx.doi.org/10.5772/intechopen.71624

Abstract

bifurcation

1. Introduction

regarding chemostat device, see [1].

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

bifurcation analysis of the system are demonstrated.

[29] Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston: Academic Press; 1993

Provisional chapter

#### **Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation in Chemostat** Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation in Chemostat

DOI: 10.5772/intechopen.71624

Zabidin Salleh and Liyana Abd Rahim

Additional information is available at the end of the chapter Zabidin Salleh and Liyana Abd Rahim

http://dx.doi.org/10.5772/intechopen.71624 Additional information is available at the end of the chapter

#### Abstract

[27] Hale J. Theory of Functional Differential Equations. New York: Springer-Verlag; 1977

ton: Academic Press; 1993

44 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

[28] Gao S, Chen L, Teng Z. Mint: Pulse vaccination of an SEIR epidemic model with time delay. Nonlinear Analysis: RWA. 2008;9:599-607. DOI: 10.1016/j.nonrwa.2006.12.004 [29] Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Bos-

> A 3-dimensional nutrient-prey-predator model with intratrophic predation is proposed and studied. Some elementary properties such as invariance of nonnegativity, boundedness and dissipativity of the system are presented. The purpose of this chapter is to study the existence and stability of equilibria along with the effects of intratrophic predation towards the positions and stability of those equilibria of the system. We also investigate the occurrence of Hopf bifurcation. In the case when there is no presence of predator organisms, intratrophic predation may not give impact on the stability of equilibria of the system. We also analysed global stability of the equilibrium point. A suitable Lyapunov function is defined for global stability analysis and some results of persistence analysis are presented for the existence of positive interior equilibrium point. Besides that, Hopf bifurcation analysis of the system are demonstrated.

Keywords: chemostat, intratrophic predation, local and global stability, Hopf bifurcation

1. Introduction

Chemostat, a piece of laboratory apparatus is frequently used in mathematical ecology. This device carries an important role for ecological studies because the mathematics are tractable, and the relevant parameters are readily measurable. Chemostat also can be used for modelling microorganism systems such as lake and wastewater treatment. For detailed information regarding chemostat device, see [1].

There are abundant of research journals and papers for analysing chemostat models. One of them is Li and Kuang [2], considered a simple food chain with distinct removal rate, which in this case,

> © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

conservation law has failed. Therefore, to overcome the problem, they constructed a Lyapunov function in global stability analysis of predator-free steady state. Local and global stability of other steady states were shown in the paper along with persistence analysis of the system.

as the bifurcation parameter. In the analysis of Hopf bifurcation in a delay model, they used a

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

Furthermore, a simple 3-dimensional food chain model in chemostat with variable yield for prey population and constant yield for predator population is proposed by Rana et al. [11]. In this model, the prey consumes the nutrient and the predator consumes the prey but the predator does not consume the nutrient. The functional response functions are assumed in Michaelis-Menton type. The stability of equilibrium points, the existence of limit cycles, the Hopf bifurcation and the positive invariant set for the system are discussed by qualitative analysis of differential equations. Finally, numerical simulations are carried out in support of

Our work is a modification of the models in Li and Kuang [2] and El-Sheikh and Mahrouf [3]. It should be emphasised that this work is different from [2, 3]. The modified model contains parameter b, known as the measure of intensity of intratrophic predation which is not in their

in the interest of studying the behaviour of the modified model. By this motivation, we analysed the stability and Hopf bifurcation of the model with intratrophic predation, as intratrophic predation analysis is rarely considered in mathematical models of populations biologically [6]. The purpose of this chapter is to study the existence and stability of equilibria along with the effects of intratrophic predation towards the positions and stability of those equilibria of the system. We shall also investigate the occurrence of Hopf bifurcation towards the system. Hence, we introduced a simple nutrient-prey-predator model in chemostat with intratrophic predation. Some notations regarding the model will be explained. Not only that, several elementary properties such as nonnegativity, boundedness and dissipativity along with definitions and several results will be presented. The local and global stability together with existence of the equilibrium points are shown. For global stability analysis, a suitable Lyapunov function is defined. Lastly, we applied Hopf bifurcation theorems (see [9]) in the

In this work, we consider a nutrient-prey-predator model with one prey organism and one predator organism in chemostat with intratrophic predation. Biologically, the predator organisms will feed upon the prey organisms, while the prey organisms will consume the nutrient in

f <sup>1</sup>ð Þs x,

β2

D1xy <sup>D</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> by � � � y,

f <sup>2</sup>ð Þx y þ

D1xy <sup>D</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> by � �,

(1)

β1

� �<sup>x</sup> � <sup>1</sup>

<sup>1</sup>þxþD1by � � are added to differential equations <sup>x</sup><sup>0</sup> and <sup>y</sup><sup>0</sup>

http://dx.doi.org/10.5772/intechopen.71624

47

computer code BIFDD to determine the stability of bifurcation solutions.

the theoretical results.

analysis of Hopf bifurcation.

the chemostat. Precisely, the modified model of [2, 3] is

ðÞ¼ <sup>t</sup> <sup>s</sup><sup>0</sup> � <sup>s</sup> � �D<sup>0</sup> � <sup>1</sup>

ðÞ¼ t f <sup>1</sup>ð Þ� s D<sup>1</sup>

ðÞ¼ t f <sup>2</sup>ð Þx y þ

s0

8 >>>>>>><

x0

y0

>>>>>>>:

2. The model

models. The parameter <sup>b</sup> and term 1 <sup>þ</sup> <sup>D</sup>1<sup>x</sup>

Another paper regarding chemostat model is by El-Sheikh and Mahrouf [3] that presented a 4-dimensional food chain in a chemostat with removal rates. They studied local and global stability of equilibria along with elementary properties including boundedness of solutions, invariance of nonnegativity, dissipativity and persistence analysis. Hopf bifurcation theory was applied.

Recently, several analysis on chemostat models are carried out, for example, by Hamra and Yadi [4] and Yang et al. [5]. In the work of [4], they studied a chemostat model with constant recycle sludge concentration. Number of parameters are reduced by considering a dimensionless model. Next, they successfully proved the existence of a positive uniform attractor for the model with different removal rates by using dissipative theory. Hence, they used methods of singular perturbation theory in order to investigate the asymptotic behaviour of the chemostat model under small pertubations. Thus, it is shown that in the case of two species in competition, the positive unique equilibrium point is globally asymptotically stable.

In the research of Yang et al. [5], piecewise chemostat models which involve control strategy with threshold window are proposed and analysed. They investigated the qualitative analysis such as existence and stability of equilibrium points of the system and it is proved that the regular equilibria and pseudo-equilibrium cannot coexist. The global stability analysis of both the regular equilibria and pseudo-equilibrium have been studied using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, the bifurcation analysis of the system is investigated with theoretical and numerical techniques.

Moreover, one of the interesting topic is the research involving intratrophic predation. Intratrophic predation is a situation where members of a trophic group consume other members of the same trophic group (for the purpose of mathematical modelling). Several previous studies based on intratrophic predation have been discovered. One of them is by Pitchford and Brindley [6]. They studied a predator-prey model with intratrophic predation and successfully shown that the model had desirable and plausible features. Furthermore, to investigate the effect of intratrophic predation towards the position and stability of equilibrium points of a model, they had developed a simple asymptotic method.

For Hopf bifurcation analysis, a study by Mada Sanjaya et al. [7] has been carried out. They introduced an ecological model of three species food chain with Holling Type-III functional response. Two equilibrium points are obtained. They proved that the system has periodic solutions around those equilibrium points. Not only that, they also investigated the dynamical behaviours of the system and found that it was sensitive when the parameter values varied.

Tee and Salleh [8] investigated Hopf bifurcation of a nonlinear modified Lorenz system using normal form theory that was the same technique used in Hassard et al. [9]. Then, the dynamics on centre manifold of the system was presented as it will be applied in the technique of normal form theory. Another research based on Hopf bifurcation analysis is carried out by [10]. They considered a three-species food chain models with group defence. It is proved that the model without delay undergone Hopf bifurcation by using the carrying capacity of the environment as the bifurcation parameter. In the analysis of Hopf bifurcation in a delay model, they used a computer code BIFDD to determine the stability of bifurcation solutions.

Furthermore, a simple 3-dimensional food chain model in chemostat with variable yield for prey population and constant yield for predator population is proposed by Rana et al. [11]. In this model, the prey consumes the nutrient and the predator consumes the prey but the predator does not consume the nutrient. The functional response functions are assumed in Michaelis-Menton type. The stability of equilibrium points, the existence of limit cycles, the Hopf bifurcation and the positive invariant set for the system are discussed by qualitative analysis of differential equations. Finally, numerical simulations are carried out in support of the theoretical results.

Our work is a modification of the models in Li and Kuang [2] and El-Sheikh and Mahrouf [3]. It should be emphasised that this work is different from [2, 3]. The modified model contains parameter b, known as the measure of intensity of intratrophic predation which is not in their models. The parameter <sup>b</sup> and term 1 <sup>þ</sup> <sup>D</sup>1<sup>x</sup> <sup>1</sup>þxþD1by � � are added to differential equations <sup>x</sup><sup>0</sup> and <sup>y</sup><sup>0</sup> in the interest of studying the behaviour of the modified model. By this motivation, we analysed the stability and Hopf bifurcation of the model with intratrophic predation, as intratrophic predation analysis is rarely considered in mathematical models of populations biologically [6].

The purpose of this chapter is to study the existence and stability of equilibria along with the effects of intratrophic predation towards the positions and stability of those equilibria of the system. We shall also investigate the occurrence of Hopf bifurcation towards the system. Hence, we introduced a simple nutrient-prey-predator model in chemostat with intratrophic predation. Some notations regarding the model will be explained. Not only that, several elementary properties such as nonnegativity, boundedness and dissipativity along with definitions and several results will be presented. The local and global stability together with existence of the equilibrium points are shown. For global stability analysis, a suitable Lyapunov function is defined. Lastly, we applied Hopf bifurcation theorems (see [9]) in the analysis of Hopf bifurcation.

#### 2. The model

conservation law has failed. Therefore, to overcome the problem, they constructed a Lyapunov function in global stability analysis of predator-free steady state. Local and global stability of other steady states were shown in the paper along with persistence analysis of the system.

46 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Another paper regarding chemostat model is by El-Sheikh and Mahrouf [3] that presented a 4-dimensional food chain in a chemostat with removal rates. They studied local and global stability of equilibria along with elementary properties including boundedness of solutions, invariance of nonnegativity, dissipativity and persistence analysis. Hopf bifurcation theory

Recently, several analysis on chemostat models are carried out, for example, by Hamra and Yadi [4] and Yang et al. [5]. In the work of [4], they studied a chemostat model with constant recycle sludge concentration. Number of parameters are reduced by considering a dimensionless model. Next, they successfully proved the existence of a positive uniform attractor for the model with different removal rates by using dissipative theory. Hence, they used methods of singular perturbation theory in order to investigate the asymptotic behaviour of the chemostat model under small pertubations. Thus, it is shown that in the case of two species in competi-

In the research of Yang et al. [5], piecewise chemostat models which involve control strategy with threshold window are proposed and analysed. They investigated the qualitative analysis such as existence and stability of equilibrium points of the system and it is proved that the regular equilibria and pseudo-equilibrium cannot coexist. The global stability analysis of both the regular equilibria and pseudo-equilibrium have been studied using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, the bifurcation analysis of

Moreover, one of the interesting topic is the research involving intratrophic predation. Intratrophic predation is a situation where members of a trophic group consume other members of the same trophic group (for the purpose of mathematical modelling). Several previous studies based on intratrophic predation have been discovered. One of them is by Pitchford and Brindley [6]. They studied a predator-prey model with intratrophic predation and successfully shown that the model had desirable and plausible features. Furthermore, to investigate the effect of intratrophic predation towards the position and stability of equilibrium points of a

For Hopf bifurcation analysis, a study by Mada Sanjaya et al. [7] has been carried out. They introduced an ecological model of three species food chain with Holling Type-III functional response. Two equilibrium points are obtained. They proved that the system has periodic solutions around those equilibrium points. Not only that, they also investigated the dynamical behaviours of the system and found that it was sensitive when the parameter values varied. Tee and Salleh [8] investigated Hopf bifurcation of a nonlinear modified Lorenz system using normal form theory that was the same technique used in Hassard et al. [9]. Then, the dynamics on centre manifold of the system was presented as it will be applied in the technique of normal form theory. Another research based on Hopf bifurcation analysis is carried out by [10]. They considered a three-species food chain models with group defence. It is proved that the model without delay undergone Hopf bifurcation by using the carrying capacity of the environment

tion, the positive unique equilibrium point is globally asymptotically stable.

the system is investigated with theoretical and numerical techniques.

model, they had developed a simple asymptotic method.

was applied.

In this work, we consider a nutrient-prey-predator model with one prey organism and one predator organism in chemostat with intratrophic predation. Biologically, the predator organisms will feed upon the prey organisms, while the prey organisms will consume the nutrient in the chemostat. Precisely, the modified model of [2, 3] is

$$\begin{cases} s'(t) = (s^0 - s)D\_0 - \frac{1}{\beta\_1} f\_1(s) \mathbf{x}, \\ \mathbf{x}'(t) = \left( f\_1(s) - D\_1 \right) \mathbf{x} - \frac{1}{\beta\_2} f\_2(\mathbf{x}) \left( y + \frac{D\_1 \mathbf{x} y}{D\_2 + \mathbf{x} + by} \right), \\ y'(t) = f\_2(\mathbf{x}) \left( y + \frac{D\_1 \mathbf{x} y}{D\_2 + \mathbf{x} + by} \right) - y, \end{cases} \tag{1}$$

where sð Þ¼ 0 s<sup>0</sup> > 0, xð Þ¼ 0 x<sup>0</sup> > 0, yð Þ¼ 0 y<sup>0</sup> > 0, Di ¼ 1; i ¼ 0, 1, 2, β<sup>j</sup> ≤ 1; j ¼ 1, 2 and 0 ≤ b ≤ 1: This leads to the following assumptions on functional response f <sup>n</sup>, n ¼ 1, 2:

$$\begin{aligned} \text{(i) } f\_n: &\mathbb{R}^+ \to \mathbb{R}^+ \text{ and } f\_{n'}n = 1, 2 \text{ are continuously differentiable equations,} \\ \text{(ii) } f\_n(0) &= 0, \\ \text{(iii) } f\_1'(\mathbf{s}) &> 0, f\_2'(\mathbf{x}) > 0 \text{ for all } \mathbf{s}, \mathbf{x} > 0. \end{aligned} \tag{2}$$

2.1. Elementary properties of system (3)

there exists a bounded subset Γ of R<sup>3</sup>

Theorem 1. Let H be the region

where Pmax <sup>¼</sup> max 1; <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

i. Η is positively invariant.

iii. The system is dissipative.

(i): First, we must show that <sup>1</sup>

Proof of Theorem 1.

Hence, this leads to

eventually attracted into region Η:

that the system becomes biologically meaningful.

Pmax <sup>¼</sup> max 1; <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

<sup>1</sup> � Pmaxð Þ <sup>s</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>y</sup> <sup>≤</sup> <sup>1</sup> � <sup>s</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

By adding those differential equations s<sup>0</sup>

s

Thus, solving the above inequality gives

Then,

on x<sup>0</sup> and Γ so that the solution of the system ϕ t; x<sup>0</sup> � � ∈Γ for t ≥ t0:

<sup>H</sup> <sup>¼</sup> ð Þ <sup>s</sup>; <sup>x</sup>; <sup>y</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup>

<sup>D</sup><sup>1</sup> � 1

ii. All nonnegative solutions of (3) with initial values in R<sup>3</sup>

respect to a region in R<sup>3</sup>

In this section, we present nonnegativity, boundedness and dissipativity of the system (3) with

Definition 1 [13]. A system with differential equations x<sup>0</sup> ¼ f xð Þ is defined to be dissipative if

<sup>þ</sup> : <sup>1</sup> Pmax

n o, Pmin <sup>¼</sup> min 1; <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

Pmax � <sup>q</sup> <sup>≤</sup> <sup>s</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>y</sup> <sup>≤</sup> <sup>1</sup>

<sup>0</sup> <sup>þ</sup> <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> � <sup>s</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

D<sup>1</sup> � 1

<sup>þ</sup>. Firstly, we consider dissipativity of the system (3).

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

� q ≤ s þ x þ y ≤

� �,

Remark 1. We must show that the solutions of system (3) are nonnegative and bounded, so

� � and Pmin <sup>¼</sup> min 1; <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

D<sup>1</sup> � 1 � � � � <sup>x</sup> <sup>þ</sup> <sup>y</sup>

, x<sup>0</sup> and y<sup>0</sup> in (3), we shall get

D<sup>1</sup> � 1 � � � � <sup>x</sup> <sup>þ</sup> <sup>y</sup> � �:

� � <sup>≤</sup> <sup>1</sup> � Pminð Þ <sup>s</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>y</sup> :

, such that there is a time t<sup>0</sup> for any x<sup>0</sup> ∈ R<sup>3</sup> which depends

1 Pmin þ q

<sup>D</sup><sup>1</sup> � 1

Pmin þ q: Let us define

n o and q is a positive constant.

http://dx.doi.org/10.5772/intechopen.71624

49

<sup>þ</sup> are uniformly bounded and they

D<sup>1</sup> � 1

� �:

Generally, functional response is a common component in predator-prey system. The term 'functional response' was first stated by Solomon [12] which defined the relationship between the rate of predation, i.e., the number of prey organisms consumed per predator organism in time, t with the density of prey organisms.

By referring to system (1), s represents the concentration of nutrient at time t while s<sup>0</sup> is the input nutrient concentration. The variables x and y represent the concentration of prey and predator at time t, respectively. Parameters β<sup>1</sup> and β<sup>2</sup> denote the yield constants, D<sup>0</sup> is the washout rate of the chemostat while D<sup>1</sup> and D<sup>2</sup> are the removal rates of x and y, respectively. Removal rate is the sum of washout rate and death rate θi, i.e., Di ¼ D<sup>0</sup> þ θi, i ¼ 1, 2: f <sup>1</sup>ð Þs and f <sup>2</sup>ð Þx denote the specific growth rate of prey x and predator y, respectively, while b is the measure of intensity of intratrophic predation in predator organisms y: These observations are based on numerical simulations. We can rescale system (1) by reducing the number of parameters using standard change of variables such as.

$$\begin{aligned} \mathfrak{T} &= \frac{s}{s^0}, & \mathfrak{T} &= \frac{\chi}{D\_0 \beta\_1 s^0}, & \tilde{y} &= \frac{y}{D\_0 D\_1 \beta\_1 \beta\_2 s^0}, \\\ f\_1(s) &= \frac{\bar{f}\_1 \left(s^0 \mathfrak{T}\right)}{D\_0}, & f\_2(x) &= \frac{\bar{f}\_2 \left(D\_0 \beta\_1 s^0 \mathfrak{T}\right)}{D\_0}, & b &= \frac{\tilde{b}}{\beta\_2}, \\\ \tilde{t} &= D\_t t, i = 0, 1, & D\_2 &= D\_0 \beta\_1 s^0. \end{aligned}$$

To make our works more convenient, we just drop the bars and tildes. So after rescaling, the system (1) becomes

$$\begin{cases} \mathbf{s}' = \mathbf{1} - \mathbf{s} - f\_1(\mathbf{s})\mathbf{x}, \\ \mathbf{x}' = \left(\frac{f\_1(\mathbf{s})}{D\_1} - \mathbf{1}\right)\mathbf{x} - f\_2(\mathbf{x})y\left(\mathbf{1} + \frac{D\_1\mathbf{x}}{\mathbf{1} + \mathbf{x} + D\_1\mathbf{b}y}\right), \\ \mathbf{y}' = f\_2(\mathbf{x})y\left(\mathbf{1} + \frac{D\_1\mathbf{x}}{\mathbf{1} + \mathbf{x} + D\_1\mathbf{b}y}\right) - y, \end{cases} \tag{3}$$

where

$$\text{Is } \mathbf{x}(0) > 0, \mathbf{x}(0) > 0, y(0) > 0 \text{ and } D\_1 > 0. \tag{4}$$

Now, the variables are non-dimensional and the discussion is in R<sup>3</sup> <sup>þ</sup> <sup>¼</sup> fð Þ <sup>s</sup>; <sup>x</sup>; <sup>y</sup> : <sup>s</sup> <sup>&</sup>gt; <sup>0</sup>; x > 0; y > 0g.

#### 2.1. Elementary properties of system (3)

In this section, we present nonnegativity, boundedness and dissipativity of the system (3) with respect to a region in R<sup>3</sup> <sup>þ</sup>. Firstly, we consider dissipativity of the system (3).

Definition 1 [13]. A system with differential equations x<sup>0</sup> ¼ f xð Þ is defined to be dissipative if there exists a bounded subset Γ of R<sup>3</sup> , such that there is a time t<sup>0</sup> for any x<sup>0</sup> ∈ R<sup>3</sup> which depends on x<sup>0</sup> and Γ so that the solution of the system ϕ t; x<sup>0</sup> � � ∈Γ for t ≥ t0:

Theorem 1. Let H be the region

$$H = \left\{ (\mathbf{s}, \mathbf{x}, \mathbf{y}) \in \mathbb{R}\_+^3 : \frac{1}{P\_{\max}} - q \le \mathbf{s} + \mathbf{x} + \mathbf{y} \le \frac{1}{P\_{\min}} + q \right\},$$

where Pmax <sup>¼</sup> max 1; <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>D</sup><sup>1</sup> � 1 n o, Pmin <sup>¼</sup> min 1; <sup>f</sup> <sup>1</sup>ð Þþ <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>D</sup><sup>1</sup> � 1 n o and q is a positive constant.

Then,

(2)

(3)

<sup>þ</sup> <sup>¼</sup> fð Þ <sup>s</sup>; <sup>x</sup>; <sup>y</sup> : <sup>s</sup> <sup>&</sup>gt; <sup>0</sup>;

where sð Þ¼ 0 s<sup>0</sup> > 0, xð Þ¼ 0 x<sup>0</sup> > 0, yð Þ¼ 0 y<sup>0</sup> > 0, Di ¼ 1; i ¼ 0, 1, 2, β<sup>j</sup> ≤ 1; j ¼ 1, 2 and

ð Þi f <sup>n</sup> : R<sup>þ</sup> ! R<sup>þ</sup> and f <sup>n</sup>, n ¼ 1, 2 are continuously differential equations,

Generally, functional response is a common component in predator-prey system. The term 'functional response' was first stated by Solomon [12] which defined the relationship between the rate of predation, i.e., the number of prey organisms consumed per predator organism in

By referring to system (1), s represents the concentration of nutrient at time t while s<sup>0</sup> is the input nutrient concentration. The variables x and y represent the concentration of prey and predator at time t, respectively. Parameters β<sup>1</sup> and β<sup>2</sup> denote the yield constants, D<sup>0</sup> is the washout rate of the chemostat while D<sup>1</sup> and D<sup>2</sup> are the removal rates of x and y, respectively. Removal rate is the sum of washout rate and death rate θi, i.e., Di ¼ D<sup>0</sup> þ θi, i ¼ 1, 2: f <sup>1</sup>ð Þs and f <sup>2</sup>ð Þx denote the specific growth rate of prey x and predator y, respectively, while b is the measure of intensity of intratrophic predation in predator organisms y: These observations are based on numerical simulations. We can rescale system (1) by reducing the number of

To make our works more convenient, we just drop the bars and tildes. So after rescaling, the

0:

<sup>~</sup><sup>f</sup> <sup>2</sup> <sup>D</sup>0β1s<sup>0</sup><sup>x</sup> � � D<sup>0</sup>

s<sup>0</sup> ¼ 1 � s � f <sup>1</sup>ð Þs x,

x � f <sup>2</sup>ð Þx y 1 þ

D1x 1 þ x þ D1by � �

D1x 1 þ x þ D1by � �

<sup>D</sup>0β1s<sup>0</sup> , <sup>y</sup> <sup>¼</sup> <sup>y</sup>

, <sup>b</sup> <sup>¼</sup> <sup>b</sup>

� y,

sð Þ0 > 0, xð Þ0 > 0, yð Þ0 > 0 and D<sup>1</sup> > 0: (4)

,

<sup>D</sup>0D1β1β2s<sup>0</sup> ,

β2 ,

0 ≤ b ≤ 1: This leads to the following assumptions on functional response f <sup>n</sup>, n ¼ 1, 2:

<sup>2</sup>ð Þx > 0 for all s, x > 0:

48 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

ð Þii f <sup>n</sup>ð Þ¼ 0 0,

<sup>1</sup>ð Þs > 0, f <sup>0</sup>

time, t with the density of prey organisms.

parameters using standard change of variables such as.

<sup>s</sup><sup>0</sup> , <sup>x</sup> <sup>¼</sup> <sup>x</sup>

t ¼ Dit, i ¼ 0, 1, D<sup>2</sup> ¼ D0β1s

, f <sup>2</sup>ð Þ¼ x

<sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> D<sup>1</sup> � 1 � �

8 >>>>><

>>>>>:

y<sup>0</sup> ¼ f <sup>2</sup>ð Þx y 1 þ

Now, the variables are non-dimensional and the discussion is in R<sup>3</sup>

ð Þ iii f 0

system (1) becomes

<sup>s</sup> <sup>¼</sup> <sup>s</sup>

f <sup>1</sup>ð Þ¼ s

<sup>~</sup><sup>f</sup> <sup>1</sup> <sup>s</sup><sup>0</sup><sup>s</sup> � � D<sup>0</sup>

where

x > 0; y > 0g.


Remark 1. We must show that the solutions of system (3) are nonnegative and bounded, so that the system becomes biologically meaningful.

#### Proof of Theorem 1.

(i): First, we must show that <sup>1</sup> Pmax � <sup>q</sup> <sup>≤</sup> <sup>s</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>y</sup> <sup>≤</sup> <sup>1</sup> Pmin þ q: Let us define

$$P\_{\max} = \max\left\{1, f\_1(s) + \frac{f\_1(s)}{D\_1} - 1\right\} \text{ and } P\_{\min} = \min\left\{1, f\_1(s) + \frac{f\_1(s)}{D\_1} - 1\right\}.$$

By adding those differential equations s<sup>0</sup> , x<sup>0</sup> and y<sup>0</sup> in (3), we shall get

$$s' + \mathbf{x'} + y' = 1 - \left[\mathbf{s} + \left(f\_1(\mathbf{s}) + \left(\frac{f\_1(\mathbf{s})}{D\_1} - 1\right)\right)\mathbf{x} + y\right].$$

Hence, this leads to

$$1 - P\_{\max}(\mathbf{s} + \mathbf{x} + \mathbf{y}) \le 1 - \left[ \mathbf{s} + \left( f\_1(\mathbf{s}) + \left( \frac{f\_1(\mathbf{s})}{D\_1} - 1 \right) \right) \mathbf{x} + \mathbf{y} \right] \le 1 - P\_{\min}(\mathbf{s} + \mathbf{x} + \mathbf{y}).$$

Thus, solving the above inequality gives

$$\frac{1}{P\_{\max}} - q \le s + x + y \le \frac{1}{P\_{\min}} + q\_f$$

equation <sup>y</sup><sup>0</sup> <sup>¼</sup> 0. Since <sup>y</sup> 6¼ <sup>0</sup>, f <sup>2</sup>ð Þ<sup>x</sup> <sup>1</sup> <sup>þ</sup> <sup>D</sup>1<sup>x</sup>

Lemma 1. Suppose that a point <sup>ζ</sup><sup>s</sup> ð Þ ; <sup>b</sup><sup>x</sup> exists in <sup>R</sup><sup>þ</sup>

approaches ∞: Then, the non-trivial equilibrium point E<sup>2</sup> ζs;

Proof. We will get two surfaces by equating system (5) to zero;

1 � s

satisfying <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup>b<sup>x</sup> � <sup>1</sup> <sup>¼</sup> 0. Hence, the equilibrium point <sup>E</sup><sup>2</sup> <sup>ζ</sup>s;

and theorem from Ref. [14] are stated to make us understand clearly.

2.3. Stability analysis of equilibrium points E1, E<sup>2</sup> and E<sup>3</sup>

� �

bD<sup>1</sup> <sup>1</sup>�<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ð Þ . Thus, the equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ is <sup>s</sup><sup>∗</sup>; <sup>1</sup>�s<sup>∗</sup>

To show the existence of E2, we let the system (3) be restricted to R<sup>þ</sup>

8 ><

>:

s<sup>0</sup> ¼ 1 � s � f <sup>1</sup>ð Þs x,

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

where sð Þ0 > 0 and xð Þ0 > 0. Thus, Lemma 1 below shows the existence of non-trivial equilib-

1 � s � f <sup>1</sup>ð Þs x ¼ 0,

<sup>x</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> and <sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> :

In this section, we analyse the stability of equilibrium points as it plays an important part in ordinary differential equations with their applications. As we cannot easily identify the positions of equilibrium point in applications of dynamical system, but only approximately, so the equilibrium point must be in stable state to be biologically meaningful [15]. Several definitions

Definition 2. [14] A fixed point x<sup>∗</sup> is Lyapunov stable or L-stable, provided that any solution <sup>ϕ</sup>ð Þ <sup>t</sup>; <sup>x</sup><sup>0</sup> stays near <sup>x</sup><sup>∗</sup> for all future time <sup>t</sup> <sup>≥</sup> 0 if the initial condition <sup>x</sup><sup>0</sup> starts near enough to <sup>x</sup><sup>∗</sup>: Specifically, a fixed point x<sup>∗</sup> is L-stable, provided that for any E > 0, there is δ > 0, such that if

� < E for all t ≥ 0.

f <sup>1</sup>ð Þs D<sup>1</sup> � 1 � �<sup>x</sup> <sup>¼</sup> <sup>0</sup>:

<sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> D<sup>1</sup> � 1 � �x,

<sup>2</sup><sup>x</sup> ð Þ <sup>∗</sup>þ<sup>1</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ�x∗�<sup>1</sup>

where <sup>x</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup>�s<sup>∗</sup>

rium point E2:

Then,

Thus, 1 � <sup>s</sup> <sup>¼</sup> <sup>D</sup>1x, i.e., <sup>x</sup> <sup>¼</sup> <sup>1</sup>�<sup>s</sup>

<sup>x</sup><sup>0</sup> � <sup>x</sup><sup>∗</sup> k k <sup>&</sup>lt; <sup>δ</sup>, then <sup>ϕ</sup>ð Þ� <sup>t</sup>; <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup> �

<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ.

<sup>1</sup>þxþD1by � � <sup>¼</sup> 1. We solve for y, and get <sup>y</sup><sup>∗</sup> <sup>¼</sup>

<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ ; <sup>2</sup><sup>x</sup> ð Þ <sup>∗</sup>þ<sup>1</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ�x∗�<sup>1</sup> bD<sup>1</sup> <sup>1</sup>�<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ð Þ � �,

(5)

51

<sup>D</sup><sup>1</sup> and

sx as

http://dx.doi.org/10.5772/intechopen.71624

sx such that <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup>b<sup>x</sup> � <sup>1</sup> <sup>¼</sup> 0 as time, <sup>t</sup>

1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � exists.

<sup>D</sup><sup>1</sup> . By taking <sup>s</sup> <sup>¼</sup> <sup>ζ</sup>s, we will have <sup>x</sup> <sup>¼</sup> <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>�ζ<sup>s</sup>

1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

� � exists. <sup>∎</sup>

as q is the positive constant

(ii) and (iii): Let 0 < s<sup>0</sup> < 1 and consider.

$$\mathbf{s}' = \mathbf{1} - \mathbf{s} - f\_1(\mathbf{s})\mathbf{x} < \mathbf{1} - \mathbf{s}.$$

Then s tð Þ <sup>&</sup>lt; <sup>1</sup> � <sup>e</sup>�<sup>t</sup> <sup>1</sup>�s<sup>0</sup> for 0 <sup>&</sup>lt; <sup>s</sup><sup>0</sup> <sup>&</sup>lt; <sup>1</sup>, and hence lim<sup>t</sup>!<sup>∞</sup> sups tð Þ <sup>&</sup>lt; 1 for 0 <sup>&</sup>lt; <sup>s</sup><sup>0</sup> <sup>&</sup>lt; <sup>1</sup>:

Now, consider the x<sup>0</sup> equation;

$$\mathbf{x}' = \left(\frac{f\_1(\mathbf{s})}{D\_1} - 1\right)\mathbf{x} - f\_2(\mathbf{x})y\left(1 + \frac{D\_1\mathbf{x}}{1 + \mathbf{x} + D\_1by}\right) < \left(f\_1(\mathbf{s}) - 1\right)\mathbf{x} < a\_1\mathbf{x}$$

where <sup>α</sup><sup>1</sup> <sup>¼</sup> max<sup>s</sup> <sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>s</sup> <sup>1</sup> : We assume that <sup>α</sup><sup>1</sup> <sup>¼</sup> max<sup>s</sup><sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>s</sup> <sup>1</sup> <sup>&</sup>lt; <sup>0</sup>: Hence, x tð Þ <sup>&</sup>lt; <sup>x</sup>0e<sup>α</sup>1<sup>t</sup> , α<sup>1</sup> < 0, and thus, lim<sup>t</sup>!<sup>∞</sup> supx tð Þ < x0, for x<sup>0</sup> > 0:

Similarly, consider the y<sup>0</sup> equation,

$$y' = f\_2(\mathbf{x}) y \left( 1 + \frac{D\_1 \mathbf{x}}{1 + \mathbf{x} + D\_1 b y} \right) - y < a\_2 y\_{\text{-}x}$$

where <sup>α</sup><sup>2</sup> <sup>¼</sup> max<sup>x</sup><sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>2</sup>ð Þþ <sup>x</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> <sup>D</sup>1<sup>x</sup> <sup>1</sup>þxþD1by : We shall assume that max<sup>x</sup><sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>2</sup>ð Þþ <sup>x</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> <sup>D</sup>1<sup>x</sup> <sup>1</sup>þxþD1by <sup>&</sup>lt; <sup>0</sup>: Then, y tð Þ <sup>&</sup>lt; <sup>y</sup>0e<sup>α</sup>2<sup>t</sup> where <sup>α</sup><sup>2</sup> <sup>&</sup>lt; <sup>0</sup>: Hence, lim<sup>t</sup>!<sup>∞</sup> sup y tð Þ <sup>&</sup>lt; <sup>y</sup><sup>0</sup> for <sup>y</sup><sup>0</sup> <sup>&</sup>gt; <sup>0</sup>: Thus, system (3) is proved to be uniformly bounded and dissipative, following Definition 1. ∎

#### 2.2. Existence of equilibrium points

According to Robinson [14], a point is termed 'equilibrium point' because of the forces are in equilibrium and the mass did not move. Therefore, in order to find equilibrium points of the system (3); E1, E<sup>2</sup> and E3, we equate the differential equations s<sup>0</sup> , x<sup>0</sup> and y<sup>0</sup> to zero and solve the resulting equations simultaneously. The possible equilibrium points are as follows;


Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation… http://dx.doi.org/10.5772/intechopen.71624 51

equation <sup>y</sup><sup>0</sup> <sup>¼</sup> 0. Since <sup>y</sup> 6¼ <sup>0</sup>, f <sup>2</sup>ð Þ<sup>x</sup> <sup>1</sup> <sup>þ</sup> <sup>D</sup>1<sup>x</sup> <sup>1</sup>þxþD1by � � <sup>¼</sup> 1. We solve for y, and get <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>2</sup><sup>x</sup> ð Þ <sup>∗</sup>þ<sup>1</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ�x∗�<sup>1</sup> bD<sup>1</sup> <sup>1</sup>�<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ð Þ . Thus, the equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ is <sup>s</sup><sup>∗</sup>; <sup>1</sup>�s<sup>∗</sup> <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ ; <sup>2</sup><sup>x</sup> ð Þ <sup>∗</sup>þ<sup>1</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ�x∗�<sup>1</sup> bD<sup>1</sup> <sup>1</sup>�<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ð Þ � �, where <sup>x</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup>�s<sup>∗</sup> <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ.

To show the existence of E2, we let the system (3) be restricted to R<sup>þ</sup> sx as

$$\begin{cases} \mathbf{s}' = \mathbf{1} - \mathbf{s} - f\_1(\mathbf{s})\mathbf{x}, \\\\ \mathbf{x}' = \left(\frac{f\_1(\mathbf{s})}{D\_1} - \mathbf{1}\right)\mathbf{x}, \end{cases} \tag{5}$$

where sð Þ0 > 0 and xð Þ0 > 0. Thus, Lemma 1 below shows the existence of non-trivial equilibrium point E2:

Lemma 1. Suppose that a point <sup>ζ</sup><sup>s</sup> ð Þ ; <sup>b</sup><sup>x</sup> exists in <sup>R</sup><sup>þ</sup> sx such that <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup>b<sup>x</sup> � <sup>1</sup> <sup>¼</sup> 0 as time, <sup>t</sup> approaches ∞: Then, the non-trivial equilibrium point E<sup>2</sup> ζs; 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � exists.

Proof. We will get two surfaces by equating system (5) to zero;

$$\begin{aligned} 1 - s - f\_1(s)x &= 0, \\\\ \left(\frac{f\_1(s)}{D\_1} - 1\right)x &= 0. \end{aligned}$$

Then,

1 Pmax

50 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

s

x � f <sup>2</sup>ð Þx y 1 þ

, α<sup>1</sup> < 0, and thus, lim<sup>t</sup>!<sup>∞</sup> supx tð Þ < x0, for x<sup>0</sup> > 0:

y<sup>0</sup> ¼ f <sup>2</sup>ð Þx y 1 þ

1þxþD1by 

system (3); E1, E<sup>2</sup> and E3, we equate the differential equations s<sup>0</sup>

as q is the positive constant

Now, consider the x<sup>0</sup> equation;

<sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> D<sup>1</sup> � 1 

Similarly, consider the y<sup>0</sup> equation,

where <sup>α</sup><sup>2</sup> <sup>¼</sup> max<sup>x</sup><sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>2</sup>ð Þþ <sup>x</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> <sup>D</sup>1<sup>x</sup>

2.2. Existence of equilibrium points

x ¼ 0 when y ¼ 0:

<sup>D</sup><sup>1</sup> ; 0 

<sup>1</sup> � <sup>s</sup> � <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>x</sup> <sup>¼</sup> 0 and <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

and <sup>s</sup> <sup>¼</sup> <sup>ζ</sup>s, we get <sup>x</sup> <sup>¼</sup> <sup>1</sup>�ζ<sup>s</sup>

ii. E<sup>2</sup> ζs; <sup>1</sup>�ζ<sup>s</sup>

Then s tð Þ <sup>&</sup>lt; <sup>1</sup> � <sup>e</sup>�<sup>t</sup>

x tð Þ <sup>&</sup>lt; <sup>x</sup>0e<sup>α</sup>1<sup>t</sup>

(ii) and (iii): Let 0 < s<sup>0</sup> < 1 and consider.

� q ≤ s þ x þ y ≤

<sup>0</sup> ¼ 1 � s � f <sup>1</sup>ð Þs x < 1 � s:

<sup>1</sup>�s<sup>0</sup> for 0 <sup>&</sup>lt; <sup>s</sup><sup>0</sup> <sup>&</sup>lt; <sup>1</sup>, and hence lim<sup>t</sup>!<sup>∞</sup> sups tð Þ <sup>&</sup>lt; 1 for 0 <sup>&</sup>lt; <sup>s</sup><sup>0</sup> <sup>&</sup>lt; <sup>1</sup>:

where <sup>α</sup><sup>1</sup> <sup>¼</sup> max<sup>s</sup> <sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>s</sup> <sup>1</sup> : We assume that <sup>α</sup><sup>1</sup> <sup>¼</sup> max<sup>s</sup><sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>s</sup> <sup>1</sup> <sup>&</sup>lt; <sup>0</sup>: Hence,

D1x 1 þ x þ D1by 

Then, y tð Þ <sup>&</sup>lt; <sup>y</sup>0e<sup>α</sup>2<sup>t</sup> where <sup>α</sup><sup>2</sup> <sup>&</sup>lt; <sup>0</sup>: Hence, lim<sup>t</sup>!<sup>∞</sup> sup y tð Þ <sup>&</sup>lt; <sup>y</sup><sup>0</sup> for <sup>y</sup><sup>0</sup> <sup>&</sup>gt; <sup>0</sup>: Thus, system (3) is proved to be uniformly bounded and dissipative, following Definition 1. ∎

According to Robinson [14], a point is termed 'equilibrium point' because of the forces are in equilibrium and the mass did not move. Therefore, in order to find equilibrium points of the

i. E1ð Þ 1; 0; 0 where no predator organism y and prey x exist. By equating system (3) to zero, we will get s ¼ 1 from the equation 1 � s � f <sup>1</sup>ð Þs x ¼ 0. Then, from equation x<sup>0</sup> ¼ 0, we get

<sup>x</sup> <sup>¼</sup> <sup>0</sup>: When <sup>x</sup> 6¼ <sup>0</sup>, <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

iii. <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ denotes the positive interior equilibrium point with <sup>s</sup><sup>∗</sup>, x<sup>∗</sup>, y<sup>∗</sup> <sup>&</sup>gt; 0. <sup>s</sup><sup>∗</sup> is a

unique solution of 1 � <sup>s</sup> � <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>x</sup> <sup>¼</sup> <sup>0</sup>: From this, we have <sup>x</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup>�s<sup>∗</sup>

resulting equations simultaneously. The possible equilibrium points are as follows;

. <sup>s</sup> <sup>¼</sup> <sup>ζ</sup><sup>s</sup> is the unique solution of <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

<sup>D</sup><sup>1</sup> � 1 

only prey organisms and not predator organisms.

Next, we find <sup>x</sup>: From equation 1 � <sup>s</sup> � <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>x</sup> <sup>¼</sup> 0, <sup>x</sup> <sup>¼</sup> <sup>1</sup>�<sup>s</sup>

D1x 1 þ x þ D1by 

<sup>&</sup>lt; <sup>f</sup> <sup>1</sup>ð Þ� <sup>s</sup> <sup>1</sup> <sup>x</sup> <sup>&</sup>lt; <sup>α</sup>1x,

1þxþD1by 

, x<sup>0</sup> and y<sup>0</sup> to zero and solve the

<sup>D</sup><sup>1</sup> � 1 ¼ 0: Let y ¼ 0, then Eq. (3) give

<sup>D</sup><sup>1</sup> � 1 ¼ 0. Therefore, f <sup>1</sup>ð Þ¼ s D1:

<sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> : By substituting <sup>f</sup> <sup>1</sup>ð Þ¼ <sup>s</sup> <sup>D</sup><sup>1</sup>

<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ. Next, let the

< 0:

� y < α2y,

: We shall assume that max<sup>x</sup><sup>∈</sup> <sup>Η</sup> <sup>f</sup> <sup>2</sup>ð Þþ <sup>x</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> <sup>D</sup>1<sup>x</sup>

<sup>D</sup><sup>1</sup> : Thus, E<sup>2</sup> is the one possible equilibrium point that consists

1 Pmin þ q,

$$\frac{1-s}{\infty} = f\_1(s) \quad \text{and} \quad D\_1 = f\_1(s).$$

Thus, 1 � <sup>s</sup> <sup>¼</sup> <sup>D</sup>1x, i.e., <sup>x</sup> <sup>¼</sup> <sup>1</sup>�<sup>s</sup> <sup>D</sup><sup>1</sup> . By taking <sup>s</sup> <sup>¼</sup> <sup>ζ</sup>s, we will have <sup>x</sup> <sup>¼</sup> <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> and satisfying <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup>b<sup>x</sup> � <sup>1</sup> <sup>¼</sup> 0. Hence, the equilibrium point <sup>E</sup><sup>2</sup> <sup>ζ</sup>s; 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � exists. <sup>∎</sup>

#### 2.3. Stability analysis of equilibrium points E1, E<sup>2</sup> and E<sup>3</sup>

In this section, we analyse the stability of equilibrium points as it plays an important part in ordinary differential equations with their applications. As we cannot easily identify the positions of equilibrium point in applications of dynamical system, but only approximately, so the equilibrium point must be in stable state to be biologically meaningful [15]. Several definitions and theorem from Ref. [14] are stated to make us understand clearly.

Definition 2. [14] A fixed point x<sup>∗</sup> is Lyapunov stable or L-stable, provided that any solution <sup>ϕ</sup>ð Þ <sup>t</sup>; <sup>x</sup><sup>0</sup> stays near <sup>x</sup><sup>∗</sup> for all future time <sup>t</sup> <sup>≥</sup> 0 if the initial condition <sup>x</sup><sup>0</sup> starts near enough to <sup>x</sup><sup>∗</sup>: Specifically, a fixed point x<sup>∗</sup> is L-stable, provided that for any E > 0, there is δ > 0, such that if <sup>x</sup><sup>0</sup> � <sup>x</sup><sup>∗</sup> k k <sup>&</sup>lt; <sup>δ</sup>, then <sup>ϕ</sup>ð Þ� <sup>t</sup>; <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup> � � � � < E for all t ≥ 0.

Another form of stability is asymptotically stable. This is stated in the following Definition 3. below. Before going further to understand this concept, we need the definition of ω-limit set as follows.

J ¼

Then,

�f 0

f 0 <sup>1</sup>ð Þs x D<sup>1</sup>

f <sup>1</sup>ð Þs D<sup>1</sup>

0 f <sup>2</sup>ð Þx y

� f <sup>2</sup>ð Þx y

its eigenvalues; <sup>λ</sup><sup>1</sup> ¼ �1, <sup>λ</sup><sup>2</sup> ¼ �1 and <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup>

then this leads to the following result.

Theorem 3. If the eigenvalue λ<sup>3</sup> such that

(ii) Stability analysis of E2:

equilibrium point E<sup>2</sup> ζs;

0

BBBBBBBBBBBBBBBB@

� f 0

> f 0

JE<sup>2</sup> ¼

<sup>1</sup>ð Þs x � 1 �f <sup>1</sup>ð Þs

D<sup>1</sup>

D<sup>1</sup>

JE<sup>1</sup> ¼

then the trivial equilibrium point E1ð Þ 1; 0; 0 is locally asymptotically stable.

f <sup>1</sup>ð Þ ζ<sup>s</sup> D<sup>1</sup>

0 0 f <sup>2</sup>

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þ 1 � ζ<sup>s</sup> D<sup>1</sup>

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þ 1 � ζ<sup>s</sup> D2 1

0

BB@

<sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by � � � <sup>D</sup>1<sup>x</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by � � � <sup>D</sup>1<sup>x</sup>

" # !

" # !

ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup>

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup>

<sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> <sup>D</sup><sup>2</sup> <sup>1</sup>bxy ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup> !

� 1 0

f <sup>2</sup>ð Þx 1 þ

�1 �f <sup>1</sup>ð Þ1 0

0 0 �1

� 1 < 0,

<sup>0</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup> D<sup>1</sup>

As Jacobian matrix JE<sup>1</sup> above is an upper triangular, therefore the diagonal is the value of all of

Next, we analyse the local stability of the system (3) that restricted to the neighbourhood of the

� 1 �f <sup>1</sup>ð Þ ζ<sup>s</sup> 0

1 � ζ<sup>s</sup> D<sup>1</sup> � � <sup>1</sup> <sup>þ</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> � � <sup>1</sup> <sup>þ</sup>

� 1 �f <sup>2</sup>

<sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup> D<sup>1</sup>

� �. The Jacobian matrix at <sup>E</sup><sup>2</sup> <sup>ζ</sup>s;

� f <sup>2</sup> 0 ð Þx y

þ f 0 <sup>2</sup>ð Þx y 1 þ

D1x <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by � � � <sup>f</sup> <sup>2</sup>ð Þ<sup>x</sup> <sup>D</sup><sup>2</sup>

1

CCA:

0

� f <sup>2</sup>ð Þx 1 þ

<sup>D</sup><sup>1</sup> � 1: If all the eigenvalues of JE<sup>1</sup> are negatives,

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � is given as

0

0

BB@

0

BB@

BB@

0

BB@

1 þ

1 � ζ<sup>s</sup>

1 � ζ<sup>S</sup> D<sup>1</sup> � �

1

CCA

1

1

1

CCCCCCCCCCCCCCCCA :

CCA

CCA

1

CCA � <sup>1</sup>

1 þ

1 � ζ<sup>s</sup>

1 � ζ<sup>S</sup> D<sup>1</sup> � �

D1x <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by � � <sup>þ</sup> <sup>1</sup> � �

http://dx.doi.org/10.5772/intechopen.71624

D1x <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by � �

D1x <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by � �

<sup>1</sup>bxy ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup> !

� 1

� 1

53

:

Definition 3. [14] A point k is an ω-limit point of the trajectory of x0, if ϕð Þ t; x<sup>0</sup> keeps coming near k as t ! ∞ (i.e., there is a sequence of times tj, with tj ! ∞ as j ! ∞ such that ϕ tj; x<sup>0</sup> converges to <sup>k</sup>). Certainly, if <sup>ϕ</sup>ð Þ� <sup>t</sup>; <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup> ! 0 as <sup>t</sup> ! <sup>∞</sup>, then <sup>x</sup><sup>∗</sup> is the only <sup>ω</sup>-limit point of <sup>x</sup>0. There can be more than one point that is an ω-limit point of x0. The set of all ω-limit points of x<sup>0</sup> is denoted by ωð Þ x<sup>0</sup> and is called the omega limit set of x0.

Definition 4. [14] A fixed point x<sup>∗</sup> is weakly asymptotically stable, if there exists δ<sup>1</sup> > 0 such that <sup>ω</sup>ð Þ¼ <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup> f g for all <sup>x</sup><sup>0</sup> � <sup>x</sup><sup>∗</sup> k k <sup>&</sup>lt; <sup>δ</sup><sup>1</sup> (i.e., <sup>ϕ</sup>ð Þ� <sup>t</sup>; <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup> ! 0 as time t ! ∞ for all <sup>x</sup><sup>0</sup> � <sup>x</sup><sup>∗</sup> k k <sup>&</sup>lt; <sup>δ</sup>1). Therefore, a fixed point is weakly asymptotically stable, if the stable manifold contains all points in a neighbourhood of the fixed point (i.e., all points are sufficiently close). A fixed point x<sup>∗</sup> is asymptotically stable if it is both L-stable and weakly asymptotically stable. An asymptotically stable fixed point is also called sink.

Moreover, Theorem 2 below clearly shows that if a fixed point x<sup>∗</sup> is hyperbolic (the real parts of all eigenvalues of the Jacobian matrix at <sup>x</sup><sup>∗</sup>, i.e., DF <sup>x</sup><sup>∗</sup> ð Þ are nonzero), then the stability type of the fixed point for the nonlinear system is the same as for the linearized system at that fixed point.

Theorem 2. [14] Let <sup>x</sup>\_ <sup>¼</sup> F xð Þ be a differential equation in n variables, with a hyperbolic fixed point x<sup>∗</sup>. Suppose thal F, <sup>∂</sup>Fi ∂xj ð Þ<sup>x</sup> and <sup>∂</sup><sup>2</sup>Fi <sup>∂</sup>xj∂xk ð Þx are all continuous. Then, the stability type of the fixed point for the nonlinear system is the same as for the linearized system at that fixed point.


Now by utilising the Theorem 2, we are going to analyse the stability of the fixed points E1, E<sup>2</sup> and E3.

(i) Stability analysis of E1:

Now, we will discuss the stability type of the equilibrium point E1ð Þ 1; 0; 0 . The Jacobian matrix of the system (3) is

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation… http://dx.doi.org/10.5772/intechopen.71624 53

$$J = \begin{bmatrix} -f\_1'(\mathbf{s})\mathbf{x} - \mathbf{1} & -f\_1(\mathbf{s}) \\ f\_1'(\mathbf{s})\mathbf{x} & f\_2(\mathbf{s}) - f\_2(\mathbf{x})g\_1'\left[\left(\frac{D\_1}{1+\mathbf{x}+D\_1\mathbf{b}y}\right) - \left(\frac{D\_1\mathbf{x}}{(1+\mathbf{x}+D\_1\mathbf{b}y)^2}\right)\right] - f\_2'(\mathbf{x})g\left(\left(\frac{D\_1\mathbf{x}}{1+\mathbf{x}+D\_1\mathbf{b}y}\right) + 1\right) - \mathbf{1} \\ 0 & f\_2(\mathbf{x})g\left[\left(\frac{D\_1}{1+\mathbf{x}+D\_1\mathbf{b}y}\right) - \left(\frac{D\_1\mathbf{x}}{(1+\mathbf{x}+D\_1\mathbf{b}y)^2}\right)\right] + f\_2'(\mathbf{x})g\left(1 + \frac{D\_1\mathbf{x}}{1+\mathbf{x}+D\_1\mathbf{b}y}\right) \\ 0 & 0 \\ \left(\frac{f\_2(\mathbf{x})D\_1^2\mathbf{b}xy}{(1+\mathbf{x}+D\_1\mathbf{b}y)^2}\right) - f\_2(\mathbf{x})\left(1 + \frac{D\_1\mathbf{x}}{1+\mathbf{x}+D\_1\mathbf{b}y}\right) \\ f\_2(\mathbf{x})\left(1 + \frac{D\_1\mathbf{x}}{1+\mathbf{x}+D\_1\mathbf{b}y}\right) - \left(\frac{f\_2(\mathbf{x})D\_1^2\mathbf{b}xy}{(1+\mathbf{x}+D\_1\mathbf{b}y)^2}\right) - 1 \end{bmatrix}$$

Then,

! 0 as time t ! ∞ for

Another form of stability is asymptotically stable. This is stated in the following Definition 3. below. Before going further to understand this concept, we need the definition of ω-limit set as

Definition 3. [14] A point k is an ω-limit point of the trajectory of x0, if ϕð Þ t; x<sup>0</sup> keeps coming near k as t ! ∞ (i.e., there is a sequence of times tj, with tj ! ∞ as j ! ∞ such that ϕ tj; x<sup>0</sup>

There can be more than one point that is an ω-limit point of x0. The set of all ω-limit points of x<sup>0</sup>

Definition 4. [14] A fixed point x<sup>∗</sup> is weakly asymptotically stable, if there exists δ<sup>1</sup> > 0

all <sup>x</sup><sup>0</sup> � <sup>x</sup><sup>∗</sup> k k <sup>&</sup>lt; <sup>δ</sup>1). Therefore, a fixed point is weakly asymptotically stable, if the stable manifold contains all points in a neighbourhood of the fixed point (i.e., all points are sufficiently close). A fixed point x<sup>∗</sup> is asymptotically stable if it is both L-stable and weakly asymptotically

Moreover, Theorem 2 below clearly shows that if a fixed point x<sup>∗</sup> is hyperbolic (the real parts of all eigenvalues of the Jacobian matrix at <sup>x</sup><sup>∗</sup>, i.e., DF <sup>x</sup><sup>∗</sup> ð Þ are nonzero), then the stability type of the fixed point for the nonlinear system is the same as for the linearized system at that fixed

Theorem 2. [14] Let <sup>x</sup>\_ <sup>¼</sup> F xð Þ be a differential equation in n variables, with a hyperbolic fixed point x<sup>∗</sup>.

a. If the real parts of all the eigenvalues of DF <sup>x</sup><sup>∗</sup> ð Þ are negative, then the fixed point is asymptotically stable for the nonlinear equation (i.e., if the origin of the linearized system is asymptotically stable, then x<sup>∗</sup> is asymptotically stable for the nonlinear equation). In this case, the basin of attraction

b. If there is at least one eigenvalue of DF <sup>x</sup><sup>∗</sup> ð Þ has positive real part, then the fixed point x<sup>∗</sup> is unstable. (For the linearized system, the fixed point can be a saddle, unstable node, unstable focus, etc.)

c. If one of the eigenvalues of DF <sup>x</sup><sup>∗</sup> ð Þ has zero real part, then the situation is more complicated. In particular, for n ¼ 2, if the fixed point is an elliptic center (eigenvalues �βi) or one eigenvalue is zero of multiplicity one, then the linearized system does not determine the stability type of the fixed

Now by utilising the Theorem 2, we are going to analyse the stability of the fixed points E1, E<sup>2</sup>

Now, we will discuss the stability type of the equilibrium point E1ð Þ 1; 0; 0 . The Jacobian matrix

! 0 as <sup>t</sup> ! <sup>∞</sup>, then <sup>x</sup><sup>∗</sup> is the only <sup>ω</sup>-limit point of <sup>x</sup>0.

<sup>∂</sup>xj∂xk ð Þx are all continuous. Then, the stability type of the fixed point for the

such that <sup>ω</sup>ð Þ¼ <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup> f g for all <sup>x</sup><sup>0</sup> � <sup>x</sup><sup>∗</sup> k k <sup>&</sup>lt; <sup>δ</sup><sup>1</sup> (i.e., <sup>ϕ</sup>ð Þ� <sup>t</sup>; <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup>

nonlinear system is the same as for the linearized system at that fixed point.

<sup>W</sup><sup>S</sup> <sup>x</sup><sup>∗</sup> ð Þ is an open set that contains some solid ball about the fixed point.

stable. An asymptotically stable fixed point is also called sink.

is denoted by ωð Þ x<sup>0</sup> and is called the omega limit set of x0.

52 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

follows.

point.

Suppose thal F, <sup>∂</sup>Fi

point.

(i) Stability analysis of E1:

of the system (3) is

and E3.

∂xj

ð Þ<sup>x</sup> and <sup>∂</sup><sup>2</sup>Fi

converges to <sup>k</sup>). Certainly, if <sup>ϕ</sup>ð Þ� <sup>t</sup>; <sup>x</sup><sup>0</sup> <sup>x</sup><sup>∗</sup>

$$J\_{E\_1} = \begin{pmatrix} -1 & -f\_1(1) & 0\\ 0 & \frac{f\_1(1)}{D\_1} - 1 & 0\\ 0 & 0 & -1 \end{pmatrix}.$$

As Jacobian matrix JE<sup>1</sup> above is an upper triangular, therefore the diagonal is the value of all of its eigenvalues; <sup>λ</sup><sup>1</sup> ¼ �1, <sup>λ</sup><sup>2</sup> ¼ �1 and <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup> <sup>D</sup><sup>1</sup> � 1: If all the eigenvalues of JE<sup>1</sup> are negatives, then this leads to the following result.

Theorem 3. If the eigenvalue λ<sup>3</sup> such that

$$
\lambda\_3 = \frac{f\_1(1)}{D\_1} - 1 < 0,
$$

then the trivial equilibrium point E1ð Þ 1; 0; 0 is locally asymptotically stable.

(ii) Stability analysis of E2:

Next, we analyse the local stability of the system (3) that restricted to the neighbourhood of the equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � �. The Jacobian matrix at <sup>E</sup><sup>2</sup> <sup>ζ</sup>s; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � is given as

$$J\_{E\_{2}} = \begin{pmatrix} -\frac{f\_{1}'(\zeta\_{s})(1-\zeta\_{s})}{D\_{1}} - 1 & -f\_{1}(\zeta\_{s}) & 0\\ \frac{f\_{1}'(\zeta\_{s})(1-\zeta\_{s})}{D\_{1}^{2}} & \frac{f\_{1}(\zeta\_{s})}{D\_{1}} - 1 & -f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}}\right)\left(1 + \left(\frac{1-\zeta\_{s}}{1+\left(\frac{1-\zeta\_{s}}{D\_{1}}\right)}\right)\right) \\\\ 0 & 0 & f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}}\right)\left(1 + \left(\frac{1-\zeta\_{s}}{1+\left(\frac{1-\zeta\_{s}}{D\_{1}}\right)}\right)\right) - 1 \end{pmatrix}.$$

We can see from both Jacobian matrices JE<sup>1</sup> and JE<sup>2</sup> that there are no parameter b involved when the predator organisms y is zero. This means that intratrophic predation does not affect the local stability and existence of E<sup>1</sup> and E<sup>2</sup> when no predator organisms involved. The characteristic equation is <sup>λ</sup><sup>3</sup> <sup>þ</sup> <sup>c</sup>1λ<sup>2</sup> <sup>þ</sup> <sup>c</sup>2<sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>3</sup> <sup>¼</sup> 0 where

Theorem 4. Suppose the assumptions such as

f 0

are satisfied, then the equilibrium point, E<sup>2</sup> ζs;

f 0

Particularly, the dimension of the stable manifold W<sup>s</sup>

, limt!<sup>∞</sup> dð Þ πð Þ x; t ; ∂E ≥ ε0:

ii. E<sup>2</sup> is a unique hyperbolic saddle point in R<sup>þ</sup>

iii. no existence of periodic orbits in the planes of R<sup>þ</sup>

then the equilibrium point E<sup>2</sup> ζs;

Theorem 5. If

by

for all x∈ E<sup>0</sup>

Theorem 6. Assume that

i. Lemma 1 being holds,

analysis.

Then

where k > 0:

exists.

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þþ ζs�1 f <sup>1</sup>ð Þ� ζ<sup>s</sup> D<sup>1</sup>

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þþ ζs�1 f <sup>1</sup>ð Þ� ζ<sup>s</sup> D<sup>1</sup>

<sup>D</sup><sup>1</sup> < 0 and

<sup>D</sup><sup>1</sup> < 0 and

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

dimW<sup>s</sup>

f 2 1�ζs D1 

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

> f 2 1�ζs D1

ð Þ¼ <sup>E</sup><sup>2</sup> <sup>2</sup> and dimW<sup>u</sup>ð Þ¼ <sup>E</sup><sup>2</sup> 1.

Proof. The results follow from inspections of the eigenvalues of the matrix JE<sup>2</sup> and Theorem 2 (see [16, 18]). ∎

Definition 4. [17] The flow F will be called uniformly persistent if there exists ε<sup>0</sup> > 0 such that

The following theorem shows the existence of the equilibrium point E<sup>2</sup> using persistence

sxy.

lim<sup>t</sup>!þ<sup>∞</sup> inf s tð Þ <sup>&</sup>gt; k, lim<sup>t</sup>!þ<sup>∞</sup> inf x tð Þ <sup>&</sup>gt; k, lim<sup>t</sup>!þ<sup>∞</sup> inf y tð Þ <sup>&</sup>gt; k,

Proof. The result follows from the Definition 4, which defines uniform persistence by Butler et al. [17] and Nani and Freedman [18]. ∎

Particularly, the system (3) exhibits uniform persistence and the equilibrium point E<sup>2</sup> ζs;

ð1þ2D1�ζs�D1ζsÞ�1 <sup>D</sup>1�ζsþ<sup>1</sup> <sup>&</sup>lt; <sup>0</sup>

is locally asymptotically stable.

ð1þ2D1�ζs�D1ζsÞ�1 <sup>D</sup>1�ζsþ<sup>1</sup> <sup>&</sup>lt; <sup>0</sup>,

is a hyperbolic saddle and is repels locally in y-direction.

ð Þ <sup>E</sup><sup>2</sup> and the unstable manifold W<sup>u</sup>ð Þ <sup>E</sup><sup>2</sup> are given

http://dx.doi.org/10.5772/intechopen.71624

55

sxy and repels locally in y-direction (as in Theorem 5),

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 

$$\begin{split} c\_{1} &= \frac{1}{D\_{1}(D\_{1}-\zeta\_{s}+1)} \Big( f\_{1}'(\zeta\_{s}) + 3D\_{1} - f\_{1}(\zeta\_{s}) + f\_{1}'(\zeta\_{s})D\_{1} - D\_{1}f\_{2}\Big(\frac{1-\zeta\_{s}}{D\_{1}}\Big) \\ &- 2\zeta\_{s}f\_{1}'(\zeta\_{s}) - 3D\_{1}\zeta\_{s} - 2D\_{1}^{2}f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}}\right) + \zeta\_{s}^{2}f\_{1}'(\zeta\_{s}) - f\_{1}(\zeta\_{s})D\_{1} + \zeta\_{s}f\_{1}(\zeta\_{s}) + 3D\_{1}^{2} - D\_{1}\zeta\_{s}f\_{1}'(\zeta\_{s}) \\ &+ D\_{1}\zeta\_{s}f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}}\right) + D\_{1}^{2}\zeta\_{s}f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}}\right) \Big), \end{split}$$

<sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>S</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup><sup>f</sup> <sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>3</sup>D<sup>1</sup> � <sup>2</sup><sup>f</sup> 0 <sup>1</sup>ð Þ� ζ<sup>s</sup> 2D1f 0 <sup>1</sup>ð Þþ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> <sup>f</sup> 0 <sup>1</sup>ð Þ ζ<sup>s</sup> þ 2D1f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ 4ζsf 0 <sup>1</sup>ð Þþ <sup>ζ</sup><sup>s</sup> <sup>3</sup>D1ζ<sup>s</sup> <sup>þ</sup> <sup>4</sup>D<sup>2</sup> 1f 2 1 � ζ<sup>s</sup> D<sup>1</sup> � <sup>2</sup>ζ<sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> þ 2D1f <sup>1</sup>ð Þ� ζ<sup>s</sup> f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � <sup>2</sup>ζsf <sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>3</sup>D<sup>2</sup> <sup>1</sup> � 2D1f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ 2D1f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ 2D1ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> � 2ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � 2D1ζsf <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> �2D<sup>2</sup> <sup>1</sup>ζsf <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � 3D1ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ D1ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ,

<sup>c</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup> <sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>D</sup><sup>1</sup> � <sup>f</sup> 0 <sup>1</sup>ð Þ� ζ<sup>s</sup> D1f 0 <sup>1</sup>ð Þþ ζ<sup>s</sup> f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ D1f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ 2ζsf 0 <sup>1</sup>ð Þþ <sup>ζ</sup><sup>s</sup> <sup>D</sup>1ζ<sup>s</sup> � <sup>ζ</sup><sup>2</sup> s f 0 <sup>1</sup>ð Þþ ζ<sup>s</sup> f <sup>1</sup>ð Þ ζ<sup>s</sup> D<sup>1</sup> � f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> � <sup>D</sup><sup>2</sup> <sup>1</sup> � 2D1f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ 2D1f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ D1ζsf 0 <sup>1</sup>ð Þ� ζ<sup>s</sup> 2ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � D1ζsf <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � <sup>D</sup><sup>2</sup> <sup>1</sup>ζsf <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> � 3D1ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þD1ζ<sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> þ D1ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> :

Then, by using MATLAB R2015b, we get the eigenvalues of JE<sup>2</sup> which are λ<sup>4</sup> ¼ �1 , <sup>λ</sup><sup>5</sup> <sup>¼</sup> <sup>f</sup> 0 <sup>1</sup>ð Þ ζ<sup>s</sup> ð Þþ ζs�1 f <sup>1</sup>ð Þ� ζ<sup>s</sup> D<sup>1</sup> <sup>D</sup><sup>1</sup> and <sup>λ</sup><sup>6</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> 1�ζs D1 ð1þ2D1�ζs�D1ζsÞ�1 <sup>D</sup>1�ζsþ<sup>1</sup> where <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> 6¼ <sup>0</sup>: We summarise the above discussion in the following theorems.

Theorem 4. Suppose the assumptions such as

We can see from both Jacobian matrices JE<sup>1</sup> and JE<sup>2</sup> that there are no parameter b involved when the predator organisms y is zero. This means that intratrophic predation does not affect the local stability and existence of E<sup>1</sup> and E<sup>2</sup> when no predator organisms involved. The

0

<sup>þ</sup> <sup>ζ</sup><sup>2</sup> s f 0

<sup>1</sup>ð Þ� ζ<sup>s</sup> 2D1f

1f 2

<sup>þ</sup> <sup>ζ</sup><sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

<sup>1</sup>ð Þ� ζ<sup>s</sup> D1f

� D1ζsf <sup>2</sup>

<sup>1</sup>ð Þþ ζ<sup>s</sup> f <sup>1</sup>ð Þ ζ<sup>s</sup> D<sup>1</sup> � f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

0 <sup>1</sup>ð Þþ ζ<sup>s</sup> f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

þ 2D1ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup>

<sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

> 0 <sup>1</sup>ð Þþ ζ<sup>s</sup> f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

� 3D1ζsf

1 � ζ<sup>s</sup> D<sup>1</sup> 

Then, by using MATLAB R2015b, we get the eigenvalues of JE<sup>2</sup> which are λ<sup>4</sup> ¼ �1 ,

ð1þ2D1�ζs�D1ζsÞ�1

1 � ζ<sup>s</sup> D<sup>1</sup> 

> 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

:

Þ,

� <sup>2</sup>ζsf <sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>3</sup>D<sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

1 � ζ<sup>s</sup> D<sup>1</sup> 

> 1 � ζ<sup>s</sup> D<sup>1</sup>

> > 0

þ ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

1 � ζ<sup>s</sup> D<sup>1</sup> 

þ D1ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

1�ζs D1 

0

<sup>1</sup>ð Þ ζ<sup>s</sup> D<sup>1</sup> � D1f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup>

<sup>1</sup> � D1ζsf

0 <sup>1</sup>ð Þ ζ<sup>s</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

> 1 � ζ<sup>s</sup> D<sup>1</sup>

,

<sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>ζ</sup>sf <sup>1</sup>ð Þþ <sup>ζ</sup><sup>s</sup> <sup>3</sup>D<sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

� <sup>2</sup>ζ<sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

1 � ζ<sup>s</sup> D<sup>1</sup> 

þ 2D1f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

<sup>1</sup> � 2D1f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

f 0 <sup>1</sup>ð Þ ζ<sup>s</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

þ D1ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

þ D1f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

� ζsf <sup>1</sup>ð Þ ζ<sup>s</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

<sup>D</sup>1�ζsþ<sup>1</sup> where <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> 6¼ <sup>0</sup>: We summarise

1 � ζ<sup>s</sup> D<sup>1</sup> 

characteristic equation is <sup>λ</sup><sup>3</sup> <sup>þ</sup> <sup>c</sup>1λ<sup>2</sup> <sup>þ</sup> <sup>c</sup>2<sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>3</sup> <sup>¼</sup> 0 where

54 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

1f 2

<sup>1</sup>ð Þþ ζ<sup>s</sup> 3D<sup>1</sup> � f <sup>1</sup>ð Þþ ζ<sup>s</sup> f

1 � ζ<sup>s</sup> D<sup>1</sup> 

1 � ζ<sup>s</sup> D<sup>1</sup> 

<sup>1</sup>ð Þþ <sup>ζ</sup><sup>s</sup> <sup>3</sup>D1ζ<sup>s</sup> <sup>þ</sup> <sup>4</sup>D<sup>2</sup>

0

<sup>þ</sup> <sup>D</sup><sup>2</sup> <sup>1</sup>ζsf <sup>2</sup>

<sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>3</sup>D1ζ<sup>s</sup> � <sup>2</sup>D<sup>2</sup>

<sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>S</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup><sup>f</sup> <sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>3</sup>D<sup>1</sup> � <sup>2</sup><sup>f</sup>

þ 4ζsf 0

> 1 � ζ<sup>s</sup> D<sup>1</sup>

� 2D1ζsf <sup>2</sup>

s f 0

1 � ζ<sup>s</sup> D<sup>1</sup> 

> � <sup>D</sup><sup>2</sup> <sup>1</sup>ζsf <sup>2</sup>

0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

<sup>D</sup><sup>1</sup> and <sup>λ</sup><sup>6</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup>

the above discussion in the following theorems.

þ 2D1f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

� 3D1ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

<sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>ζ</sup><sup>s</sup> <sup>D</sup><sup>1</sup> � <sup>f</sup>

<sup>1</sup>ð Þþ <sup>ζ</sup><sup>s</sup> <sup>D</sup>1ζ<sup>s</sup> � <sup>ζ</sup><sup>2</sup>

<sup>1</sup> � 2D1f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

<sup>1</sup>ð Þ� ζ<sup>s</sup> 2ζsf

1 � ζ<sup>s</sup> D<sup>1</sup> 

1 � ζ<sup>s</sup> D<sup>1</sup> 

1 � ζ<sup>s</sup> D<sup>1</sup> 

> 1 � ζ<sup>s</sup> D<sup>1</sup>

> > 1 � ζ<sup>s</sup> D<sup>1</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> 

þ 2D1f <sup>1</sup>ð Þ� ζ<sup>s</sup> f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

<sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

� 2ζsf 0

<sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

þ 2D1f <sup>2</sup>

þ ζsf <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

� 2ζsf 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

�2D<sup>2</sup> <sup>1</sup>ζsf <sup>2</sup>

<sup>c</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>

þ 2ζsf 0

� <sup>D</sup><sup>2</sup>

þ D1ζsf 0

<sup>þ</sup> <sup>ζ</sup><sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

þD1ζ<sup>2</sup> s f 0 <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þþ ζs�1 f <sup>1</sup>ð Þ� ζ<sup>s</sup> D<sup>1</sup>

<sup>λ</sup><sup>5</sup> <sup>¼</sup> <sup>f</sup> 0

þ D1ζsf <sup>2</sup>

<sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup>

$$\frac{f\_1'(\zeta\_\*) (\zeta\_\* - 1) + f\_1(\zeta\_\*) - D\_1}{D\_1} < 0 \text{ and } \frac{f\_2 \left(\frac{1 - \zeta\_\*}{D\_1}\right) (1 + 2D\_1 - \zeta\_\* - D\_1 \zeta\_\*) - 1}{D\_1 - \zeta\_\* + 1} < 0$$

are satisfied, then the equilibrium point, E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 is locally asymptotically stable. Theorem 5. If

$$\frac{f\_1'(\zeta\_\*)(\zeta\_\*-1) + f\_1(\zeta\_\*) - D\_1}{D\_1} < 0 \text{ and } \frac{f\_2\left(\frac{1-\zeta\_\*}{D\_1}\right)(1+2D\_1-\zeta\_\*-D\_1\zeta\_\*)-1}{D\_1-\zeta\_\*+1} < 0,$$

then the equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 is a hyperbolic saddle and is repels locally in y-direction. Particularly, the dimension of the stable manifold W<sup>s</sup> ð Þ <sup>E</sup><sup>2</sup> and the unstable manifold W<sup>u</sup>ð Þ <sup>E</sup><sup>2</sup> are given by

$$
\dim \mathcal{W}^s(E\_2) \quad = 2 \text{ and } \dim \mathcal{W}^u(E\_2) = 1.
$$

Proof. The results follow from inspections of the eigenvalues of the matrix JE<sup>2</sup> and Theorem 2 (see [16, 18]). ∎

Definition 4. [17] The flow F will be called uniformly persistent if there exists ε<sup>0</sup> > 0 such that for all x∈ E<sup>0</sup> , limt!<sup>∞</sup> dð Þ πð Þ x; t ; ∂E ≥ ε0:

The following theorem shows the existence of the equilibrium point E<sup>2</sup> using persistence analysis.

Theorem 6. Assume that

i. Lemma 1 being holds,

ii. E<sup>2</sup> is a unique hyperbolic saddle point in R<sup>þ</sup> sxy and repels locally in y-direction (as in Theorem 5),

iii. no existence of periodic orbits in the planes of R<sup>þ</sup> sxy.

Then

$$\lim\_{t \to +\infty} \inf s(t) > k, \lim\_{t \to +\infty} \inf x(t) > k, \lim\_{t \to +\infty} \inf y(t) > k.$$

where k > 0:

Particularly, the system (3) exhibits uniform persistence and the equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 exists.

Proof. The result follows from the Definition 4, which defines uniform persistence by Butler et al. [17] and Nani and Freedman [18]. ∎ (iii) Stability analysis of <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ:

#### The Jacobian matrix at E<sup>3</sup> is

$$J\_{E\_{3}} = \begin{bmatrix} -f\_{1}(s^{\*})x^{\*} - 1 & -f\_{1}(s^{\*}) \\ f\_{1}(s^{\*})x^{\*} & f\_{1}(s^{\*}) \\ D\_{1} & D\_{1} - f\_{2}(x^{\*})y^{\*} \left[ \left(\frac{D\_{1}}{1+x^{\*}+D\_{1}by^{\*}}\right) - \left(\frac{D\_{1}x^{\*}}{(1+x^{\*}+D\_{1}by^{\*})^{2}}\right) \right] - f\_{2}(x^{\*})y^{\*} \left(\frac{D\_{1}x^{\*}}{1+x^{\*}+D\_{1}by^{\*}} + 1\right) - 1 \\ 0 & f\_{2}(x^{\*})y^{\*} \left[ \left(\frac{D\_{1}}{1+x^{\*}+D\_{1}by^{\*}}\right) - \left(\frac{D\_{1}x^{\*}}{(1+x^{\*}+D\_{1}by^{\*})^{2}}\right) \right] + f\_{2}(x^{\*})y^{\*} \left(\frac{D\_{1}x^{\*}}{1+x^{\*}+D\_{1}by^{\*}} + 1\right) \\ 0 & 0 \\ \left(\frac{f\_{2}(x^{\*})D\_{1}^{2}bx^{\*}y^{\*}}{(1+x^{\*}+D\_{1}by^{\*})^{2}}\right) - f\_{2}(x^{\*}) \left(\frac{D\_{1}x^{\*}}{1+x^{\*}+D\_{1}by^{\*}} + 1\right) \\ f\_{2}(x^{\*}) \left(\frac{D\_{1}x^{\*}}{1+x^{\*}+D\_{1}by^{\*}} + 1\right) - \left(\frac{f\_{2}(x^{\*})D\_{1}^{2}bx^{\*}y^{\*}}{(1+x^{\*}+D\_{1}by^{\*})^{2}}\right) - 1 \end{bmatrix}.$$

The eigenvalues of JE<sup>3</sup> are resulted from the characteristic equation below

$$c\lambda^3 + c\_4\lambda^2 + c\_5\lambda + c\_6 = 0,\tag{6}$$

<sup>λ</sup><sup>8</sup> ¼ � <sup>1</sup> 2D<sup>1</sup>

0

BBBBBB@

þ

þf 0 1 s <sup>∗</sup> ð ÞD1x<sup>∗</sup>� ,

<sup>λ</sup><sup>9</sup> ¼ � <sup>1</sup> 2D<sup>1</sup>

0

BBBBBB@

�

þf 0 1 s <sup>∗</sup> ð ÞD1x<sup>∗</sup>� :

stable.

E<sup>2</sup> ζs;

1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � in <sup>R</sup><sup>þ</sup>

where <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>�b<sup>s</sup>

Lyapunov function <sup>L</sup> <sup>¼</sup> <sup>1</sup>

D<sup>1</sup> f 0 1 s

> 0 B@

D<sup>1</sup> f 0 1 s

> 0 B@

D<sup>1</sup>

vuuuuuuuut

D<sup>1</sup>

vuuuuuuuut

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>þ</sup> <sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>2</sup> � � � <sup>f</sup> <sup>1</sup> <sup>s</sup> <sup>∗</sup> ð Þ �

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>þ</sup> <sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>2</sup> � � � <sup>f</sup> <sup>1</sup> <sup>s</sup> <sup>∗</sup> ð Þ �

<sup>Q</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>QR <sup>þ</sup> <sup>2</sup>QV � <sup>2</sup>QZ � <sup>2</sup>Q f <sup>1</sup> <sup>s</sup>

These results lead to the following theorem.

use the similar technique as in [3, 18].

(i) Global stability analysis of E<sup>2</sup> ζs;

Consider system (3) restricted to R<sup>þ</sup>

nents of <sup>E</sup>2ð Þ <sup>b</sup>s; <sup>b</sup><sup>x</sup> , i.e., <sup>b</sup><sup>s</sup> <sup>¼</sup> <sup>ζ</sup><sup>s</sup> and <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>�ζ<sup>s</sup>

positive definite function with respect to E<sup>2</sup> in R<sup>þ</sup>

By differentiating L with respect to time t, we get

<sup>D</sup><sup>1</sup> . From (5), we have

2.4. Global stability and uniform persistence analysis

<sup>2</sup> <sup>n</sup>1ð Þ <sup>s</sup> �b<sup>s</sup>

dL

Now we will analyse the global stability of the equilibrium point E<sup>2</sup> ζs;

1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � �

<sup>2</sup> <sup>þ</sup> <sup>n</sup>2ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup>

dt <sup>¼</sup> <sup>n</sup>1ð Þ <sup>s</sup> �b<sup>s</sup> <sup>s</sup>

<sup>Q</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>QR <sup>þ</sup> <sup>2</sup>QV � <sup>2</sup>QZ � <sup>2</sup>Q f <sup>1</sup> <sup>s</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

f <sup>1</sup> s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f <sup>1</sup> s

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>2</sup><sup>Q</sup> <sup>þ</sup> <sup>2</sup><sup>R</sup> <sup>þ</sup> <sup>2</sup><sup>V</sup> � <sup>2</sup><sup>Z</sup> � <sup>f</sup>

<sup>∗</sup> ð Þ <sup>2</sup>D1<sup>R</sup> � <sup>2</sup>D1<sup>V</sup> � <sup>2</sup>D1<sup>Z</sup> � <sup>f</sup>

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>2</sup><sup>Q</sup> <sup>þ</sup> <sup>2</sup><sup>R</sup> <sup>þ</sup> <sup>2</sup><sup>V</sup> � <sup>2</sup><sup>Z</sup> � <sup>f</sup>

<sup>∗</sup> ð Þ <sup>2</sup>D1<sup>R</sup> � <sup>2</sup>D1<sup>V</sup> � <sup>2</sup>D1<sup>Z</sup> � <sup>f</sup>

sx as in (5). Let N be a neighbourhood of equilibrium point

<sup>D</sup><sup>1</sup> , while n<sup>1</sup> and n<sup>2</sup> are positive constants. Note that L is a

sx and a Lyapunov function for system (5) in N .

, (7)

sx. To analyse the global stability of the equilibrium point E2, a suitable

<sup>0</sup> <sup>þ</sup> <sup>n</sup>2ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup> <sup>x</sup><sup>0</sup>

<sup>2</sup> � � is used, where <sup>b</sup><sup>s</sup> and <sup>b</sup><sup>x</sup> denote the compo-

0 1 s

http://dx.doi.org/10.5772/intechopen.71624

<sup>2</sup>RV � <sup>2</sup>RZ <sup>þ</sup> <sup>V</sup><sup>2</sup> � <sup>2</sup>VZ <sup>þ</sup> <sup>Z</sup><sup>2</sup>

0 1 s

<sup>2</sup>RV � <sup>2</sup>RZ <sup>þ</sup> <sup>V</sup><sup>2</sup> � <sup>2</sup>VZ <sup>þ</sup> <sup>Z</sup><sup>2</sup>

0 1 s

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � and the

0 1 s þ

1 CA<sup>þ</sup> 1

57

CCCCCCA

<sup>∗</sup> ð Þ<sup>2</sup>

þ

1 CA<sup>þ</sup> 1

CCCCCCA

<sup>∗</sup> ð Þ<sup>2</sup>

<sup>∗</sup> ð Þx<sup>∗</sup> � � <sup>þ</sup> <sup>R</sup><sup>2</sup>

<sup>∗</sup> ð Þx<sup>∗</sup> � � <sup>þ</sup> <sup>f</sup> <sup>1</sup> <sup>s</sup>

<sup>∗</sup> ð Þx<sup>∗</sup> � � <sup>þ</sup> <sup>R</sup><sup>2</sup>

<sup>∗</sup> ð Þx<sup>∗</sup> � � <sup>þ</sup> <sup>f</sup> <sup>1</sup> <sup>s</sup>

<sup>∗</sup> ð Þ� <sup>f</sup> 0 1 s

<sup>∗</sup> ð Þ� <sup>f</sup> 0 1 s

Theorem 7. Suppose that λ<sup>8</sup> < 0 and λ<sup>9</sup> < 0, then the equilibrium point E<sup>3</sup> is locally asymptotically

existence of positive interior equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ. For global stability analysis, we

where

$$\begin{aligned} c\_4 &= f\_1'(s^\*) \mathbf{x}^\* + Q + R + V - Z + 3 - \frac{f\_1(s^\*)}{D\_1}, \\ c\_5 &= f\_1'(s^\*) \mathbf{x}^\* (Q + R + V - Z + 2) + \frac{f\_1(s^\*)(Z - R - 2)}{D\_1} + 2Q + 2R + 2V - 2Z + 3, \\ c\_6 &= f\_1'(s^\*) \mathbf{x}^\* (Q + R + V - Z + 1) + \frac{f\_1(s^\*)(Z - R - 1)}{D\_1} + Q + R + V - Z + 1, \end{aligned}$$

and

$$\begin{aligned} Q &= f\_2(\mathbf{x}^\*) y^\* \left[ \left( \frac{D\_1}{1 + \mathbf{x}^\* + D\_1 by^\*} \right) - \left( \frac{D\_1 \mathbf{x}^\*}{(1 + \mathbf{x}^\* + D\_1 by^\*)^2} \right) \right], \\ R &= \left( \frac{f\_2(\mathbf{x}^\*) D\_1^2 b \mathbf{x}^\* y^\*}{(1 + \mathbf{x}^\* + D\_1 by^\*)^2} \right), \\ V &= f\_2'(\mathbf{x}^\*) y^\* \left( \frac{D\_1 \mathbf{x}^\*}{1 + \mathbf{x}^\* + D\_1 by^\*} + 1 \right), \\ Z &= f\_2(\mathbf{x}^\*) \left( \frac{D\_1 \mathbf{x}^\*}{1 + \mathbf{x}^\* + D\_1 by^\*} + 1 \right). \end{aligned}$$

From (6), we obtain the eigenvalues; λ<sup>7</sup> ¼ �1, and λ8, λ<sup>9</sup> are

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation… http://dx.doi.org/10.5772/intechopen.71624 57

$$\begin{aligned} \boldsymbol{\theta}\_{8} &= -\frac{1}{2D\_{1}}\Bigl(D\_{1}\big(f\_{1}^{\star}(\mathbf{s}^{\star})\mathbf{x}^{\star} + Q + \mathcal{R} + V - \mathcal{Z} + 2\big) - f\_{1}(\mathbf{s}^{\star})\Bigr) \\ &+ \begin{pmatrix} \left(D\_{1}\left(Q^{2} + 2Q\mathcal{R} + 2QV - 2Q\mathcal{Z} - 2Qf\_{1}(\mathbf{s}^{\star}) - f\_{1}^{\star}(\mathbf{s}^{\star})\mathbf{x}^{\star}\big(2Q + 2\mathcal{R} + 2V - 2\mathcal{Z} - f\_{1}^{\star}(\mathbf{s}^{\star})\mathbf{x}^{\star}\big) + R^{2}\right) + \mathbf{1}\_{\star}\right) \\ &\mathcal{R}V - 2R\mathcal{Z} + V^{2} - 2\mathcal{V}\mathcal{Z} + \mathcal{Z}^{2} \Biggr) \\ &+ f\_{1}(\mathbf{s}^{\star})D\_{1}\mathbf{x}^{\star}\bigr), \\ &\mathcal{I}\_{1}(\mathbf{s}^{\star})D\_{1}\mathbf{x}^{\star}\bigr), \\ \boldsymbol{\theta}\_{8} &= -\frac{1}{2D\_{1}}\Bigl(D\_{1}(f\_{1}^{\star}(\mathbf{s}^{\star})\mathbf{x}^{\star} + Q + \mathcal{R} + V - \mathcal{Z} + 2\right) - f\_{1}(\mathbf{s}^{\star}) \\ &- \begin{pmatrix} \left(D\_{1}\left(\boldsymbol{\big$$

These results lead to the following theorem.

:

(iii) Stability analysis of <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ:

<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ D<sup>1</sup>

<sup>0</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þy<sup>∗</sup> <sup>D</sup><sup>1</sup>

<sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þx<sup>∗</sup> � <sup>1</sup> �<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ

1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � �

1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � �

The eigenvalues of JE<sup>3</sup> are resulted from the characteristic equation below

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>þ</sup> <sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>3</sup> � <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ

<sup>Q</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þy<sup>∗</sup> <sup>D</sup><sup>1</sup>

<sup>R</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð ÞD<sup>2</sup>

V ¼ f 0

<sup>∗</sup> ð Þx<sup>∗</sup>ð<sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>2</sup>Þ þ <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þð Þ <sup>Z</sup> � <sup>R</sup> � <sup>2</sup>

<sup>∗</sup> ð Þx<sup>∗</sup>ð<sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>1</sup>Þ þ <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þð Þ <sup>Z</sup> � <sup>R</sup> � <sup>1</sup>

1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � �

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

,

1bx<sup>∗</sup>y<sup>∗</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> !

<sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þy<sup>∗</sup> <sup>D</sup>1x<sup>∗</sup>

<sup>Z</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ <sup>D</sup>1x<sup>∗</sup>

From (6), we obtain the eigenvalues; λ<sup>7</sup> ¼ �1, and λ8, λ<sup>9</sup> are

� <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup>

� <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup>

<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð ÞD<sup>2</sup>

<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ <sup>D</sup>1x<sup>∗</sup>

1bx<sup>∗</sup>y<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> !

> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>λ</sup><sup>3</sup> <sup>þ</sup> <sup>c</sup>4λ<sup>2</sup> <sup>þ</sup> <sup>c</sup>5<sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>6</sup> <sup>¼</sup> <sup>0</sup>, (6)

� f <sup>2</sup>

þ f 0 <sup>0</sup> <sup>x</sup><sup>∗</sup> ð Þy<sup>∗</sup> <sup>D</sup>1x<sup>∗</sup>

<sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þy<sup>∗</sup> <sup>D</sup>1x<sup>∗</sup>

� <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ <sup>D</sup>1x<sup>∗</sup>

þ 2Q þ 2R þ 2V � 2Z þ 3,

,

þ Q þ R þ V � Z þ 1,

� <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð ÞD<sup>2</sup>

0

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

1bx<sup>∗</sup>y<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> ! � 1

� 1

" # !

" # !

D<sup>1</sup> ,

D<sup>1</sup>

D<sup>1</sup>

" # !

,

:

� <sup>D</sup>1x<sup>∗</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup>

� <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þy<sup>∗</sup> <sup>D</sup><sup>1</sup>

56 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

The Jacobian matrix at E<sup>3</sup> is

� f 0

f 0 <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þx<sup>∗</sup> D<sup>1</sup>

JE<sup>3</sup> ¼

where

and

c<sup>4</sup> ¼ f 0 1 s

c<sup>5</sup> ¼ f 0 1 s

c<sup>6</sup> ¼ f 0 1 s Theorem 7. Suppose that λ<sup>8</sup> < 0 and λ<sup>9</sup> < 0, then the equilibrium point E<sup>3</sup> is locally asymptotically stable.

#### 2.4. Global stability and uniform persistence analysis

Now we will analyse the global stability of the equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � and the existence of positive interior equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ. For global stability analysis, we use the similar technique as in [3, 18].

(i) Global stability analysis of E<sup>2</sup> ζs; 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � �

Consider system (3) restricted to R<sup>þ</sup> sx as in (5). Let N be a neighbourhood of equilibrium point E<sup>2</sup> ζs; 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 � � in <sup>R</sup><sup>þ</sup> sx. To analyse the global stability of the equilibrium point E2, a suitable Lyapunov function <sup>L</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>n</sup>1ð Þ <sup>s</sup> �b<sup>s</sup> <sup>2</sup> <sup>þ</sup> <sup>n</sup>2ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup> <sup>2</sup> � � is used, where <sup>b</sup><sup>s</sup> and <sup>b</sup><sup>x</sup> denote the components of <sup>E</sup>2ð Þ <sup>b</sup>s; <sup>b</sup><sup>x</sup> , i.e., <sup>b</sup><sup>s</sup> <sup>¼</sup> <sup>ζ</sup><sup>s</sup> and <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> , while n<sup>1</sup> and n<sup>2</sup> are positive constants. Note that L is a positive definite function with respect to E<sup>2</sup> in R<sup>þ</sup> sx and a Lyapunov function for system (5) in N . By differentiating L with respect to time t, we get

$$\frac{d\mathbf{L}}{dt} = n\_1(\mathbf{s} - \widehat{\mathbf{s}})\mathbf{s}' + n\_2(\mathbf{x} - \widehat{\mathbf{x}})\mathbf{x}',\tag{7}$$

where <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>�b<sup>s</sup> <sup>D</sup><sup>1</sup> . From (5), we have

$$1 = \widehat{\mathbf{s}} + \widehat{\mathbf{x}} f\_1(\widehat{\mathbf{s}}) \text{ and } D\_1 = f\_1(\widehat{\mathbf{s}}).$$

Hence (7) can be written as

$$\begin{split} L' &= n\_1(\mathbf{s} - \widehat{\mathbf{s}}) \left( \widehat{\mathbf{s}} + \widehat{\mathbf{x}} f\_1(\widehat{\mathbf{s}}) - \mathbf{s} - f\_1(\mathbf{s}) \mathbf{x} \right) + n\_2(\mathbf{x} - \widehat{\mathbf{x}}) \left( \frac{f\_1(\mathbf{s})}{f\_1(\widehat{\mathbf{s}})} - 1 \right) \mathbf{x} \\ &= -n\_1(\mathbf{s} - \widehat{\mathbf{s}})^2 + n\_1(\mathbf{s} - \widehat{\mathbf{s}}) \left( f\_1(\widehat{\mathbf{s}}) \widehat{\mathbf{x}} - f\_1(\mathbf{s}) \mathbf{x} \right) + n\_2(\mathbf{x} - \widehat{\mathbf{x}}) \mathbf{x} \left( \frac{f\_1(\mathbf{s})}{f\_1(\widehat{\mathbf{s}})} - 1 \right) \\ &= n\_{11}(\mathbf{s} - \widehat{\mathbf{s}})^2 + \frac{1}{2} n\_{12}(\mathbf{s} - \widehat{\mathbf{s}}) (\mathbf{x} - \widehat{\mathbf{x}}) + \frac{1}{2} n\_{21}(\mathbf{s} - \widehat{\mathbf{s}}) (\mathbf{x} - \widehat{\mathbf{x}}) + n\_{22}(\mathbf{x} - \widehat{\mathbf{x}})^2. \end{split}$$

where

$$n\_{11} = -n\_1 < 0,$$

$$n\_{12} = n\_{21} = n\_1 \frac{f\_1(\widehat{\mathbf{s}})\widehat{\mathbf{x}} - f\_1(\mathbf{s})\mathbf{x}}{\mathbf{x} - \widehat{\mathbf{x}}},$$

$$n\_{22} = n\_2 \frac{\mathbf{x}(f\_1(\mathbf{s}) - f\_1(\widehat{\mathbf{s}}))}{(\mathbf{x} - \widehat{\mathbf{x}})f\_1(\widehat{\mathbf{s}})}.$$

Clearly that <sup>L</sup><sup>0</sup> can be written as <sup>L</sup><sup>0</sup> <sup>¼</sup> XTNX, which <sup>T</sup> denotes the transpose and the matrix <sup>N</sup> is particularly a real, symmetric 2 � 2 matrix, where X and N can be represented by

$$X = \begin{pmatrix} \upsilon\_1 \\ \upsilon\_2 \end{pmatrix} = \begin{pmatrix} s - \widehat{s} \\ \varkappa - \widehat{\chi} \end{pmatrix} \text{ and } N = \begin{pmatrix} n\_{11} & \frac{1}{2} n\_{12} \\ 1 & n\_{21} \\ \frac{1}{2} n\_{21} & n\_{22} \end{pmatrix}.$$

Thus, it leads to the following theorem.

Theorem 8. The equilibrium point E<sup>2</sup> is global asymptotically stable with respect to solution trajectories are initiated from int R<sup>þ</sup> sx if the assumptions n<sup>22</sup> < 0 and det N > 0 are satisfied.

Proof. By using the Frobenius Theorem in ([18], Lemma 6.2), we can see that n<sup>22</sup> and det N are the leading principal minors of the matrix N: It is shown that matrix N is negative definite if

$$n\_{22} < 0, \text{ and } \det N = \det \begin{pmatrix} n\_{11} & \frac{1}{2} n\_{12} \\ 1 \\ \frac{1}{2} n\_{21} & n\_{22} \end{pmatrix} > 0.1$$

This completes the proof of the theorem. ∎

(ii) Existence of positive interior equilibrium point E<sup>3</sup>

In this subsection, we present some results of persistence analysis, including uniform persistence and state the necessary conditions for the existence of positive equilibrium point E3. The following lemma from Ref. [19] is applied to obtain the persistence results.

Lemma 2. [19] Let G be an isolated hyperbolic equilibrium point in the omega limit set, ωð Þ X of

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

Proof. This proof strictly depends on Lemma 2. Suppose Η is the region as stated in Theorem 1. It showed that region Η is positive invariant set and any solutions of system (3) emanating at

> ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

ð Þ E<sup>2</sup> \ Ef g<sup>2</sup> such that J

<sup>þ</sup>: Thus, the equilibrium point <sup>E</sup><sup>2</sup> is not in the omega limit set of <sup>P</sup>; <sup>E</sup>2∉ωð Þ <sup>P</sup> : Next, we

the closure of the orbit of the point J, i.e., clð Þ O Jð Þ must either contains E<sup>2</sup> or unbounded. This is

that the boundary flow is isolated and a cyclic with respect to region Η. Thus, the system (3) undergoes uniform persistence and implies that a positive interior equilibrium point

In this section, we investigate Hopf bifurcation on the system (3) with a bifurcation real parameter, σ: Particularly, σ is selected in such a way that the growth rate function f <sup>2</sup> is a

ð Þ <sup>E</sup><sup>2</sup> ; W<sup>s</sup>

is empty which is a contradiction for the positive invariance property of

<sup>þ</sup> is uniformly bounded. Despites that, the only compact invariant set on <sup>∂</sup>R<sup>3</sup>

<sup>þ</sup> and J

ð Þ <sup>G</sup> and <sup>W</sup><sup>u</sup>ð Þ <sup>G</sup> denote stable and unstable manifolds of <sup>G</sup>.

is a hyperbolic saddle point and is repels locally in y direction (as in Theorem <sup>5</sup>),

is globally asymptotically stable with respect to <sup>R</sup><sup>þ</sup>

� in ωð Þ X with J

http://dx.doi.org/10.5772/intechopen.71624

<sup>þ</sup>, i.e., <sup>P</sup><sup>∈</sup> int <sup>R</sup><sup>3</sup>

<sup>¼</sup> <sup>∅</sup>, and for unstable mani-

: This completes the proof. ∎

<sup>1</sup> ∈ ωð Þ P by Lemma 2. But

<sup>þ</sup> and <sup>J</sup> <sup>∈</sup> <sup>ω</sup>ð Þ <sup>P</sup> : Then

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 

sx indicates

<sup>þ</sup> belongs to <sup>ω</sup>ð Þ <sup>P</sup> , the omega limit set of <sup>P</sup>: Now,

<sup>∉</sup>ωð Þ <sup>P</sup> : Assume that <sup>E</sup><sup>2</sup> <sup>∈</sup> <sup>ω</sup>ð Þ <sup>P</sup> is true.

<sup>þ</sup> <sup>∩</sup> <sup>ω</sup>ð Þ¼ <sup>P</sup> <sup>∅</sup>: We deduce that if <sup>E</sup><sup>2</sup> <sup>ζ</sup>s;

with respect to <sup>R</sup><sup>þ</sup>

þ

þ

<sup>þ</sup> <sup>∩</sup> <sup>ω</sup>ð Þ <sup>P</sup> 6¼ <sup>∅</sup>, and let <sup>J</sup> <sup>∈</sup> <sup>∂</sup>R<sup>3</sup>

6¼ <sup>∅</sup>. Therefore, the result of uniform persistence follows

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

ð Þ <sup>E</sup><sup>2</sup> <sup>∩</sup> int <sup>R</sup><sup>3</sup>

þ

sxy are uniformly bounded and

sx.

<sup>þ</sup> ∈ W<sup>s</sup>

ð Þ G

59

<sup>þ</sup> is

<sup>þ</sup>. We shall

the orbit Oð Þ X : Then either ωð Þ¼ X G or there exist points J

ii. system (3) is dissipative and all solutions with initial values in R<sup>þ</sup>

. Let <sup>P</sup> <sup>¼</sup> <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ belongs to the interior of <sup>R</sup><sup>3</sup>

þ <sup>1</sup> ∈ W<sup>s</sup>

<sup>þ</sup> <sup>∩</sup> <sup>ω</sup>ð Þ¼ <sup>P</sup> <sup>∅</sup>: Assume that <sup>∂</sup>R<sup>3</sup>

þ

since the omega limit set of <sup>P</sup>, <sup>ω</sup>ð Þ <sup>P</sup> must be in int <sup>R</sup><sup>3</sup>

Remark 2. Global stability of equilibrium point E<sup>2</sup> ζs;

function of x and σ: Therefore, system (3) takes of the form

<sup>þ</sup> where <sup>∂</sup>R<sup>3</sup>

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

attracted into region Η (as in Theorem 1), and

and J

i. E<sup>2</sup> ζs;

a point in R<sup>3</sup>

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

ð Þ <sup>E</sup><sup>2</sup> <sup>∩</sup> <sup>R</sup><sup>3</sup>

shall show ∂R<sup>3</sup>

E<sup>2</sup> ζs;

W<sup>s</sup>

Η ⊂ R<sup>3</sup>

� <sup>∈</sup> <sup>W</sup><sup>u</sup>ð Þ <sup>G</sup> , where <sup>W</sup><sup>s</sup>

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0

iii. equilibrium point E<sup>2</sup> ζs;

Then, the system (3) is uniformly persistence.

show that there is no points Ji ∈∂R<sup>3</sup>

Then, there is a point J

<sup>þ</sup>\ Ef g<sup>2</sup>

fold <sup>W</sup><sup>u</sup>ð Þ <sup>E</sup><sup>2</sup> ; W<sup>u</sup>ð Þ <sup>E</sup><sup>2</sup> <sup>∩</sup> int <sup>R</sup><sup>3</sup>

<sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ exists (see [20]).

2.5. Hopf bifurcation

we prove that the equilibrium point E<sup>2</sup> ζs;

a contradiction, and hence it is proved that ∂R<sup>3</sup>

is unstable, then, for stable manifold W<sup>s</sup>

Theorem 9. Assume that

Lemma 2. [19] Let G be an isolated hyperbolic equilibrium point in the omega limit set, ωð Þ X of the orbit Oð Þ X : Then either ωð Þ¼ X G or there exist points J <sup>þ</sup> and J � in ωð Þ X with J <sup>þ</sup> ∈ W<sup>s</sup> ð Þ G and J � <sup>∈</sup> <sup>W</sup><sup>u</sup>ð Þ <sup>G</sup> , where <sup>W</sup><sup>s</sup> ð Þ <sup>G</sup> and <sup>W</sup><sup>u</sup>ð Þ <sup>G</sup> denote stable and unstable manifolds of <sup>G</sup>.

Theorem 9. Assume that

<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>s</sup> <sup>þ</sup> <sup>b</sup>xf <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> and <sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> :

<sup>2</sup> <sup>þ</sup> <sup>n</sup>1ð Þ <sup>s</sup> �b<sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> <sup>b</sup><sup>x</sup> � <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>x</sup> � � <sup>þ</sup> <sup>n</sup>2ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup> <sup>x</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

1 2

n<sup>11</sup> ¼ �n<sup>1</sup> < 0,

n<sup>12</sup> ¼ n<sup>21</sup> ¼ n<sup>1</sup>

Clearly that <sup>L</sup><sup>0</sup> can be written as <sup>L</sup><sup>0</sup> <sup>¼</sup> XTNX, which <sup>T</sup> denotes the transpose and the matrix <sup>N</sup> is

Theorem 8. The equilibrium point E<sup>2</sup> is global asymptotically stable with respect to solution trajecto-

Proof. By using the Frobenius Theorem in ([18], Lemma 6.2), we can see that n<sup>22</sup> and det N are the leading principal minors of the matrix N: It is shown that matrix N is negative definite if

This completes the proof of the theorem. ∎

In this subsection, we present some results of persistence analysis, including uniform persistence and state the necessary conditions for the existence of positive equilibrium point E3. The

sx if the assumptions n<sup>22</sup> < 0 and det N > 0 are satisfied.

n<sup>11</sup>

1 2

0 B@ 1 2 n<sup>12</sup> 1

CA <sup>&</sup>gt; <sup>0</sup>:

n<sup>21</sup> n<sup>22</sup>

particularly a real, symmetric 2 � 2 matrix, where X and N can be represented by

<sup>¼</sup> <sup>s</sup> �b<sup>s</sup> <sup>x</sup> � <sup>b</sup><sup>x</sup> � �

n<sup>22</sup> < 0, and det N ¼ det

following lemma from Ref. [19] is applied to obtain the persistence results.

n<sup>22</sup> ¼ n<sup>2</sup>

and N ¼

<sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> � <sup>1</sup> � �

<sup>n</sup>21ð Þ <sup>s</sup> �b<sup>s</sup> ð Þþ <sup>x</sup> � <sup>b</sup><sup>x</sup> <sup>n</sup>22ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup>

<sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> <sup>b</sup><sup>x</sup> � <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>x</sup> <sup>x</sup> � <sup>b</sup><sup>x</sup> ,

x f <sup>1</sup>ð Þ� <sup>s</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> � � ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> :

n<sup>11</sup>

1 2

0 B@ 1 2 n<sup>12</sup> 1 CA:

n<sup>21</sup> n<sup>22</sup>

x

<sup>f</sup> <sup>1</sup>ð Þ <sup>b</sup><sup>s</sup> � <sup>1</sup> � �

> 2 ,

<sup>L</sup><sup>0</sup> <sup>¼</sup> <sup>n</sup>1ð Þ <sup>s</sup> �b<sup>s</sup> <sup>b</sup><sup>s</sup> <sup>þ</sup> <sup>b</sup>xf <sup>1</sup>ð Þ� <sup>b</sup><sup>s</sup> <sup>s</sup> � <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> <sup>x</sup> � � <sup>þ</sup> <sup>n</sup>2ð Þ <sup>x</sup> � <sup>b</sup><sup>x</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup>

<sup>n</sup>12ð Þ <sup>s</sup> �b<sup>s</sup> ð Þþ <sup>x</sup> � <sup>b</sup><sup>x</sup>

Hence (7) can be written as

where

¼ �n1ð Þ <sup>s</sup> �b<sup>s</sup>

<sup>¼</sup> <sup>n</sup>11ð Þ <sup>s</sup> �b<sup>s</sup>

58 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>X</sup> <sup>¼</sup> <sup>v</sup><sup>1</sup> v2 � �

(ii) Existence of positive interior equilibrium point E<sup>3</sup>

Thus, it leads to the following theorem.

ries are initiated from int R<sup>þ</sup>


Then, the system (3) is uniformly persistence.

Proof. This proof strictly depends on Lemma 2. Suppose Η is the region as stated in Theorem 1. It showed that region Η is positive invariant set and any solutions of system (3) emanating at a point in R<sup>3</sup> <sup>þ</sup> is uniformly bounded. Despites that, the only compact invariant set on <sup>∂</sup>R<sup>3</sup> <sup>þ</sup> is E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 . Let <sup>P</sup> <sup>¼</sup> <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ belongs to the interior of <sup>R</sup><sup>3</sup> <sup>þ</sup>, i.e., <sup>P</sup><sup>∈</sup> int <sup>R</sup><sup>3</sup> <sup>þ</sup>. We shall show that there is no points Ji ∈∂R<sup>3</sup> <sup>þ</sup> where <sup>∂</sup>R<sup>3</sup> <sup>þ</sup> belongs to <sup>ω</sup>ð Þ <sup>P</sup> , the omega limit set of <sup>P</sup>: Now, we prove that the equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 <sup>∉</sup>ωð Þ <sup>P</sup> : Assume that <sup>E</sup><sup>2</sup> <sup>∈</sup> <sup>ω</sup>ð Þ <sup>P</sup> is true. Then, there is a point J þ <sup>1</sup> ∈ W<sup>s</sup> ð Þ E<sup>2</sup> \ Ef g<sup>2</sup> such that J þ <sup>1</sup> ∈ ωð Þ P by Lemma 2. But W<sup>s</sup> ð Þ <sup>E</sup><sup>2</sup> <sup>∩</sup> <sup>R</sup><sup>3</sup> <sup>þ</sup>\ Ef g<sup>2</sup> is empty which is a contradiction for the positive invariance property of Η ⊂ R<sup>3</sup> <sup>þ</sup>: Thus, the equilibrium point <sup>E</sup><sup>2</sup> is not in the omega limit set of <sup>P</sup>; <sup>E</sup>2∉ωð Þ <sup>P</sup> : Next, we shall show ∂R<sup>3</sup> <sup>þ</sup> <sup>∩</sup> <sup>ω</sup>ð Þ¼ <sup>P</sup> <sup>∅</sup>: Assume that <sup>∂</sup>R<sup>3</sup> <sup>þ</sup> <sup>∩</sup> <sup>ω</sup>ð Þ <sup>P</sup> 6¼ <sup>∅</sup>, and let <sup>J</sup> <sup>∈</sup> <sup>∂</sup>R<sup>3</sup> <sup>þ</sup> and <sup>J</sup> <sup>∈</sup> <sup>ω</sup>ð Þ <sup>P</sup> : Then the closure of the orbit of the point J, i.e., clð Þ O Jð Þ must either contains E<sup>2</sup> or unbounded. This is a contradiction, and hence it is proved that ∂R<sup>3</sup> <sup>þ</sup> <sup>∩</sup> <sup>ω</sup>ð Þ¼ <sup>P</sup> <sup>∅</sup>: We deduce that if <sup>E</sup><sup>2</sup> <sup>ζ</sup>s; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 is unstable, then, for stable manifold W<sup>s</sup> ð Þ <sup>E</sup><sup>2</sup> ; W<sup>s</sup> ð Þ <sup>E</sup><sup>2</sup> <sup>∩</sup> int <sup>R</sup><sup>3</sup> þ <sup>¼</sup> <sup>∅</sup>, and for unstable manifold <sup>W</sup><sup>u</sup>ð Þ <sup>E</sup><sup>2</sup> ; W<sup>u</sup>ð Þ <sup>E</sup><sup>2</sup> <sup>∩</sup> int <sup>R</sup><sup>3</sup> þ 6¼ <sup>∅</sup>. Therefore, the result of uniform persistence follows since the omega limit set of <sup>P</sup>, <sup>ω</sup>ð Þ <sup>P</sup> must be in int <sup>R</sup><sup>3</sup> þ : This completes the proof. ∎

Remark 2. Global stability of equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0 with respect to <sup>R</sup><sup>þ</sup> sx indicates that the boundary flow is isolated and a cyclic with respect to region Η. Thus, the system (3) undergoes uniform persistence and implies that a positive interior equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ exists (see [20]).

#### 2.5. Hopf bifurcation

In this section, we investigate Hopf bifurcation on the system (3) with a bifurcation real parameter, σ: Particularly, σ is selected in such a way that the growth rate function f <sup>2</sup> is a function of x and σ: Therefore, system (3) takes of the form

$$\begin{cases} \mathbf{s}' = \mathbf{1} - \mathbf{s} - f\_1(\mathbf{s})\mathbf{x}, \\\\ \mathbf{x}' = \left(\frac{f\_1(\mathbf{s})}{D\_1} - \mathbf{1}\right) \mathbf{x} - f\_2(\mathbf{x}; \sigma) y \left(\mathbf{1} + \frac{D\_1 \mathbf{x}}{1 + \mathbf{x} + D\_1 \mathbf{b} y}\right), \\\\ \mathbf{y}' = f\_2(\mathbf{x}; \sigma) y \left(\mathbf{1} + \frac{D\_1 \mathbf{x}}{1 + \mathbf{x} + D\_1 \mathbf{b} y}\right) - y, \end{cases} \tag{8}$$

where sð Þ0 > 0, xð Þ0 > 0, yð Þ0 > 0: Next, we do linearization on the system (8). First, let

$$\begin{cases} \mathbf{S} = \mathbf{s} - h\_1; \\\\ \mathbf{X} = \mathbf{x} - h\_2; \quad \Rightarrow \\\\ \mathbf{Y} = \mathbf{y} - h\_3; \end{cases} \Rightarrow \begin{cases} \mathbf{s} = \mathbf{S} + h\_1; \\\\ \mathbf{x} = \mathbf{X} + h\_2; \\\\ \mathbf{y} = \mathbf{Y} + h\_3; \end{cases}$$

where ð Þ h1; h2; h<sup>3</sup> is the non-trivial equilibrium point. Then, we obtain the following differential equations

$$\begin{cases} \begin{aligned} S' &= 1 - S - h\_1 - f\_1(S + h\_1)(X + h\_2), \\ X' &= \left(\frac{f\_1(S + h\_1)}{D\_1} - 1\right)(X + h\_2) - f\_2(X + h\_2; \sigma)(Y + h\_3) \left(1 + \frac{D\_1(X + h\_2)}{1 + (X + h\_2) + D\_1b(Y + h\_3)}\right), \\ &\quad Y' = f\_2(X + h\_2; \sigma)(Y + h\_3) \left(1 + \frac{D\_1(X + h\_2)}{1 + (X + h\_2) + D\_1b(Y + h\_3)}\right) - (Y + h\_3). \end{aligned} \end{cases} \tag{9}$$

The Jacobian matrix of system (8) is given by

$$J\_{4} = \begin{bmatrix} -f\_{1}'(\mathbf{s})\mathbf{x} - \mathbf{1} & -f\_{1}(\mathbf{s})\\ \frac{f\_{1}'(\mathbf{s})\mathbf{x}}{D\_{1}} & \frac{f\_{1}(\mathbf{s})}{D\_{1}} - f\_{2}(\mathbf{x};\sigma)y\left[\left(\frac{D\_{1}}{\mathbf{1}+\mathbf{x}+D\_{1}\mathbf{b}}\right) - \left(\frac{D\_{1}\mathbf{x}}{\left(\mathbf{1}+\mathbf{x}+D\_{1}\mathbf{b}\right)^{2}}\right)\right] - f\_{2}'(\mathbf{x};\sigma)y\left(\left(\frac{D\_{1}\mathbf{x}}{\mathbf{1}+\mathbf{x}+D\_{1}\mathbf{b}}\right) + 1\right) - 1\\ 0 & f\_{2}(\mathbf{x};\sigma)y\left[\left(\frac{D\_{1}}{\mathbf{1}+\mathbf{x}+D\_{1}\mathbf{b}}\mathbf{y}\right) - \left(\frac{D\_{1}\mathbf{x}}{\left(\mathbf{1}+\mathbf{x}+D\_{1}\mathbf{b}\right)^{2}}\right)\right] + f\_{2}'(\mathbf{x};\sigma)y\left(\mathbf{1} + \frac{D\_{1}\mathbf{x}}{\mathbf{1}+\mathbf{x}+D\_{1}\mathbf{b}}\right) \end{bmatrix}$$

$$\begin{aligned} f\_2(\mathbf{x}; \sigma) \left[ \left( \frac{D\_1^2 b \mathbf{x} \mathbf{y}}{\left(1 + \mathbf{x} + D\_1 b \mathbf{y}\right)^2} \right) - \left( \frac{D\_1 \mathbf{x}}{1 + \mathbf{x} + D\_1 b \mathbf{y}} + 1 \right) \right] \\ f\_2(\mathbf{x}; \sigma) \left[ \left( 1 + \frac{D\_1 \mathbf{x}}{1 + \mathbf{x} + D\_1 b \mathbf{y}} \right) - \left( \frac{D\_1^2 b \mathbf{x} \mathbf{y}}{\left(1 + \mathbf{x} + D\_1 b \mathbf{y}\right)^2} \right) \right] - 1 \end{aligned}$$

:

J<sup>σ</sup>ð Þ <sup>E</sup><sup>2</sup> ¼

where

� f 0

0

BBBBBBBBBBBBBBBB@

f 0

<sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

þ D1f2

<sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

� f <sup>2</sup>

� f <sup>2</sup>

obtain the matrices

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þ 1 � ζ<sup>s</sup> D<sup>1</sup>

<sup>1</sup>ð Þ ζ<sup>s</sup> ð Þ 1 � ζ<sup>s</sup> D2 1

The characteristic equation of (10) is given as

<sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup>

1 � ζ<sup>s</sup> D<sup>1</sup>

<sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � � <sup>2</sup>D<sup>1</sup> <sup>þ</sup> <sup>4</sup>D<sup>2</sup>

; σ � �

� f <sup>1</sup>ð Þ ζ<sup>s</sup> 2 þ 2D<sup>1</sup> � 2ζ<sup>s</sup> þ f <sup>2</sup>

<sup>c</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>

�f <sup>2</sup>

�f <sup>2</sup>

� 1 �f <sup>1</sup>ð Þ ζ<sup>s</sup> 0

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � � <sup>1</sup> <sup>þ</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � � <sup>1</sup> <sup>þ</sup> 1 � ζ<sup>s</sup>

http://dx.doi.org/10.5772/intechopen.71624

1 � ζ<sup>s</sup> D<sup>1</sup> � �

1

CCA

1

1

1

61

CCCCCCCCCCCCCCCCA :

(10)

CCA

CCA

1

CCA � <sup>1</sup>

1 þ

1 � ζ<sup>s</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> � �

0

0

BB@

0

BB@

<sup>s</sup> � D1ζ<sup>s</sup> � � <sup>þ</sup> f1 <sup>ζ</sup><sup>s</sup> ð Þð Þ <sup>ζ</sup><sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>1</sup> �

ðζ<sup>s</sup> þ D1ζ<sup>s</sup> � 2D<sup>1</sup> � 1Þ þ 3D1ð ÞÞ 1 þ D<sup>1</sup> � ζ<sup>s</sup> ,

<sup>s</sup> þ 2 þ 2D<sup>1</sup> � 4ζ<sup>s</sup> � 2D1ζ<sup>s</sup>

� ��

� �

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �

<sup>1</sup> � <sup>2</sup>D1ζ<sup>s</sup> � <sup>2</sup>D<sup>2</sup>

0

; σ

We applied Routh Hurwitz criterion (see [21, 22]) onto the characteristic equation (11) and

� �

<sup>s</sup> � <sup>3</sup>D1ζ<sup>s</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup>

1ζs � � <sup>þ</sup> <sup>3</sup>D1ζ<sup>s</sup> � <sup>3</sup>D<sup>2</sup>

<sup>s</sup> � <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup>

<sup>1</sup> <sup>ζ</sup><sup>s</sup> ð Þ <sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

� ��

� � <sup>ζ</sup><sup>s</sup> � <sup>1</sup> � <sup>2</sup>D<sup>1</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>s</sup> <sup>ð</sup> Þ þ <sup>1</sup> � <sup>ζ</sup><sup>s</sup> � �

ð Þþ <sup>D</sup><sup>1</sup> � <sup>D</sup>1ζ<sup>s</sup> <sup>D</sup>1ζ<sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>D</sup><sup>2</sup>

BB@

0

BB@

<sup>λ</sup><sup>3</sup> <sup>þ</sup> <sup>c</sup>1λ<sup>2</sup> <sup>þ</sup> <sup>c</sup>2<sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>, (11)

s

ð Þ ζ<sup>s</sup> � 2D<sup>1</sup> � 1 þ D1ζ<sup>s</sup>

s

<sup>s</sup> � D1ζ<sup>s</sup>

1Þ:

<sup>1</sup> � 3D1Þ,

1 þ

� 1 �f <sup>2</sup>

<sup>1</sup> <sup>ζ</sup><sup>s</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

f <sup>1</sup>ð Þ ζ<sup>s</sup> D<sup>1</sup>

0 0 f <sup>2</sup>

0

0 <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>2</sup>ζ<sup>2</sup>

� �

� � <sup>1</sup> <sup>þ</sup> <sup>2</sup>D<sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

<sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup>

; σ � � <sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

; σ � �

1 � ζ<sup>s</sup> D<sup>1</sup>

1 � ζ<sup>s</sup> D<sup>1</sup>

1 � ζ<sup>s</sup> D<sup>1</sup>

�f <sup>1</sup> ζ<sup>s</sup> ð Þ f <sup>2</sup>

Thus, the Jacobian matrix of system (8) about E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ � � is Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation… http://dx.doi.org/10.5772/intechopen.71624 61

$$f\_{o(E\_{2})} = \begin{pmatrix} -\frac{f\_{1}'(\zeta\_{s})(1-\zeta\_{s})}{D\_{1}} - 1 & -f\_{1}(\zeta\_{s}) & 0\\ \frac{f\_{1}'(\zeta\_{s})(1-\zeta\_{s})}{D\_{1}^{2}} & \frac{f\_{1}(\zeta\_{s})}{D\_{1}} - 1 & -f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}};\sigma\right)\left(1 + \left(\frac{1-\zeta\_{s}}{1+\left(\frac{1-\zeta\_{s}}{D\_{1}}\right)}\right)\right) \\\\ 0 & 0 & f\_{2}\left(\frac{1-\zeta\_{s}}{D\_{1}};\sigma\right)\left(1 + \left(\frac{1-\zeta\_{s}}{1+\left(\frac{1-\zeta\_{s}}{D\_{1}}\right)}\right)\right) - 1\\\\ \end{pmatrix}.\tag{10}$$

The characteristic equation of (10) is given as

$$
\lambda^3 + c\_1 \lambda^2 + c\_2 \lambda + c\_3 = 0,\tag{11}
$$

where

s<sup>0</sup> ¼ 1 � s � f <sup>1</sup>ð Þs x,

where sð Þ0 > 0, xð Þ0 > 0, yð Þ0 > 0: Next, we do linearization on the system (8). First, let

)

where ð Þ h1; h2; h<sup>3</sup> is the non-trivial equilibrium point. Then, we obtain the following differential

S<sup>0</sup> ¼ 1 � S � h<sup>1</sup> � f <sup>1</sup>ð Þ S þ h<sup>1</sup> ð Þ X þ h<sup>2</sup> ,

ð Þ� X þ h<sup>2</sup> f <sup>2</sup>ð Þ X þ h2; σ ð Þ Y þ h<sup>3</sup> 1 þ

x � f <sup>2</sup>ð Þ x; σ y 1 þ

D1x 1 þ x þ D1by � �

> 8 >>>>><

> >>>>>:

D1ð Þ X þ h<sup>2</sup> 1 þ ð Þþ X þ h<sup>2</sup> D1b Yð Þ þ h<sup>3</sup> � �

> � <sup>D</sup>1<sup>x</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup>

� <sup>D</sup>1<sup>x</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup>

<sup>f</sup> <sup>2</sup>ð Þ <sup>x</sup>; <sup>σ</sup> <sup>D</sup><sup>2</sup>

f <sup>2</sup>ð Þ x; σ 1 þ

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ � �

" # !

" # !

s ¼ S þ h1;

x ¼ X þ h2;

y ¼ Y þ h3;

D1x 1 þ x þ D1by � �

� y,

,

D1ð Þ X þ h<sup>2</sup> 1 þ ð Þþ X þ h<sup>2</sup> D1b Yð Þ þ h<sup>3</sup> � �

� ð Þ Y þ h<sup>3</sup> :

D1x 1 þ x þ D1by � �

� �

D1x 1 þ x þ D1by � �

� f <sup>2</sup> 0 ð Þ x; σ y

þ f 0 <sup>2</sup>ð Þ x; σ y 1 þ

0

" # � �

" # !

� <sup>D</sup>1<sup>x</sup>

� <sup>D</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>þ</sup> <sup>1</sup>

<sup>1</sup>bxy ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup>

<sup>1</sup>bxy ð Þ <sup>1</sup> <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>D</sup>1by <sup>2</sup> !

D1x 1 þ x þ D1by � �

is

(8)

,

(9)

� 1

þ 1

� 1

:

<sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> D<sup>1</sup> � 1 � �

60 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

8 >>>>><

>>>>>:

8

>>>>>>>><

>>>>>>>>:

� 1

The Jacobian matrix of system (8) is given by

� f <sup>2</sup>ð Þ x; σ y

Thus, the Jacobian matrix of system (8) about E<sup>2</sup> ζs;

f <sup>1</sup>ð Þs D<sup>1</sup>

0 f <sup>2</sup>ð Þ x; σ y

<sup>1</sup>ð Þs x � 1 �f <sup>1</sup>ð Þs

D<sup>1</sup> 1 þ x þ D1by � �

D<sup>1</sup> 1 þ x þ D1by � �

Y<sup>0</sup> ¼ f <sup>2</sup>ð Þ X þ h2; σ ð Þ Y þ h<sup>3</sup> 1 þ

� �

equations

8

>>>>>>>>><

>>>>>>>>>:

J<sup>σ</sup> ¼

�f 0

f 0 <sup>1</sup>ð Þs x D<sup>1</sup>

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>S</sup> <sup>þ</sup> <sup>h</sup><sup>1</sup> D<sup>1</sup>

y<sup>0</sup> ¼ f <sup>2</sup>ð Þ x; σ y 1 þ

S ¼ s � h1;

X ¼ x � h2;

Y ¼ y � h3;

<sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup> 0 <sup>1</sup> <sup>ζ</sup><sup>s</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> <sup>s</sup> � D1ζ<sup>s</sup> � � <sup>þ</sup> f1 <sup>ζ</sup><sup>s</sup> ð Þð Þ <sup>ζ</sup><sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>1</sup> � þ D1f2 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �ðζ<sup>s</sup> þ D1ζ<sup>s</sup> � 2D<sup>1</sup> � 1Þ þ 3D1ð ÞÞ 1 þ D<sup>1</sup> � ζ<sup>s</sup> , <sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>D</sup>1ð Þ <sup>D</sup><sup>1</sup> � <sup>ζ</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup> 0 <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>2</sup>ζ<sup>2</sup> <sup>s</sup> þ 2 þ 2D<sup>1</sup> � 4ζ<sup>s</sup> � 2D1ζ<sup>s</sup> � � � f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>D<sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> <sup>s</sup> � <sup>3</sup>D1ζ<sup>s</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup> s � �� � f <sup>1</sup>ð Þ ζ<sup>s</sup> 2 þ 2D<sup>1</sup> � 2ζ<sup>s</sup> þ f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �ð Þ ζ<sup>s</sup> � 2D<sup>1</sup> � 1 þ D1ζ<sup>s</sup> � � � f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � � <sup>2</sup>D<sup>1</sup> <sup>þ</sup> <sup>4</sup>D<sup>2</sup> <sup>1</sup> � <sup>2</sup>D1ζ<sup>s</sup> � <sup>2</sup>D<sup>2</sup> 1ζs � � <sup>þ</sup> <sup>3</sup>D1ζ<sup>s</sup> � <sup>3</sup>D<sup>2</sup> <sup>1</sup> � 3D1Þ,

$$\begin{split} c\_{3} &= \frac{1}{D\_{1}(D\_{1}-\zeta\_{\sf s}+1)} \left( f\_{1}^{\prime}(\zeta\_{\sf s}) \left( 1-2\zeta\_{\sf s}+D\_{1}+\zeta\_{\sf s}^{2}-D\_{1}\zeta\_{\sf s} \right) \\ &- f\_{2} \left( \frac{1-\zeta\_{\sf s}}{D\_{1}};\sigma \right) \left( 1-2\zeta\_{\sf s}+\zeta\_{\sf s}^{2}-D\_{1}+D\_{1}\zeta\_{\sf s}^{2} \right) \right) \\ &- f\_{1}(\zeta\_{\sf s}) \left( f\_{2}\left( \frac{1-\zeta\_{\sf s}}{D\_{1}};\sigma \right) \left( \zeta\_{\sf s}-1-2D\_{1}+D\_{1}\zeta\_{\sf s} \right)+1-\zeta\_{\sf s} \right) \\ &- f\_{2} \left( \frac{1-\zeta\_{\sf s}}{D\_{1}};\sigma \right) \left( D\_{1}-D\_{1}\zeta\_{\sf s} \right) + D\_{1}\zeta\_{\sf s}-D\_{1}-D\_{1}^{2} \right). \end{split}$$

We applied Routh Hurwitz criterion (see [21, 22]) onto the characteristic equation (11) and obtain the matrices

$$M\_1 = [c\_1]; M\_2 = \begin{bmatrix} c\_1 & 1\\ c\_3 & c\_2 \end{bmatrix}; \ M\_3 = \begin{bmatrix} c\_1 & 1 & 0\\ c\_3 & c\_2 & c\_1\\ 0 & 0 & c\_3 \end{bmatrix}.$$

Thus, the characteristic equation (11) has all negative real parts of λ if and only if

$$\begin{cases} c\_1 > 0 \\ c\_3 > 0 \\ c\_1 c\_2 - c\_3 > 0. \end{cases} \tag{12}$$

Re dλ<sup>j</sup> dσ � �

Kð Þ σ α<sup>0</sup>

Lð Þ σ α<sup>0</sup>

<sup>K</sup>ð Þ¼ <sup>σ</sup> <sup>3</sup>α<sup>2</sup>

<sup>M</sup>ð Þ¼ <sup>σ</sup> <sup>α</sup><sup>2</sup>

1 � �N σ<sup>∗</sup> 1

> <sup>σ</sup>¼σ<sup>∗</sup> 1

Theorem 10. Suppose that the equilibrium point E<sup>2</sup> ζs;

<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ D<sup>1</sup>

<sup>0</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup><sup>1</sup>

Re dλ<sup>j</sup> dσ � �

1

where

Since K σ<sup>∗</sup>

and λ<sup>3</sup> σ<sup>∗</sup> 1 � � ¼ �c<sup>1</sup> <sup>σ</sup><sup>∗</sup>

from E<sup>2</sup> ζs;

J<sup>σ</sup>ð Þ <sup>E</sup><sup>3</sup> ¼

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ

� f 0

> f 0 <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þx<sup>∗</sup> D<sup>1</sup>

1 � �M σ<sup>∗</sup> 1 � � <sup>þ</sup> <sup>L</sup> <sup>σ</sup><sup>∗</sup>

and calculating the implicit derivative, we obtain the following equations

ð Þ σ c 0 <sup>1</sup>ð Þþ σ c 0

� � 6¼ <sup>0</sup>, we have

� � for some values of <sup>σ</sup> in the neighbourhood of <sup>σ</sup><sup>∗</sup>

<sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þx<sup>∗</sup> � <sup>1</sup> �<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ

1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � �

1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � �

matrix of the system (8) about the equilibrium point E<sup>3</sup> is given by

� <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup><sup>1</sup>

Hence, the characteristic equation for the Jacobian matrix J<sup>σ</sup>ð Þ <sup>E</sup><sup>3</sup> is

Nð Þ¼ σ 2α σð Þβ σð Þc

<sup>¼</sup> <sup>K</sup> <sup>σ</sup><sup>∗</sup> 1 � �M σ<sup>∗</sup> 1 � � <sup>þ</sup> <sup>L</sup> <sup>σ</sup><sup>∗</sup>

<sup>σ</sup>¼σ<sup>∗</sup> 1

ð Þ� σ Lð Þ σ β<sup>0</sup>

ð Þþ σ Kð Þ σ β<sup>0</sup>

Lð Þ¼ σ 6α σð Þβ σð Þþ 2c1ð Þ σ β σð Þ;

K<sup>2</sup> σ<sup>∗</sup> 1 � � <sup>þ</sup> <sup>L</sup><sup>2</sup> <sup>σ</sup><sup>∗</sup>

0 <sup>1</sup>ð Þþ σ c 0 <sup>2</sup>ð Þ σ β σð Þ:

By substituting λ1ð Þ¼ σ α σð Þþ β σð Þi and λ2ð Þ¼ σ α σð Þ� β σð Þi into characteristic equation (11)

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

ð Þþ σ Mð Þ¼ σ 0,

ð Þþ σ Nð Þ¼ σ 0,

ð Þþ <sup>σ</sup> <sup>2</sup>c1ð Þ <sup>σ</sup> α σð Þþ <sup>c</sup>2ð Þ� <sup>σ</sup> <sup>3</sup>β<sup>2</sup>

<sup>2</sup>ð Þ σ α σð Þ� c<sup>0</sup>

1 � �N σ<sup>∗</sup> 1 � �

� � 6¼ <sup>0</sup>: We conclude the details above in the following theorem.

similar as in (2) for the system (3) together with hypothesis H<sup>1</sup> until H<sup>3</sup> hold. Then the system (8) undergoes Hopf bifurcation in the first octant, which leads to a family of periodic solutions bifurcating

Next, we determine the Hopf bifurcation at the equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> : The Jacobian

� <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup>

� <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup>

<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>D</sup><sup>2</sup>

<sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>D</sup>1x<sup>∗</sup>

<sup>λ</sup><sup>3</sup> <sup>þ</sup> <sup>c</sup>4λ<sup>2</sup> <sup>þ</sup> <sup>c</sup>5<sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>6</sup> <sup>¼</sup> <sup>0</sup>, (16)

" # !

" # !

1

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ

6¼ 0, j ¼ 1, 2, 3: (14)

http://dx.doi.org/10.5772/intechopen.71624

(15)

63

�

<sup>1</sup>ð Þ <sup>σ</sup> <sup>β</sup><sup>2</sup>

� � 6¼ <sup>0</sup>, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>,

1:

� f <sup>2</sup>

þ f 0

1bx<sup>∗</sup>y<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> � <sup>D</sup>1x<sup>∗</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> � <sup>D</sup><sup>2</sup>

<sup>0</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup>1x<sup>∗</sup>

<sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup>1x<sup>∗</sup>

0

!

!

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> � <sup>1</sup>

1bx<sup>∗</sup>y<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>1</sup> � 1

� 1

ð Þ σ ;

ð Þþ σ c<sup>0</sup>

<sup>3</sup>ð Þ σ ;

� � exists and those assumptions

When the assumptions on functional responses f <sup>1</sup>ð Þ¼ s f <sup>1</sup>ð Þ ζ<sup>s</sup> and f <sup>2</sup>ð Þ¼ x f <sup>2</sup> 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; σ � � as in (2) for the system (3), together with the hypotheses H<sup>1</sup> until H3;

H<sup>1</sup> : D<sup>1</sup> � ζ<sup>s</sup> þ 1 > 0, H<sup>2</sup> : f 0 <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>1</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> <sup>s</sup> � D1ζ<sup>s</sup> � � <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> ð Þ <sup>ζ</sup><sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>1</sup> þ D1f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �ðζ<sup>s</sup> þ D1ζ<sup>s</sup> � 2D<sup>1</sup> � 1Þ þ 3D1ð Þ 1 þ D<sup>1</sup> � ζ<sup>s</sup> > 0, H<sup>3</sup> : f 0 <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> <sup>s</sup> � D1ζ<sup>s</sup> � f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � � <sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> <sup>s</sup> � <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup> s � � � � � f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �ðζ<sup>s</sup> � 1 � 2D<sup>1</sup> þ D1ζsÞ þ 1 � ζ<sup>s</sup> � � � f <sup>2</sup> 1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �ð Þþ <sup>D</sup><sup>1</sup> � <sup>D</sup>1ζ<sup>s</sup> <sup>D</sup>1ζ<sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>D</sup><sup>2</sup> <sup>1</sup> > 0,

hold, we will have c<sup>1</sup> > 0 and c<sup>3</sup> > 0: We shall obtain two pure imaginary roots for the characteristic equation (11) if and only if <sup>c</sup>1c<sup>2</sup> <sup>¼</sup> <sup>c</sup><sup>3</sup> for some values of <sup>σ</sup>, say, <sup>σ</sup><sup>∗</sup> 1:

Since at <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup> <sup>1</sup>, there exists an interval σ<sup>∗</sup> <sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup> � � containing <sup>σ</sup><sup>∗</sup> <sup>1</sup> for some ε > 0 such that σ∈ σ<sup>∗</sup> <sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup> � �. Thus, for <sup>σ</sup><sup>∈</sup> <sup>σ</sup><sup>∗</sup> <sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup> � �, the characteristic equation (11) cannot have positive real roots. For <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup> <sup>1</sup>, we acquire (see [3, 10])

$$(\lambda^2 + c\_2)(\lambda + c\_1) = 0,\tag{13}$$

that consist of three roots; <sup>λ</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffi c2 <sup>p</sup> i, <sup>λ</sup><sup>2</sup> ¼ � ffiffiffiffi c2 <sup>p</sup> i, and <sup>λ</sup><sup>3</sup> ¼ �c1: As for <sup>σ</sup><sup>∈</sup> <sup>σ</sup><sup>∗</sup> <sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup> � �, all the roots are in general of the form;

$$
\lambda\_1(\sigma) = \alpha(\sigma) + \beta(\sigma)i,
$$

$$
\lambda\_2(\sigma) = \alpha(\sigma) - \beta(\sigma)i,
$$

$$
\lambda\_3(\sigma) = -\mathbf{c}\_1(\sigma).
$$

In order to apply the Hopf bifurcation theorem as stated in [9, 23] towards the system (8), we must verify the transversality condition

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation… http://dx.doi.org/10.5772/intechopen.71624 63

$$\operatorname{Re}\left[\frac{d\lambda\_j}{d\sigma}\right]\_{\sigma=\sigma\_1^\*} \neq 0, \qquad j=1,2,3. \tag{14}$$

By substituting λ1ð Þ¼ σ α σð Þþ β σð Þi and λ2ð Þ¼ σ α σð Þ� β σð Þi into characteristic equation (11) and calculating the implicit derivative, we obtain the following equations

$$\begin{aligned} K(\sigma)\alpha'(\sigma) - L(\sigma)\beta'(\sigma) + M(\sigma) &= 0, \\ L(\sigma)\alpha'(\sigma) + K(\sigma)\beta'(\sigma) + N(\sigma) &= 0, \end{aligned} \tag{15}$$

where

<sup>M</sup><sup>1</sup> <sup>¼</sup> ½ � <sup>c</sup><sup>1</sup> ; M<sup>2</sup> <sup>¼</sup> <sup>c</sup><sup>1</sup> <sup>1</sup>

62 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

c<sup>3</sup> c<sup>2</sup> � �

Thus, the characteristic equation (11) has all negative real parts of λ if and only if

When the assumptions on functional responses f <sup>1</sup>ð Þ¼ s f <sup>1</sup>ð Þ ζ<sup>s</sup> and f <sup>2</sup>ð Þ¼ x f <sup>2</sup>

for the system (3), together with the hypotheses H<sup>1</sup> until H3;

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �

<sup>1</sup>, there exists an interval σ<sup>∗</sup>

<sup>s</sup> � D1ζ<sup>s</sup> � � <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> ð Þ <sup>ζ</sup><sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>1</sup>

<sup>s</sup> � D1ζ<sup>s</sup> � f <sup>2</sup>

� �

ð Þþ <sup>D</sup><sup>1</sup> � <sup>D</sup>1ζ<sup>s</sup> <sup>D</sup>1ζ<sup>s</sup> � <sup>D</sup><sup>1</sup> � <sup>D</sup><sup>2</sup>

characteristic equation (11) if and only if <sup>c</sup>1c<sup>2</sup> <sup>¼</sup> <sup>c</sup><sup>3</sup> for some values of <sup>σ</sup>, say, <sup>σ</sup><sup>∗</sup>

<sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup>

<sup>1</sup>, we acquire (see [3, 10])

<sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup>

<sup>p</sup> i, <sup>λ</sup><sup>2</sup> ¼ � ffiffiffiffi

c2

H<sup>1</sup> : D<sup>1</sup> � ζ<sup>s</sup> þ 1 > 0,

<sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>1</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

<sup>1</sup>ð Þ <sup>ζ</sup><sup>s</sup> <sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �

<sup>1</sup> <sup>þ</sup> <sup>ε</sup> � �. Thus, for <sup>σ</sup><sup>∈</sup> <sup>σ</sup><sup>∗</sup>

� f <sup>1</sup>ð Þ ζ<sup>s</sup> f <sup>2</sup>

� f <sup>2</sup>

þ D1f <sup>2</sup>

H<sup>2</sup> : f 0

H<sup>3</sup> : f 0

Since at <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup>

<sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup>

positive real roots. For <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup>

that consist of three roots; <sup>λ</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffi

the roots are in general of the form;

must verify the transversality condition

σ∈ σ<sup>∗</sup>

; M<sup>3</sup> ¼

c<sup>1</sup> > 0 c<sup>3</sup> > 0 9 >=

>;

ðζ<sup>s</sup> þ D1ζ<sup>s</sup> � 2D<sup>1</sup> � 1Þ þ 3D1ð Þ 1 þ D<sup>1</sup> � ζ<sup>s</sup> > 0,

<sup>1</sup> � <sup>2</sup>ζ<sup>s</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>ε</sup> � �, the characteristic equation (11) cannot have

� �ð Þ¼ <sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>0</sup>, (13)

<sup>p</sup> i, and <sup>λ</sup><sup>3</sup> ¼ �c1: As for <sup>σ</sup><sup>∈</sup> <sup>σ</sup><sup>∗</sup>

<sup>1</sup> > 0,

<sup>1</sup> <sup>þ</sup> <sup>ε</sup> � � containing <sup>σ</sup><sup>∗</sup>

c1c<sup>2</sup> � c<sup>3</sup> > 0:

1 � ζ<sup>s</sup> D<sup>1</sup> ; σ � �

� � � �

ðζ<sup>s</sup> � 1 � 2D<sup>1</sup> þ D1ζsÞ þ 1 � ζ<sup>s</sup>

hold, we will have c<sup>1</sup> > 0 and c<sup>3</sup> > 0: We shall obtain two pure imaginary roots for the

<sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup>

c2

λ1ð Þ¼ σ α σð Þþ β σð Þi,

λ2ð Þ¼ σ α σð Þ� β σð Þi,

λ3ð Þ¼� σ c1ð Þ σ :

In order to apply the Hopf bifurcation theorem as stated in [9, 23] towards the system (8), we

2 6 4

c<sup>1</sup> 1 0 c<sup>3</sup> c<sup>2</sup> c<sup>1</sup> 0 0 c<sup>3</sup> 3 7 5:

(12)

as in (2)

1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; σ � �

<sup>s</sup> � <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>D</sup>1ζ<sup>2</sup>

s

1:

<sup>1</sup> for some ε > 0 such that

<sup>1</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup> � �, all

$$\begin{aligned} K(\sigma) &= 3\alpha^2(\sigma) + 2c\_1(\sigma)\alpha(\sigma) + c\_2(\sigma) - 3\beta^2(\sigma); \\\\ M(\sigma) &= \alpha^2(\sigma)c\_1'(\sigma) + c\_2'(\sigma)\alpha(\sigma) - c\_1'(\sigma)\beta^2(\sigma) + c\_3'(\sigma); \\\\ L(\sigma) &= 6\alpha(\sigma)\beta(\sigma) + 2c\_1(\sigma)\beta(\sigma); \\\\ N(\sigma) &= 2\alpha(\sigma)\beta(\sigma)c\_1'(\sigma) + c\_2'(\sigma)\beta(\sigma). \end{aligned}$$

Since K σ<sup>∗</sup> 1 � �M σ<sup>∗</sup> 1 � � <sup>þ</sup> <sup>L</sup> <sup>σ</sup><sup>∗</sup> 1 � �N σ<sup>∗</sup> 1 � � 6¼ <sup>0</sup>, we have

$$\operatorname{Re}\left[\frac{d\lambda\_{\rangle}}{d\sigma}\right]\_{\sigma=\sigma\_1^\*} = \frac{K\left(\sigma\_1^\*\right)M\left(\sigma\_1^\*\right) + L\left(\sigma\_1^\*\right)N\left(\sigma\_1^\*\right)}{K^2\left(\sigma\_1^\*\right) + L^2\left(\sigma\_1^\*\right)} \neq 0, \quad j = 1, 2, 3\ldots$$

and λ<sup>3</sup> σ<sup>∗</sup> 1 � � ¼ �c<sup>1</sup> <sup>σ</sup><sup>∗</sup> 1 � � 6¼ <sup>0</sup>: We conclude the details above in the following theorem. Theorem 10. Suppose that the equilibrium point E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ � � exists and those assumptions similar as in (2) for the system (3) together with hypothesis H<sup>1</sup> until H<sup>3</sup> hold. Then the system (8) undergoes Hopf bifurcation in the first octant, which leads to a family of periodic solutions bifurcating

$$\text{from } E\_2\left(\zeta\_s, \frac{(1-\zeta\_s)}{D\_1}, 0; \sigma\right) \text{ for some values of } \sigma \text{ in the neighborhood of } \sigma\_1^\*.$$

Next, we determine the Hopf bifurcation at the equilibrium point <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> : The Jacobian matrix of the system (8) about the equilibrium point E<sup>3</sup> is given by

J<sup>σ</sup>ð Þ <sup>E</sup><sup>3</sup> ¼ � f 0 <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þx<sup>∗</sup> � <sup>1</sup> �<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ f 0 <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þx<sup>∗</sup> D<sup>1</sup> <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ D<sup>1</sup> � <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup><sup>1</sup> 1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � � � <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> " # ! � f <sup>2</sup> <sup>0</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � � � 1 <sup>0</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup><sup>1</sup> 1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � � � <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> " # ! þ f 0 <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � � 2 6 6 6 6 6 6 6 4 0 <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>D</sup><sup>2</sup> 1bx<sup>∗</sup>y<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> � <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> � <sup>1</sup> ! <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>D</sup>1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> � <sup>D</sup><sup>2</sup> 1bx<sup>∗</sup>y<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>1</sup> ! � 1 3 7 7 7 7 7 7 7 5 :

Hence, the characteristic equation for the Jacobian matrix J<sup>σ</sup>ð Þ <sup>E</sup><sup>3</sup> is

$$
\lambda^3 + c\_4 \lambda^2 + c\_5 \lambda + c\_6 = 0,\tag{16}
$$

where

$$\begin{aligned} c\_4 &= f\_1'(s^\*) \mathbf{x}^\* + Q + R + V - Z + 3 - \frac{f\_1(s^\*)}{D\_1}, \\ c\_5 &= f\_1'(s^\*) \mathbf{x}^\* (Q + R + V - Z + 2) + \frac{f\_1(s^\*)(Z - R - 2)}{D\_1} + 2Q + 2R + 2V - 2Z + 3, \\ c\_6 &= f\_1'(s^\*) \mathbf{x}^\* (Q + R + V - Z + 1) + \frac{f\_1(s^\*)(Z - R - 1)}{D\_1} + Q + R + V - Z + 1, \end{aligned}$$

and

$$\begin{aligned} Q &= f\_2(\mathbf{x}^\*; \sigma) y^\* \left[ \left( \frac{D\_1}{1 + \mathbf{x}^\* + D\_1 b y^\*} \right) - \left( \frac{D\_1 \mathbf{x}^\*}{(1 + \mathbf{x}^\* + D\_1 b y^\*)^2} \right) \right], \\ R &= \left( \frac{f\_2(\mathbf{x}^\*; \sigma) D\_1^2 b \mathbf{x}^\* y^\*}{(1 + \mathbf{x}^\* + D\_1 b y^\*)^2} \right), \\ V &= f\_2'(\mathbf{x}^\*; \sigma) y^\* \left( \frac{D\_1 \mathbf{x}^\*}{1 + \mathbf{x}^\* + D\_1 b y^\*} + 1 \right), \\ Z &= f\_2(\mathbf{x}^\*; \sigma) \left( \frac{D\_1 \mathbf{x}^\*}{1 + \mathbf{x}^\* + D\_1 b y^\*} + 1 \right). \end{aligned}$$

By applying the Routh-Hurwitz criterion (see [21, 22]) towards the characteristic equation (16), we obtain the following Hurwitz matrices

$$M\_4 = [c\_4];\ M\_5 = \begin{bmatrix} c\_4 & 1\\ c\_6 & c\_5 \end{bmatrix};\ M\_6 = \begin{bmatrix} c\_4 & 1 & 0\\ c\_6 & c\_5 & c\_4\\ 0 & 0 & c\_6 \end{bmatrix}.$$

Thus, the characteristic equation (16) has all negative real parts of λ if and only if

$$\begin{cases} c\_4 > 0 \\ c\_6 > 0 \\ c\_4c\_5 - c\_6 > 0. \end{cases} \tag{17}$$

<sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>c</sup><sup>5</sup>

<sup>p</sup> i, <sup>λ</sup><sup>2</sup> ¼ � ffiffiffiffi

<sup>σ</sup>¼σ<sup>∗</sup> 2

ð Þ� σ A2ð Þ σ β<sup>0</sup>

ð Þþ σ A1ð Þ σ β<sup>0</sup>

0 <sup>1</sup>ð Þþ σ c 0 <sup>2</sup>ð Þ σ β σð Þ:

� � 6¼ <sup>0</sup>, we have

∗; <sup>1</sup> � <sup>s</sup><sup>∗</sup>

exists and those assumptions similar as in (2) for the system (3) together with hypotheses H<sup>4</sup> until H<sup>6</sup> hold. Then the system (8) undergoes Hopf bifurcation in the first octant, which leads to a family of periodic solutions bifurcating from E<sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> for some values of <sup>σ</sup> in the neighbourhood of <sup>σ</sup><sup>∗</sup>

We have proposed and analysed a simple nutrient-predator-prey model in a chemostat with intratrophic predation. This system consisted of the nutrient s, prey organisms x and predator organisms y. We conclude that intratrophic predation denoted as b does not affected the local stability and existence of E<sup>1</sup> and E<sup>2</sup> when no predator organisms involved. Next, we have

A<sup>1</sup> σ<sup>∗</sup> 2 � � � � <sup>2</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>σ</sup><sup>∗</sup>

c5

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

λ1ð Þ¼ σ α σð Þþ β σð Þi, λ2ð Þ¼ σ α σð Þ� β σð Þi, λ3ð Þ¼� σ c4ð Þ σ ,

By substituting λ1ð Þ¼ σ α σð Þþ β σð Þi and λ2ð Þ¼ σ α σð Þ� β σð Þi into characteristic equation (18)

ð Þþ <sup>σ</sup> <sup>2</sup>c1ð Þ <sup>σ</sup> α σð Þþ <sup>c</sup>2ð Þ� <sup>σ</sup> <sup>3</sup>β<sup>2</sup>

<sup>2</sup>ð Þ σ α σð Þ� c

2 � �B<sup>2</sup> σ<sup>∗</sup> 2 � �

2

� � 6¼ <sup>0</sup>: We summarise the discussion above in the following theorem.

<sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ ; <sup>2</sup><sup>x</sup> ð Þ <sup>∗</sup> <sup>þ</sup> <sup>1</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ� <sup>x</sup><sup>∗</sup> � <sup>1</sup>

!

ð Þþ σ B1ð Þ¼ σ 0,

ð Þþ σ B2ð Þ¼ σ 0,

0 <sup>1</sup>ð Þ <sup>σ</sup> <sup>β</sup><sup>2</sup>

c5

To establish Hopf bifurcation towards system (8), we must show that

and calculating the implicit derivative, we get the following equations;

A2ð Þ¼ σ 6α σð Þβ σð Þþ 2c1ð Þ σ β σð Þ;

ð Þ σ c 0 <sup>1</sup>ð Þþ σ c 0

B2ð Þ¼ σ 2α σð Þβ σð Þc

<sup>¼</sup> <sup>A</sup><sup>1</sup> <sup>σ</sup><sup>∗</sup> 2 � �B<sup>1</sup> σ<sup>∗</sup> 2 � � <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>σ</sup><sup>∗</sup>

<sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ¼ ; <sup>σ</sup> <sup>E</sup><sup>3</sup> <sup>s</sup>

Re dλ<sup>j</sup> dσ � �

A1ð Þ σ α<sup>0</sup>

A2ð Þ σ α<sup>0</sup>

<sup>A</sup>1ð Þ¼ <sup>σ</sup> <sup>3</sup>α<sup>2</sup>

<sup>B</sup>1ð Þ¼ <sup>σ</sup> <sup>α</sup><sup>2</sup>

2 � �B<sup>2</sup> σ<sup>∗</sup> 2

<sup>σ</sup>¼σ<sup>∗</sup> 2

Consisting of three roots; <sup>λ</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffi

roots are in general of the form;

where

Since A<sup>1</sup> σ<sup>∗</sup>

and λ<sup>3</sup> σ<sup>∗</sup> 2 � � ¼ �c<sup>4</sup> <sup>σ</sup><sup>∗</sup>

2.6. Discussion

2 � �B<sup>1</sup> σ<sup>∗</sup> 2 � � <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>σ</sup><sup>∗</sup>

> Re dλ<sup>j</sup> dσ � �

> > 2

E<sup>3</sup> s

Theorem 11. Suppose that the equilibrium point

� �ð Þ¼ <sup>λ</sup> <sup>þ</sup> <sup>c</sup><sup>4</sup> <sup>0</sup>, (18)

6¼ 0, j ¼ 1, 2, 3: (19)

�

ð Þ σ ;

ð Þþ σ c 0 <sup>3</sup>ð Þ σ ;

� � � � <sup>2</sup> 6¼ <sup>0</sup>, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>,

bD<sup>1</sup> <sup>1</sup> � <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ � � ; <sup>σ</sup>

<sup>2</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>2</sup> <sup>þ</sup> <sup>ε</sup> � �, all

http://dx.doi.org/10.5772/intechopen.71624

(20)

65

2:

<sup>p</sup> i, <sup>λ</sup><sup>3</sup> ¼ �c4: As for <sup>σ</sup><sup>∈</sup> <sup>σ</sup><sup>∗</sup>

Suppose the assumptions of functional response <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ and <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> similar as in (2) for the system (3), together with the hypotheses H<sup>4</sup> until H6;

$$\begin{aligned} H\_4: &f\_1'(s^\*) \mathbf{x}^\* + Q + \mathbf{R} + V + \mathbf{3} > \frac{f\_1(s^\*)}{D\_1} + \mathbf{Z}, \\ H\_5: &Q + \mathbf{R} + V + \mathbf{1} > \mathbf{Z}, \\ H\_6: &\mathbf{Z} > \mathbf{R} + \mathbf{1}, \end{aligned}$$

hold, then clearly c<sup>4</sup> > 0 and c<sup>6</sup> > 0. In particular, we shall have two pure imaginary roots for the characteristic equation (16) if and only if <sup>c</sup>4c<sup>5</sup> <sup>¼</sup> <sup>c</sup><sup>6</sup> for some values of <sup>σ</sup>, say, <sup>σ</sup><sup>∗</sup> <sup>2</sup>: Since at <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup> 2, there exists an interval σ<sup>∗</sup> <sup>2</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>2</sup> <sup>þ</sup> <sup>ε</sup> � � containing <sup>σ</sup><sup>∗</sup> <sup>2</sup> for some ε > 0. Then, for σ∈ σ<sup>∗</sup> <sup>2</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> 2 � <sup>þ</sup>εÞ, the characteristic Eq. (16) cannot have positive real roots. For <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup> <sup>2</sup>, we get (see [3, 10])

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation… http://dx.doi.org/10.5772/intechopen.71624 65

$$(\lambda^2 + c\_5)(\lambda + c\_4) = 0,\tag{18}$$

Consisting of three roots; <sup>λ</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffi c5 <sup>p</sup> i, <sup>λ</sup><sup>2</sup> ¼ � ffiffiffiffi c5 <sup>p</sup> i, <sup>λ</sup><sup>3</sup> ¼ �c4: As for <sup>σ</sup><sup>∈</sup> <sup>σ</sup><sup>∗</sup> <sup>2</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> <sup>2</sup> <sup>þ</sup> <sup>ε</sup> � �, all roots are in general of the form;

$$\begin{array}{l} \lambda\_1(\sigma) = \alpha(\sigma) + \beta(\sigma)\dot{\iota}, \\ \lambda\_2(\sigma) = \alpha(\sigma) - \beta(\sigma)\dot{\iota}, \\ \lambda\_3(\sigma) = -c\_4(\sigma), \end{array}$$

To establish Hopf bifurcation towards system (8), we must show that

$$\operatorname{Re}\left[\frac{d\lambda\_j}{d\sigma}\right]\_{\sigma=\sigma\_2^\*} \neq 0, \qquad j=1,2,3. \tag{19}$$

By substituting λ1ð Þ¼ σ α σð Þþ β σð Þi and λ2ð Þ¼ σ α σð Þ� β σð Þi into characteristic equation (18) and calculating the implicit derivative, we get the following equations;

$$\begin{cases} A\_1(\sigma)\alpha'(\sigma) - A\_2(\sigma)\beta'(\sigma) + B\_1(\sigma) = 0, \\ A\_2(\sigma)\alpha'(\sigma) + A\_1(\sigma)\beta'(\sigma) + B\_2(\sigma) = 0, \end{cases} \tag{20}$$

where

where

and

c<sup>4</sup> ¼ f 0 1 s

c<sup>5</sup> ¼ f 0 1 s

c<sup>6</sup> ¼ f 0 1 s <sup>∗</sup> ð Þx<sup>∗</sup>

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>þ</sup> <sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>3</sup> � <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ

64 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>Q</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup><sup>1</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup> !

<sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>y</sup><sup>∗</sup> <sup>D</sup>1x<sup>∗</sup>

<sup>M</sup><sup>4</sup> <sup>¼</sup> ½ � <sup>c</sup><sup>4</sup> ; M<sup>5</sup> <sup>¼</sup> <sup>c</sup><sup>4</sup> <sup>1</sup>

<sup>R</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> <sup>D</sup><sup>2</sup>

V ¼ f 0

we obtain the following Hurwitz matrices

<sup>Z</sup> <sup>¼</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup>

system (3), together with the hypotheses H<sup>4</sup> until H6;

there exists an interval σ<sup>∗</sup>

H<sup>4</sup> : f 0 1 s

<sup>2</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup>

<sup>∗</sup> ð Þx<sup>∗</sup>ð<sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>2</sup>Þ þ <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þð Þ <sup>Z</sup> � <sup>R</sup> � <sup>2</sup>

<sup>ð</sup><sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> � <sup>Z</sup> <sup>þ</sup> <sup>1</sup>Þ þ <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þð Þ <sup>Z</sup> � <sup>R</sup> � <sup>1</sup>

1 þ x<sup>∗</sup> þ D1by<sup>∗</sup> � �

,

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

By applying the Routh-Hurwitz criterion (see [21, 22]) towards the characteristic equation (16),

c<sup>6</sup> c<sup>5</sup> � �

Thus, the characteristic equation (16) has all negative real parts of λ if and only if

H<sup>5</sup> : Q þ R þ V þ 1 > Z,

characteristic equation (16) if and only if <sup>c</sup>4c<sup>5</sup> <sup>¼</sup> <sup>c</sup><sup>6</sup> for some values of <sup>σ</sup>, say, <sup>σ</sup><sup>∗</sup>

<sup>2</sup> <sup>þ</sup> <sup>ε</sup> � � containing <sup>σ</sup><sup>∗</sup>

H<sup>6</sup> : Z > R þ 1,

<sup>þ</sup>εÞ, the characteristic Eq. (16) cannot have positive real roots. For <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup>

1bx<sup>∗</sup>y<sup>∗</sup>

D1x<sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> <sup>þ</sup> <sup>1</sup> � �

D<sup>1</sup> ,

D<sup>1</sup>

D<sup>1</sup>

" # !

,

2 6 4

9 >=

>;

:

; M<sup>6</sup> ¼

c<sup>4</sup> > 0 c<sup>6</sup> > 0 c4c<sup>5</sup> � c<sup>6</sup> > 0:

<sup>∗</sup> ð Þx<sup>∗</sup> <sup>þ</sup> <sup>Q</sup> <sup>þ</sup> <sup>R</sup> <sup>þ</sup> <sup>V</sup> <sup>þ</sup> <sup>3</sup> <sup>&</sup>gt; <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ

Suppose the assumptions of functional response <sup>f</sup> <sup>1</sup> <sup>s</sup><sup>∗</sup> ð Þ and <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> similar as in (2) for the

hold, then clearly c<sup>4</sup> > 0 and c<sup>6</sup> > 0. In particular, we shall have two pure imaginary roots for the

� <sup>D</sup>1x<sup>∗</sup>

c<sup>4</sup> 1 0 c<sup>6</sup> c<sup>5</sup> c<sup>4</sup> 0 0 c<sup>6</sup>

D<sup>1</sup>

þ Z,

<sup>2</sup> for some ε > 0. Then, for σ∈ σ<sup>∗</sup>

3 7 5:

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>∗</sup> <sup>þ</sup> <sup>D</sup>1by<sup>∗</sup> ð Þ<sup>2</sup>

þ 2Q þ 2R þ 2V � 2Z þ 3,

,

(17)

2,

<sup>2</sup> � <sup>ε</sup>; <sup>σ</sup><sup>∗</sup> 2

<sup>2</sup>: Since at <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>∗</sup>

�

<sup>2</sup>, we get (see [3, 10])

þ Q þ R þ V � Z þ 1,

$$\begin{aligned} A\_1(\sigma) &= 3\alpha^2(\sigma) + 2c\_1(\sigma)\alpha(\sigma) + c\_2(\sigma) - 3\beta^2(\sigma); \\ A\_2(\sigma) &= 6\alpha(\sigma)\beta(\sigma) + 2c\_1(\sigma)\beta(\sigma); \\\\ B\_1(\sigma) &= \alpha^2(\sigma)c\_1'(\sigma) + c\_2'(\sigma)\alpha(\sigma) - c\_1'(\sigma)\beta^2(\sigma) + c\_3'(\sigma); \\ B\_2(\sigma) &= 2\alpha(\sigma)\beta(\sigma)c\_1'(\sigma) + c\_2'(\sigma)\beta(\sigma). \end{aligned}$$

Since A<sup>1</sup> σ<sup>∗</sup> 2 � �B<sup>1</sup> σ<sup>∗</sup> 2 � � <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>σ</sup><sup>∗</sup> 2 � �B<sup>2</sup> σ<sup>∗</sup> 2 � � 6¼ <sup>0</sup>, we have

$$\operatorname{Re}\left[\frac{d\lambda\_j}{d\sigma}\right]\_{\sigma=\sigma\_2^\*} = \frac{A\_1(\sigma\_2^\*)B\_1(\sigma\_2^\*) + A\_2(\sigma\_2^\*)B\_2(\sigma\_2^\*)}{\left(A\_1(\sigma\_2^\*)\right)^2 + \left(A\_2(\sigma\_2^\*)\right)^2} \neq 0, \quad j = 1, 2, 3.$$

and λ<sup>3</sup> σ<sup>∗</sup> 2 � � ¼ �c<sup>4</sup> <sup>σ</sup><sup>∗</sup> 2 � � 6¼ <sup>0</sup>: We summarise the discussion above in the following theorem. Theorem 11. Suppose that the equilibrium point

$$E\_3(s^\*, \mathfrak{x}^\*, y^\*; \sigma) = E\_3\left(s^\*, \frac{1 - s^\*}{f\_1(s^\*)}, \frac{(2\mathfrak{x}^\* + 1)f\_2(\mathfrak{x}^\*) - \mathfrak{x}^\* - 1}{bD\_1(1 - f\_2(\mathfrak{x}^\*))}; \sigma\right)$$

exists and those assumptions similar as in (2) for the system (3) together with hypotheses H<sup>4</sup> until H<sup>6</sup> hold. Then the system (8) undergoes Hopf bifurcation in the first octant, which leads to a family of periodic solutions bifurcating from E<sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> for some values of <sup>σ</sup> in the neighbourhood of <sup>σ</sup><sup>∗</sup> 2:

#### 2.6. Discussion

We have proposed and analysed a simple nutrient-predator-prey model in a chemostat with intratrophic predation. This system consisted of the nutrient s, prey organisms x and predator organisms y. We conclude that intratrophic predation denoted as b does not affected the local stability and existence of E<sup>1</sup> and E<sup>2</sup> when no predator organisms involved. Next, we have shown that the washout equilibrium point E1ð Þ 1; 0; 0 (no prey and predator organisms present) is locally asymptotically stable if <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup> <sup>D</sup><sup>1</sup> � 1 < 0 holds. For stability of E<sup>2</sup> and E<sup>3</sup> of the system (3), some sufficient criteria or conditions are derived and satisfied. The points E<sup>2</sup> and E<sup>3</sup> are said to be asymptotically stable if all of their eigenvalues are less than zero.

[6] Pitchford J, Brindley J. Intratrophic predation in simple predator-prey models. Bulletin of

Stability and Hopf Bifurcation Analysis of a Simple Nutrient-Prey-Predator Model with Intratrophic Predation…

http://dx.doi.org/10.5772/intechopen.71624

67

[7] Mada Sanjaya WS, Salleh Z, Mamat M. Mathematical model of three species food chain with Holling Type-III functional response. International Journal of Pure and Applied

[8] Tee LS, Salleh Z. Hopf bifurcation of a nonlinear system derived from Lorenz system using normal form theory. International Journal of Applied Mathematics and Statistics.

[9] Hassard BD, Kazarinoff ND, Wan YH. Theory and Applications of Hopf Bifurcation.

[10] Freedman HI, Ruan S. Hopf bifurcation in three-species food chain models with group

[11] Rana SMS, Nasrin F, Hossain MI. Dynamics of a predator-prey interaction in chemostat with variable yield. Journal of Sustainability Science and Management. 2015;10(2):16-23

[12] Solomon ME. The natural control of animal populations. The Journal of Animal Ecology.

[14] Robinson CR. An Introduction to Dynamical Systems: Continuous and Discrete. Pearson

[15] Hirsch MW, Smale S, Devaney RL. Differential Equations, Dynamical Systems & an Introduction to Chaos. 2nd ed. Elsevier Academic Press, San Diego California: Elsevier; 2004

[16] Freedman HI, Mathsen RM. Persistence in predator-prey systems with ratio-dependent

[17] Butler GJ, Hsu SB, Waltman P. Coexistence of competing predators in a chemostat. Jour-

[18] Nani F, Freedman HI. A mathematical model of cancer treatment by immunotherapy.

[19] Freedman HI, Waltman P. Persistence in models of three interacting predator-prey

[20] Freedman HI, Ruan S. Uniform persistence in functional differential equations. Mathe-

[22] Tee LS, Salleh Z. Hopf bifurcation analysis of Zhou system. International Journal of Pure

[23] Marsden JE, McKracken M. The Hopf Bifurcation and its Applications. New York:

[21] Gantmacher FR. The Theory of Matrices. USA: Chelsea Publishing Company; 1964

predator influence. Bulletin of Mathematical Biology. 1993;55(4):817-827

[13] Hale JK, Kocak H. Dynamics and Bifurcations. New York: Springer; 1991

Prentice Hall, New Jersey: Pearson Education, Inc; 2004

nal of Mathematical Biology. 1983;17:133-151

Mathematical Biosciences. 2000;163:159-199

matical Biosciences. 1995;115:173-192

Springer-Verlag; 1976

and Applied Mathematics. 2015;104(1):1-18

populations. Mathematical Biosciences. 1984;68:213-231

Mathematical Biology. 1998;60:937-953; bu980053

USA: Cambridge University Press; 1981

defense. Mathematical Biosciences. 1992;111:73-87

2016;55(3):122-132

1949;18:1-32

Mathematics. 2013;89(5):647-657. DOI: 10.12737/ijpam.v89i5.1

In particular, we investigated the global stability analysis for E2: A suitable Lyapunov function L is defined and E<sup>2</sup> is globally asymptotically stable if and only if the conditions in Theorem 8 holds. Global stability of E<sup>2</sup> indicates that predator organisms y might be washout in the chemostat despites the initial prey and predator organisms' density levels. Next, in the study of the existence of positive interior equilibrium point E3, we presented some results regarding uniform persistence analysis. It has shown that the system (3) is uniformly persistence and thus, the positive interior equilibrium point E<sup>3</sup> exists.

In the analysis of occurrence of Hopf bifurcation, Hopf bifurcation theorems in Hassard et al. [9] are applied. We have shown that the system (8) undergoes Hopf bifurcation in the first octant, which leads to a family of periodic solutions bifurcating from E<sup>2</sup> ζs; ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ and <sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> for some values of <sup>σ</sup> in the neighbourhoods of <sup>σ</sup><sup>∗</sup> <sup>1</sup> and σ<sup>∗</sup> <sup>2</sup>, respectively. The method used to obtain these results is similar to the method used in [3, 10].

#### Author details

Zabidin Salleh\* and Liyana Abd Rahim

\*Address all correspondence to: zabidin@umt.edu.my

School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, Kuala Nerus, Terengganu, Malaysia

#### References


[6] Pitchford J, Brindley J. Intratrophic predation in simple predator-prey models. Bulletin of Mathematical Biology. 1998;60:937-953; bu980053

shown that the washout equilibrium point E1ð Þ 1; 0; 0 (no prey and predator organisms present)

system (3), some sufficient criteria or conditions are derived and satisfied. The points E<sup>2</sup> and

In particular, we investigated the global stability analysis for E2: A suitable Lyapunov function L is defined and E<sup>2</sup> is globally asymptotically stable if and only if the conditions in Theorem 8 holds. Global stability of E<sup>2</sup> indicates that predator organisms y might be washout in the chemostat despites the initial prey and predator organisms' density levels. Next, in the study of the existence of positive interior equilibrium point E3, we presented some results regarding uniform persistence analysis. It has shown that the system (3) is uniformly persistence and

In the analysis of occurrence of Hopf bifurcation, Hopf bifurcation theorems in Hassard et al. [9] are applied. We have shown that the system (8) undergoes Hopf bifurcation in the first

E<sup>3</sup> are said to be asymptotically stable if all of their eigenvalues are less than zero.

octant, which leads to a family of periodic solutions bifurcating from E<sup>2</sup> ζs;

School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu,

tion. New York: Cambridge University Press; 1995. pp. 1-7

[1] Smith HL, Waltman P. The Theory of the Chemostat: Dynamics of Microbial Competi-

[2] Li B, Kuang Y. Simple food chain in a chemostat with distinct removal rates. Journal of Mathematical Analysis and Applications. 2000;242:75-92. DOI: 10.1006 jmaa.1999.6655 [3] El-Sheikh MMA, Mahrouf SAA. Stability and bifurcation of a simple food chain in a chemostat with removal rates. Chaos, Solitons & Fractals. 2005;23:1475-1489. DOI: 10.1016/j.

[4] Hamra MA, Yadi K. Asymptotic behavior of a chemostat model with constant recycle sludge concentration. Acta Biotheoretica. 2017;65:233-252. DOI: 10.1007/s10441-017-9309-4

[5] Yang J, Tang G, Tang S. Modelling the regulatory system of a chemostat model with a threshold window. Mathematics and Computers in Simulation. 2017;132:220-235. DOI:

method used to obtain these results is similar to the method used in [3, 10].

<sup>E</sup><sup>3</sup> <sup>s</sup><sup>∗</sup>; <sup>x</sup><sup>∗</sup>; <sup>y</sup><sup>∗</sup> ð Þ ; <sup>σ</sup> for some values of <sup>σ</sup> in the neighbourhoods of <sup>σ</sup><sup>∗</sup>

<sup>D</sup><sup>1</sup> � 1 < 0 holds. For stability of E<sup>2</sup> and E<sup>3</sup> of the

ð Þ 1�ζ<sup>s</sup> <sup>D</sup><sup>1</sup> ; 0; σ 

<sup>2</sup>, respectively. The

<sup>1</sup> and σ<sup>∗</sup>

and

is locally asymptotically stable if <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup>

66 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

thus, the positive interior equilibrium point E<sup>3</sup> exists.

Author details

References

Zabidin Salleh\* and Liyana Abd Rahim

Kuala Nerus, Terengganu, Malaysia

chaos.2004.06.079

10.1016/j.matcom.2016.08.005

\*Address all correspondence to: zabidin@umt.edu.my


**Chapter 4**

Provisional chapter

**Sensitivity Analysis: A Useful Tool for Bifurcation**

DOI: 10.5772/intechopen.72345

Sensitivity analysis and bifurcation analysis are closely related to each other. In sensitivity analysis, especially global sensitivity analysis the effects of input parameter spaces on output quantities of interest are studied. On the other hand, in bifurcation analysis the critical points within feasible regions of parameters are detected where the long-term dynamics changes qualitatively. Prior to bifurcation analysis, it is important to identify the bifurcation parameters. In complex and computationally expensive problems which consist plenty of uncertain parameters, it is essential to find a set of bifurcation parameters before bifurcation analysis. Global sensitivity analysis is a powerful tool to identify the bifurcation parameters which contribute most on output uncertainty. Global sensitivity analysis is the first step toward bifurcation analysis which helps in dimension reduction during bifurcation analysis. As an example, in this chapter, a multi compartment, lumpedparameter model of an arm artery is considered and global sensitivity analysis (Sobol's

method) is applied to identify the bifurcation parameters of the arm arteries.

analysis, bifurcation parameters, Sobol's method

Keywords: lumped parameter model, arm arteries, sensitivity analysis, bifurcation

Sensitivity analysis and bifurcation analysis are closely related to each other. In sensitivity analysis, we study how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be apportioned to different sources of uncertainty in its inputs [1]. On the other hand, in bifurcation analysis the critical points within the feasible regions of parameters are detected where the long-term dynamics changes qualitatively [2]. Prior to the bifurcation analysis, it is important to identify the bifurcation parameters in complex and

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Sensitivity Analysis: A Useful Tool for Bifurcation

**Analysis**

Analysis

Raheem Gul and Stefan Bernhard

Raheem Gul and Stefan Bernhard

http://dx.doi.org/10.5772/intechopen.72345

Abstract

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Sensitivity Analysis: A Useful Tool for Bifurcation Analysis** Sensitivity Analysis: A Useful Tool for Bifurcation Analysis

DOI: 10.5772/intechopen.72345

Raheem Gul and Stefan Bernhard Raheem Gul and Stefan Bernhard

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72345

#### Abstract

Sensitivity analysis and bifurcation analysis are closely related to each other. In sensitivity analysis, especially global sensitivity analysis the effects of input parameter spaces on output quantities of interest are studied. On the other hand, in bifurcation analysis the critical points within feasible regions of parameters are detected where the long-term dynamics changes qualitatively. Prior to bifurcation analysis, it is important to identify the bifurcation parameters. In complex and computationally expensive problems which consist plenty of uncertain parameters, it is essential to find a set of bifurcation parameters before bifurcation analysis. Global sensitivity analysis is a powerful tool to identify the bifurcation parameters which contribute most on output uncertainty. Global sensitivity analysis is the first step toward bifurcation analysis which helps in dimension reduction during bifurcation analysis. As an example, in this chapter, a multi compartment, lumpedparameter model of an arm artery is considered and global sensitivity analysis (Sobol's method) is applied to identify the bifurcation parameters of the arm arteries.

Keywords: lumped parameter model, arm arteries, sensitivity analysis, bifurcation analysis, bifurcation parameters, Sobol's method

#### 1. Introduction

Sensitivity analysis and bifurcation analysis are closely related to each other. In sensitivity analysis, we study how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be apportioned to different sources of uncertainty in its inputs [1]. On the other hand, in bifurcation analysis the critical points within the feasible regions of parameters are detected where the long-term dynamics changes qualitatively [2]. Prior to the bifurcation analysis, it is important to identify the bifurcation parameters in complex and

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

computationally expensive problems that consist plenty of uncertain parameters. Sensitivity analysis is a powerful tool to identify the bifurcation parameters which contribute most on output uncertainty. Also, sensitivity analysis helps in dimension reduction during the bifurcation analysis by fixing less influential parameters on their nominal values.

In this chapter, the main questions of interest are:

2. Lumped-parameter model of the arm arteries

Pressure and flow equations at non-terminal nodes:

q\_

q\_

<sup>p</sup>\_<sup>i</sup> <sup>¼</sup> qi � qiþ<sup>1</sup> Ci

<sup>p</sup>\_ <sup>6</sup> <sup>¼</sup> <sup>q</sup><sup>6</sup> � <sup>q</sup><sup>11</sup> � <sup>q</sup><sup>7</sup> C6

Pressure and flow equations at terminal nodes:

<sup>q</sup>\_<sup>i</sup> <sup>¼</sup> pi�<sup>1</sup> � pi � Riqi Li

<sup>11</sup> <sup>¼</sup> <sup>p</sup><sup>6</sup> � <sup>p</sup><sup>11</sup> � <sup>R</sup>11q<sup>11</sup> L<sup>11</sup>

<sup>13</sup> <sup>¼</sup> <sup>p</sup><sup>11</sup> � <sup>p</sup><sup>13</sup> � <sup>R</sup>13q<sup>13</sup> L<sup>13</sup>

Flow equation:

Pressure equation:

1. How to identify the bifurcation parameters in a model having plenty of input parameters? 2. Which parameters could be exempted from the bifurcation analysis (dimension reduction)? This chapter seeks to answers the above-mentioned questions using a simplified 5-steps procedure of uncertainty and sensitivity analysis. As an example, a multi-compartment, lumpedparameter model of arm arteries is considered [4] and global sensitivity analysis (Sobol's method) is applied to identify the bifurcation parameters (electrical) of the arm arteries.

In this section, the major arteries of the arm are divided in to number of non-terminal and terminal arterial segments (nodes). The total number of arterial segments, N<sup>s</sup> ¼ 15 including 12 non-terminal and 3 terminal segments. Each non-terminal and terminal arterial segment is

Applying Kirchhoff's current and voltage laws on electrical representation of arm arteries, the

, i ¼ 1, 2, 3, …15 and i 6¼ 6, 11

, <sup>p</sup>\_ <sup>11</sup> <sup>¼</sup> <sup>q</sup><sup>11</sup> � <sup>q</sup><sup>12</sup> � <sup>q</sup><sup>13</sup>

<sup>q</sup>\_in <sup>¼</sup> <sup>2</sup>pin � <sup>2</sup>pi � Riqin Li

<sup>q</sup>\_out <sup>¼</sup> <sup>2</sup>pi � <sup>2</sup>pout � <sup>2</sup>Rbqout Li

<sup>p</sup>\_<sup>i</sup> <sup>¼</sup> qin � qout Ci

C<sup>11</sup>

, i ¼ 1, 2, 3, …15 and i 6¼ 11, 13

, i ¼ 10, 12, 15

ð Þ at bifurcation

Sensitivity Analysis: A Useful Tool for Bifurcation Analysis

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71

(1)

(2)

(3)

represented by its corresponding non-terminal and terminal electrical circuit.

following mathematical equations for pressure and flow are obtained:

Sensitivity analysis can be divided into two categories, local sensitivity analysis (LSA) and global sensitivity analysis (GSA). In LSA a parameter value is perturbed around its nominal values at a time, keeping other parameters fixed on their nominal values. The procedure is repeated for all parameters one by one to study their impact on output variables. LSA techniques are simple, easy to implement and computationally less expensive. On the other hand, LSA is not suitable for non-linear models and does not explore the impact of entire parameter spaces and their interactions effects on output variables [3, 4]. In order to overcome the limitations associated with the LSA, GSA can be used. In GSA, the analysis is performed over entire feasible regions of the input parameters and quantifies the impact of parameter interactions on output variables. The only deficiency related to the GSA is its computational cost [5–12] (Figure 1).

Figure 1. A simplified 5-step procedure to identify the bifurcation parameters using global sensitivity analysis.

In this chapter, the main questions of interest are:

1. How to identify the bifurcation parameters in a model having plenty of input parameters?

2. Which parameters could be exempted from the bifurcation analysis (dimension reduction)?

This chapter seeks to answers the above-mentioned questions using a simplified 5-steps procedure of uncertainty and sensitivity analysis. As an example, a multi-compartment, lumpedparameter model of arm arteries is considered [4] and global sensitivity analysis (Sobol's method) is applied to identify the bifurcation parameters (electrical) of the arm arteries.

#### 2. Lumped-parameter model of the arm arteries

In this section, the major arteries of the arm are divided in to number of non-terminal and terminal arterial segments (nodes). The total number of arterial segments, N<sup>s</sup> ¼ 15 including 12 non-terminal and 3 terminal segments. Each non-terminal and terminal arterial segment is represented by its corresponding non-terminal and terminal electrical circuit.

Applying Kirchhoff's current and voltage laws on electrical representation of arm arteries, the following mathematical equations for pressure and flow are obtained:

Pressure and flow equations at non-terminal nodes:

Flow equation:

computationally expensive problems that consist plenty of uncertain parameters. Sensitivity analysis is a powerful tool to identify the bifurcation parameters which contribute most on output uncertainty. Also, sensitivity analysis helps in dimension reduction during the bifurca-

Sensitivity analysis can be divided into two categories, local sensitivity analysis (LSA) and global sensitivity analysis (GSA). In LSA a parameter value is perturbed around its nominal values at a time, keeping other parameters fixed on their nominal values. The procedure is repeated for all parameters one by one to study their impact on output variables. LSA techniques are simple, easy to implement and computationally less expensive. On the other hand, LSA is not suitable for non-linear models and does not explore the impact of entire parameter spaces and their interactions effects on output variables [3, 4]. In order to overcome the limitations associated with the LSA, GSA can be used. In GSA, the analysis is performed over entire feasible regions of the input parameters and quantifies the impact of parameter interactions on output variables. The only deficiency related to the GSA is its computational cost [5–12]

Figure 1. A simplified 5-step procedure to identify the bifurcation parameters using global sensitivity analysis.

tion analysis by fixing less influential parameters on their nominal values.

70 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

(Figure 1).

$$\begin{aligned} \dot{q}\_{i} &= \frac{p\_{i-1} - p\_i - R\_i q\_i}{L\_i}, \quad i = 1, 2, 3, \dots 15 \text{ and } i \neq 11, 13\\ \dot{q}\_{11} &= \frac{p\_6 - p\_{11} - R\_{11} q\_{11}}{L\_{11}}\\ \dot{q}\_{13} &= \frac{p\_{11} - p\_{13} - R\_{13} q\_{13}}{L\_{13}} \end{aligned} \tag{1}$$

Pressure equation:

$$\begin{aligned} \dot{p}\_i &= \frac{q\_i - q\_{i+1}}{\mathcal{C}\_i}, \quad i = 1, 2, 3, \dots 15 \text{ and } i \neq 6, 11\\ \dot{p}\_6 &= \frac{q\_6 - q\_{11} - q\_7}{\mathcal{C}\_6}, \quad \dot{p}\_{11} = \frac{q\_{11} - q\_{12} - q\_{13}}{\mathcal{C}\_{11}} \end{aligned} \tag{2}$$

Pressure and flow equations at terminal nodes:

$$\begin{aligned} \dot{q}\_{in} &= \frac{2p\_{in} - 2p\_i - R\_i q\_{in}}{L\_i} \\ \dot{p}\_i &= \frac{q\_{in} - q\_{out}}{C\_i} \\ \dot{q}\_{out} &= \frac{2p\_i - 2p\_{out} - 2R\_b q\_{out}}{L\_i}, \quad i = 10, 12, 15 \end{aligned} \tag{3}$$


The value of boundary resistance ð Þ Rb on three terminal nodes is 3:<sup>24</sup> � <sup>10</sup><sup>9</sup> kgs�<sup>1</sup> <sup>m</sup>�4, <sup>ν</sup> <sup>¼</sup> <sup>0</sup>:<sup>004</sup> kgs�<sup>1</sup> m�<sup>1</sup> and <sup>ρ</sup> <sup>¼</sup> <sup>1050</sup> kgm�<sup>3</sup> [4, 5, 13, 14].

Table 1. Numerical values of parameters for each node of the arm arteries (shown in Figure 2).

where, Ri, Ci and Li is the blood flow resistance, compliance of the vessel and blood inertia of i th segment of the arm arteries respectively. The electrical parameters Ri; Ci ð Þ ; Li of <sup>i</sup> th segments are related with structural parameters Ei; li; di ð Þ ; hi as,

$$R\_i = \frac{8\nu l\_i}{\pi \left(\frac{d}{2}\right)^{4'}}, \quad \mathcal{C}\_i = \frac{\rho l\_i}{\pi \left(\frac{d}{2}\right)^2}, \quad L\_i = \frac{2\pi \left(\frac{d}{2}\right)^2 l\_i}{E\_i l\_i} \tag{4}$$

outputs caused by uncertainty of its inputs. Whereas, sensitivity analysis study the impact of input quantities of interest (QoI) on output quantities of interest (QoI). In this study, the input (QoI) are electrical parameters (Ri, Ci, Li) and output (QoI) are pressure and flow at each node of the arm arteries. Further, for uncertainty analysis Latin hypercube sampling (LHS) is used and variance-decomposition method (Sobol's method) is used for global sensitivity analysis (GSA). Compared to the high-dimensional cardiovascular models (3D, 2D, 1D), lumped-parameter models of the cardiovascular system (CVS) are computationally less expensive, therefore they are suitable for GSA. In our previous studies, we found that for lumped-parameter models of

Figure 2. Model geometry of arm artery (A), with total number of arterial segments, Ns = 15, including 12 non-terminal and 3 terminal segments. Each non-terminal and terminal segment is represented by its corresponding non-terminal (C) and terminal electrical circuits (D). Pressure waves is used as an input boundary condition (B) and pout = 15 mmHg which is mean venous pressure used to calculate boundary outflow. The parameter values of each arterial segment are given in Table 1.

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73

where, Ei is the Young modulus, li denotes length of the vessel, d is the diameter of the vessel and hi represents the wall thickness of i th segment of the vessel. Moreover, ν (0.004 Pa s) is the blood viscosity and ρ (1050 kgm�<sup>3</sup> ) is the blood density. The nominal values of all parameters of arm segments are given in Table 1. The geometry along with the values of the parameters is taken from [13, 14].

#### 3. Uncertainty and sensitivity analysis

Uncertainty analysis (UA) and sensitivity analysis (SA) are closely related; however they represent two different disciplines. Uncertainty analysis assesses the uncertainty in model

Figure 2. Model geometry of arm artery (A), with total number of arterial segments, Ns = 15, including 12 non-terminal and 3 terminal segments. Each non-terminal and terminal segment is represented by its corresponding non-terminal (C) and terminal electrical circuits (D). Pressure waves is used as an input boundary condition (B) and pout = 15 mmHg which is mean venous pressure used to calculate boundary outflow. The parameter values of each arterial segment are given in Table 1.

where, Ri, Ci and Li is the blood flow resistance, compliance of the vessel and blood inertia of

π <sup>d</sup> 2

where, Ei is the Young modulus, li denotes length of the vessel, d is the diameter of the vessel

of arm segments are given in Table 1. The geometry along with the values of the parameters is

Uncertainty analysis (UA) and sensitivity analysis (SA) are closely related; however they represent two different disciplines. Uncertainty analysis assesses the uncertainty in model

<sup>2</sup> , Li <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>d</sup>

2 <sup>2</sup> li Eili

<sup>m</sup>�<sup>4</sup> � 106 kg�<sup>1</sup>

<sup>s</sup><sup>2</sup>m<sup>4</sup> � <sup>10</sup>�<sup>11</sup> kgm�<sup>4</sup> � 106

th segment of the vessel. Moreover, ν (0.004 Pa s) is the

) is the blood density. The nominal values of all parameters

th segments

m�<sup>1</sup> and

<sup>m</sup>�4, <sup>ν</sup> <sup>¼</sup> <sup>0</sup>:<sup>004</sup> kgs�<sup>1</sup>

(4)

th segment of the arm arteries respectively. The electrical parameters Ri; Ci ð Þ ; Li of <sup>i</sup>

Nodes E l d h R C L

1 4 6.1 7.28 6.2 3.539 7.454 1.539 2 4 5.6 6.28 5.7 5.868 4.778 1.898 3 4 6.3 5.64 5.5 10.15 4.035 2.648 4 4 6.3 5.32 5.3 12.82 3.514 2.976 5 4 6.3 5 5.2 16.43 2.974 3.369 6 4 4.6 4.72 5 15.10 1.9 2.76 7 8 7.1 3.48 4.4 78.90 0.667 7.838 8 8 7.1 3.24 4.3 105 0.531 9.042 9 8 7.1 3 4.2 142.9 0.448 10.55 10 8 2 2.84 4.1 55.11 0.1207 3.647 11 8 2 4.3 4.9 31.94 1.067 4.844 12 16 6.7 1.82 2.8 1173 0.0834 31.88 13 8 7.9 4.06 4.7 40.19 0.9366 5.434 14 8 6.7 3.48 4.6 50.22 0.80 6.075 15 8 3.7 3.66 4.5 33.60 0.3958 3.693

<sup>s</sup>�<sup>2</sup> �10<sup>5</sup> <sup>m</sup> �10�<sup>2</sup> <sup>m</sup> �10�<sup>3</sup> <sup>m</sup> �10�<sup>4</sup> kgs�<sup>1</sup>

72 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>4</sup> , Ci <sup>¼</sup> <sup>ρ</sup>li

are related with structural parameters Ei; li; di ð Þ ; hi as,

and hi represents the wall thickness of i

3. Uncertainty and sensitivity analysis

blood viscosity and ρ (1050 kgm�<sup>3</sup>

taken from [13, 14].

<sup>ρ</sup> <sup>¼</sup> <sup>1050</sup> kgm�<sup>3</sup> [4, 5, 13, 14].

Ri <sup>¼</sup> <sup>8</sup>νli π <sup>d</sup> 2

The value of boundary resistance ð Þ Rb on three terminal nodes is 3:<sup>24</sup> � <sup>10</sup><sup>9</sup> kgs�<sup>1</sup>

Table 1. Numerical values of parameters for each node of the arm arteries (shown in Figure 2).

i

units kgm�<sup>2</sup>

outputs caused by uncertainty of its inputs. Whereas, sensitivity analysis study the impact of input quantities of interest (QoI) on output quantities of interest (QoI). In this study, the input (QoI) are electrical parameters (Ri, Ci, Li) and output (QoI) are pressure and flow at each node of the arm arteries. Further, for uncertainty analysis Latin hypercube sampling (LHS) is used and variance-decomposition method (Sobol's method) is used for global sensitivity analysis (GSA).

Compared to the high-dimensional cardiovascular models (3D, 2D, 1D), lumped-parameter models of the cardiovascular system (CVS) are computationally less expensive, therefore they are suitable for GSA. In our previous studies, we found that for lumped-parameter models of the CVS, the Sobol's method is computationally less expensive as compared to the other variance-decomposition methods, like FAST and sparse grid stochastic collocation method based on Smolyak algorithm [5, 12].

where, V is the variance operator. The relationship between functions and partial variance are

ð Þ xi � �

� � � � � Vi � <sup>V</sup> <sup>¼</sup> V f ij xi; xj

� � � �

Sensitivity Analysis: A Useful Tool for Bifurcation Analysis

http://dx.doi.org/10.5772/intechopen.72345

� � <sup>þ</sup> … <sup>þ</sup> <sup>S</sup>1,2,3,…,Kð Þ <sup>x</sup>1; <sup>x</sup>2; <sup>x</sup>3;…; xK (10)

ð Þ <sup>Y</sup>j<sup>x</sup> � <sup>i</sup> � �

(9)

75

(11)

th parameter on output

th parameters on output uncertainty.

<sup>V</sup> (12)

th parameter means main effect

th parameter. In this study, the interaction

ð Þ¼ Ex�<sup>i</sup>ð Þ Yjxi V f <sup>i</sup>

Vij ¼ Vxi,xj Ex�ij Yjxi; xj

X k

X k

Sij xi; xj

Si <sup>¼</sup> Vij

Sij <sup>¼</sup> Vij V

<sup>V</sup> , and

th and j

<sup>V</sup> <sup>¼</sup> <sup>1</sup> � Vx�<sup>i</sup> Exi

In general, the main effect is used identify the most influential parameters (bifurcation parameters) and the total effect is taken into account for those parameters which are exempted from

effects of parameters on the output (QoI) are negligible, therefore the main effects are used for

In this section, a detailed working algorithm is presented to compute the main effect, Si using

1. Generate a random numbers matrix of row dimension 2K and column length N (the sample size) and split into two independent sampling matrices, Að Þ N;K and Bð Þ N;K by

i<j

where, Si is the main effect (first order sensitivity index) of the i

ð Þ <sup>Y</sup>j<sup>x</sup> � <sup>i</sup> � �

i

given by,

Where,

Vi ¼ Vxi

…

Dividing both sides of the Eq. (8) by V, we get:

Sið Þþ xi

uncertainty and Sij is the interaction effect of i

Further, the total sensitivity index, STi can be calculated as,

STi <sup>¼</sup> Ex�<sup>i</sup> Vxi

bifurcation analysis (factor fixing). The total effect, STi of the i

the Monte Carlo simulations, we follow the steps, given in [1, 15].

using LHS. Where, K is the number of uncertain model parameters.

plus higher-order effect due to interactions of the i

factor fixing and ranking of bifurcation parameters.

3.2. Algorithm to compute sensitivity indices

<sup>1</sup> <sup>¼</sup> <sup>X</sup> k

i

#### 3.1. The method of Sobol

The method of Sobol is the variance-decomposition method used for global sensitivity analysis. The method decomposes the output variance of a system or model into fractions and assigns them to the inputs factors. For example, given a model of the form Y ¼ f Xð Þ¼ f xð Þ <sup>1</sup>; x2;…; xk , where X is the vector of K uncertain parameters, which are independently generated within a unit hypercube i.e. xi <sup>∈</sup>½ � <sup>0</sup>; <sup>1</sup> <sup>k</sup> for <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>,…, K. Compared to the other GSA methods, the Sobol's method is one of the most commonly used variance-decomposition method, because of its ease of implementation. The method is primarily based on the decomposition of output Y into summands of elementary functions in terms of increasing dimensionality [1, 8],

$$f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_k) = f\_0 + \sum\_{i}^k f\_i(\mathbf{x}\_i) + \sum\_{i}^k \sum\_{i$$

In Eq. (5), f is integrable, f <sup>0</sup> is a constant, f <sup>i</sup> is a function of xi, f ij is a function of xi and xj and so on. Furthermore, all the terms in the functional decomposition are orthogonal, which leads toward the following definitions of the terms of the functional decomposition in term of conditional expected values.

$$\begin{aligned} f\_0 &= E(Y) \\ f\_i(\mathbf{x}\_i) &= E\_{\mathbf{x} \sim i}(Y|\mathbf{x}\_i) - f\_0 \\ f\_{ij}(\mathbf{x}\_i, \mathbf{x}\_j) &= E\_{\mathbf{x} \sim \vec{\eta}}(Y|\mathbf{x}\_i, \mathbf{x}\_j) - f\_0 - f\_i - f\_j \\ \dots & \dots \end{aligned} \tag{6}$$

where, E describes the mathematical expectation and x�<sup>i</sup> denotes all parameters except xi and so on. The total unconditional variance can be obtained by,

$$V = \int\_{\Omega} f^2(X)dx - f\_0^2 \tag{7}$$

From Eq. (7), the total unconditional variance can be decomposed in a similar manner like in Eq. (5) as,

$$V = \sum\_{i}^{k} V\_{i}(\mathbf{x}\_{i}) + \sum\_{i}^{k} \sum\_{i$$

where, V is the variance operator. The relationship between functions and partial variance are given by,

$$\begin{aligned} V\_i &= V\_{\mathbf{x}\_i}(\mathbb{E}\_{\mathbf{x}\sim i}(Y|\mathbf{x}\_i)) = V\left(f\_i(\mathbf{x}\_i)\right) \\ V\_{\vec{\eta}} &= V\_{\mathbf{x}\_i\mathbf{x}\_{\vec{\eta}}}\left(\mathbb{E}\_{\mathbf{x}\sim \vec{\eta}}\left(Y|\mathbf{x}\_i, \mathbf{x}\_{\vec{\eta}}\right)\right) - V\_i - V = V\left(f\_{\vec{\eta}}(\mathbf{x}\_i, \mathbf{x}\_{\vec{\eta}})\right) \\ &\dots \end{aligned} \tag{9}$$

Dividing both sides of the Eq. (8) by V, we get:

$$\mathbf{1} = \sum\_{i}^{k} \mathbf{S}\_{i}(\mathbf{x}\_{i}) + \sum\_{i}^{k} \sum\_{i$$

Where,

the CVS, the Sobol's method is computationally less expensive as compared to the other variance-decomposition methods, like FAST and sparse grid stochastic collocation method

The method of Sobol is the variance-decomposition method used for global sensitivity analysis. The method decomposes the output variance of a system or model into fractions and assigns them to the inputs factors. For example, given a model of the form Y ¼ f Xð Þ¼ f xð Þ <sup>1</sup>; x2;…; xk , where X is the vector of K uncertain parameters, which are independently generated within a unit hypercube i.e. xi <sup>∈</sup>½ � <sup>0</sup>; <sup>1</sup> <sup>k</sup> for <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>,…, K. Compared to the other GSA methods, the Sobol's method is one of the most commonly used variance-decomposition method, because of its ease of implementation. The method is primarily based on the decomposition of output Y into summands of elementary functions in terms of increasing dimension-

based on Smolyak algorithm [5, 12].

f xð Þ¼ <sup>1</sup>; <sup>x</sup>2;…; xk <sup>f</sup> <sup>0</sup> <sup>þ</sup><sup>X</sup>

conditional expected values.

<sup>V</sup> <sup>¼</sup> <sup>X</sup> k

i

Við Þþ xi

X k

X k

i<j

i

k

74 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

X k

X k

f ij xi; xj

� � � <sup>f</sup> <sup>0</sup> � <sup>f</sup> <sup>i</sup> � <sup>f</sup> <sup>j</sup>

2

� � <sup>þ</sup> … <sup>þ</sup> <sup>f</sup> <sup>1</sup>,2,3,…, <sup>k</sup>ð Þ <sup>x</sup>1; <sup>x</sup>2; <sup>x</sup>3;…; xk (5)

<sup>0</sup> (7)

� � <sup>þ</sup> … <sup>þ</sup> <sup>f</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>,…, <sup>k</sup>ð Þ <sup>x</sup>1; <sup>x</sup>2; <sup>x</sup>3;…; xk (8)

(6)

i<j

In Eq. (5), f is integrable, f <sup>0</sup> is a constant, f <sup>i</sup> is a function of xi, f ij is a function of xi and xj and so on. Furthermore, all the terms in the functional decomposition are orthogonal, which leads toward the following definitions of the terms of the functional decomposition in term of

where, E describes the mathematical expectation and x�<sup>i</sup> denotes all parameters except xi and

From Eq. (7), the total unconditional variance can be decomposed in a similar manner like in

ð Þ X dx � f

i

ð Þ¼ xi Ex�<sup>i</sup>ð Þ� Yjxi f <sup>0</sup>

V ¼ ð

Ω<sup>K</sup> f 2

Vij xi; xj

� � <sup>¼</sup> Ex�ij <sup>Y</sup>jxi; xj

i f i ð Þþ xi

f <sup>0</sup> ¼ E Yð Þ

f ij xi; xj

f i

…

so on. The total unconditional variance can be obtained by,

3.1. The method of Sobol

ality [1, 8],

Eq. (5) as,

$$\begin{aligned} S\_i &= \frac{V\_{\vec{ij}}}{V}, \text{ and} \\ S\_{\vec{ij}} &= \frac{V\_{\vec{ij}}}{V} \end{aligned} \tag{11}$$

where, Si is the main effect (first order sensitivity index) of the i th parameter on output uncertainty and Sij is the interaction effect of i th and j th parameters on output uncertainty. Further, the total sensitivity index, STi can be calculated as,

$$\mathbf{S}\_{T\_i} = \frac{E\_{\mathbf{x} \sim i} \left( V\_{\mathbf{x}\_i}(\mathbf{Y}|\mathbf{x} \sim i) \right)}{V} = 1 - \frac{V\_{\mathbf{x} \sim i} \left( E\_{\mathbf{x}\_i}(\mathbf{Y}|\mathbf{x} \sim i) \right)}{V} \tag{12}$$

In general, the main effect is used identify the most influential parameters (bifurcation parameters) and the total effect is taken into account for those parameters which are exempted from bifurcation analysis (factor fixing). The total effect, STi of the i th parameter means main effect plus higher-order effect due to interactions of the i th parameter. In this study, the interaction effects of parameters on the output (QoI) are negligible, therefore the main effects are used for factor fixing and ranking of bifurcation parameters.

#### 3.2. Algorithm to compute sensitivity indices

In this section, a detailed working algorithm is presented to compute the main effect, Si using the Monte Carlo simulations, we follow the steps, given in [1, 15].

1. Generate a random numbers matrix of row dimension 2K and column length N (the sample size) and split into two independent sampling matrices, Að Þ N;K and Bð Þ N;K by using LHS. Where, K is the number of uncertain model parameters.

$$A(\mathbf{N}, K) = \begin{bmatrix} \mathbf{x}\_{11} & \mathbf{x}\_{12} & \dots & \mathbf{x}\_{1K} \\ \mathbf{x}\_{21} & \mathbf{x}\_{22} & \dots & \mathbf{x}\_{2K} \\ \dots & \dots & \dots & \dots \\ \mathbf{x}\_{N1} & \mathbf{x}\_{N2} & \dots & \mathbf{x}\_{NN} \end{bmatrix} \tag{13}$$
 
$$\begin{bmatrix} \mathbf{x}\_{1(K+1)} & \mathbf{x}\_{1(K+2)} & \dots & \mathbf{x}\_{1(2K)} \end{bmatrix}$$

5. Finally, the main effect, Si of each parameter on the state variables is calculated.

X Nt

ð Þ t; j; t , i ¼ 1, 2, …, K (19)

Sensitivity Analysis: A Useful Tool for Bifurcation Analysis

http://dx.doi.org/10.5772/intechopen.72345

77

t¼0 Sti

In Eq. (19), NTs is the number of output variables (pressure and flow time series at all

The results of the UA and SA are greatly affected by the choice of input parameters distributions. In principle, the parameters distributions should be estimated using medical data. Unfortunately, the medical data is not easy to obtained. The input parameters distributions could be chosen according to the expert opinion or using the data from the literature. Due to limited data availability, here in this work the input parameters are randomized within �10%

The method of Sobol requires Nð Þ K þ 2 number of model simulations to compute Si. The main effect, Si is computed for N = [500, 1000, 2000, 3000, 4000] model runs. It is observed that, when the total number of simulations run N increases from 3000 then the sensitivity indices (Si) become stable [18]. Therefore, the minimum number of simulations for each parameter to

In this section, the sensitivity results based on main effect Si are presented. In order to calculate sensitivity time series, the method of Sobol is applied on each time point of the output QoI i.e. pressure and flow waves at each location of the arm arteries. For each parameter, there are two sensitivity time series at each segment of the arm arteries, one for the pressure and one for the flow. In total, K � NTs ¼ 45 � 33 ¼ 1485 sensitivity time series are obtained. In order to represent the sensitivity results in a compact way, mean absolute values of each pressure and flow sensitivity time series per parameter is taken. In this way, a matrix of dimension 45 � 33 is acquired, where each entry of the matrix represents the mean absolute values of pressure and flow sensitivity time series per parameter, see Figure 3. The numbers in the boxes show the impact (%) on the output (pressure and flow) when input parameters Ri;Ci ð Þ ; Li are randomized within the feasible ranges of �10%. The parameters having main effect, Si > 10% on output QoI are not shown in the Figure 3. Each row in Figure 3 represents the ranking of influential (bifurcation) parameters. For convenience, the electrical parameters Ri;Ci ð Þ ; Li , i ¼ 1, 2, 3, …, 15 that have impact greater than 10% on pressure and flow are considered as bifurcation parameters which further can be used in bifurcation analysis. For example, for pressure

j¼1

range of their base (nominal) values using Latin hypercube sampling (LHS).

1 Nt X NTs

Si <sup>¼</sup> <sup>1</sup> NTs

locations) and N<sup>t</sup> is the number of time-points [12].

achieve convergence of sensitivity indices is around 3000.

3.3. Input parameters distribution

3.4. Convergence of sensitivity indices

4. Results and discussion

$$B(\mathbf{N}, K) = \begin{bmatrix} \mathbf{x}\_{1(\mathsf{K}+1)} & \mathbf{x}\_{1(\mathsf{K}+2)} & \dots & \mathbf{x}\_{1(\mathsf{K})} \\ \mathbf{x}\_{2(\mathsf{K}+1)} & \mathbf{x}\_{2(\mathsf{K}+2)} & \dots & \mathbf{x}\_{2(2\mathsf{K})} \\ \dots & \dots & \dots & \dots \\ \mathbf{x}\_{\mathsf{N}(\mathsf{K}+1)} & \mathbf{x}\_{\mathsf{N}(\mathsf{K}+2)} & \dots & \mathbf{x}\_{\mathsf{N}(2\mathsf{K})} \end{bmatrix} \tag{14}$$

2. Define matrix Ci, which is matrix A except the i th column of matrix B.

$$\mathbf{C}\_{i}(\mathbf{N},K) = \begin{bmatrix} \mathbf{x}\_{11} & \mathbf{x}\_{12} & \dots & \mathbf{x}\_{1}(K+i) & \dots & \mathbf{x}\_{1K} \\ \mathbf{x}\_{21} & \mathbf{x}\_{22} & \dots & \mathbf{x}\_{2}(K+i) & \dots & \mathbf{x}\_{2K} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ \mathbf{x}\_{N1} & \mathbf{x}\_{N2} & \dots & \mathbf{x}\_{N}(K+i) & \dots & \mathbf{x}\_{NK} \end{bmatrix} \tag{15}$$


$$\begin{split} S\_{i} = \frac{V\_{i}}{V} = \frac{V\_{\text{xi}i}(E\_{\text{x}\sim i}(\mathbf{Y}|\mathbf{x}\_{i}))}{V} = \frac{V - \frac{1}{2\mathbf{N}}\sum\_{n=1}^{\text{N}} \left( Y\_{\text{B}}^{(n)} - Y\_{\text{C}\_{i}}^{(n)} \right)^{2}}{V} \\ = 1 - \frac{1}{2\mathbf{N} \times V} \sum\_{n=1}^{\text{N}} \left( Y\_{\text{B}}^{(n)} - Y\_{\text{C}\_{i}}^{(n)} \right)^{2} \end{split} \tag{16}$$

where,

$$V = \frac{1}{N} \sum\_{n=1}^{N} \left( Y\_{\mathcal{B}}^{(n)} \right)^{2} - E^{2} \tag{17}$$

and

$$E = \left(\frac{1}{\mathcal{N}} \sum\_{n=1}^{N} Y\_{\mathcal{B}}^{(n)}\right)^2 \tag{18}$$

The total variance ð Þ V and the expectation ð Þ E are also calculated at each time-point of pressure and flow waves with respect to each parameter.

5. Finally, the main effect, Si of each parameter on the state variables is calculated.

$$S\_{l} = \frac{1}{\mathbf{N}\_{T\_{s}}} \frac{1}{\mathbf{N}\_{l}} \sum\_{j=1}^{\mathbf{N}\_{l}} \sum\_{t=0}^{\mathbf{N}\_{l}} S\_{l\natural}(t, j, t), i = 1, 2, \dots, K \tag{19}$$

In Eq. (19), NTs is the number of output variables (pressure and flow time series at all locations) and N<sup>t</sup> is the number of time-points [12].

#### 3.3. Input parameters distribution

Að Þ¼ N;K

Bð Þ¼ N;K

76 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

<sup>V</sup> <sup>¼</sup> Vxið Þ Ex�<sup>i</sup>ð Þ <sup>Y</sup>jxi

pressure and flow waves with respect to each parameter.

V ¼

<sup>V</sup> <sup>¼</sup> <sup>1</sup> N X N

n¼1

<sup>E</sup> <sup>¼</sup> <sup>1</sup> N X N

2. Define matrix Ci, which is matrix A except the i

Cið Þ¼ N;K

model runs Nð Þ ¼ 4000 .

where,

and

estimator offered by Jansen [15–17].

Sti <sup>¼</sup> Vi

x<sup>11</sup> x<sup>12</sup> … x1<sup>K</sup> x<sup>21</sup> x<sup>22</sup> … x2<sup>K</sup> … … …… xN1 xN2 … xN<sup>K</sup>

x<sup>1</sup>ð Þ <sup>K</sup>þ<sup>1</sup> x<sup>1</sup>ð Þ <sup>K</sup>þ<sup>2</sup> … x1 2ð Þ <sup>K</sup> x<sup>2</sup>ð Þ <sup>K</sup>þ<sup>1</sup> x<sup>2</sup>ð Þ <sup>K</sup>þ<sup>2</sup> … x2 2ð Þ <sup>K</sup> … … …… x<sup>N</sup>ð Þ <sup>K</sup>þ<sup>1</sup> x<sup>N</sup>ð Þ <sup>K</sup>þ<sup>2</sup> … xN 2ð Þ <sup>K</sup>

x<sup>11</sup> x<sup>12</sup> … x1ð Þ K þ i … x1<sup>K</sup> x<sup>21</sup> x<sup>22</sup> … x2ð Þ K þ i … x2<sup>K</sup> … … …… …… xN1 xN2 … xNð Þ K þ i … xN<sup>K</sup>

3. Compute and save model runs for all parameter spaces using matrices A, B and Ci i.e. YA t; Ts ð Þ¼ ; N f Að Þ, YB t; Ts ð Þ¼ ; N f Bð Þ and YCi t; Ts ð Þ¼ ; N;K f Cð Þ<sup>i</sup> , where, t are the time points for one heart beat with period tp ¼ 0:8s, Ts represents the state variables (pressure and flow time series at six locations of arm artery Nð Þ Ts ¼ 15 and Nis the total number of

4. For the time dependent model outputs, we compute the time dependent main sensitivity index, of each parameter at each time-point of the pressure and flow waves, using the

> <sup>V</sup> � <sup>1</sup> 2N P N n¼1

<sup>¼</sup> <sup>1</sup> � <sup>1</sup>

Yð Þ <sup>n</sup> B � �<sup>2</sup>

n¼1

The total variance ð Þ V and the expectation ð Þ E are also calculated at each time-point of

Yð Þ <sup>n</sup> B !<sup>2</sup>

2N � V

Yð Þ <sup>n</sup> <sup>B</sup> � <sup>Y</sup>ð Þ <sup>n</sup> Ci � �<sup>2</sup>

> X N

Yð Þ <sup>n</sup> <sup>B</sup> � <sup>Y</sup>ð Þ <sup>n</sup> Ci � �<sup>2</sup>

� <sup>E</sup><sup>2</sup> (17)

n¼1

V

th column of matrix B.

(13)

(14)

(15)

(16)

(18)

The results of the UA and SA are greatly affected by the choice of input parameters distributions. In principle, the parameters distributions should be estimated using medical data. Unfortunately, the medical data is not easy to obtained. The input parameters distributions could be chosen according to the expert opinion or using the data from the literature. Due to limited data availability, here in this work the input parameters are randomized within �10% range of their base (nominal) values using Latin hypercube sampling (LHS).

#### 3.4. Convergence of sensitivity indices

The method of Sobol requires Nð Þ K þ 2 number of model simulations to compute Si. The main effect, Si is computed for N = [500, 1000, 2000, 3000, 4000] model runs. It is observed that, when the total number of simulations run N increases from 3000 then the sensitivity indices (Si) become stable [18]. Therefore, the minimum number of simulations for each parameter to achieve convergence of sensitivity indices is around 3000.

#### 4. Results and discussion

In this section, the sensitivity results based on main effect Si are presented. In order to calculate sensitivity time series, the method of Sobol is applied on each time point of the output QoI i.e. pressure and flow waves at each location of the arm arteries. For each parameter, there are two sensitivity time series at each segment of the arm arteries, one for the pressure and one for the flow. In total, K � NTs ¼ 45 � 33 ¼ 1485 sensitivity time series are obtained. In order to represent the sensitivity results in a compact way, mean absolute values of each pressure and flow sensitivity time series per parameter is taken. In this way, a matrix of dimension 45 � 33 is acquired, where each entry of the matrix represents the mean absolute values of pressure and flow sensitivity time series per parameter, see Figure 3. The numbers in the boxes show the impact (%) on the output (pressure and flow) when input parameters Ri;Ci ð Þ ; Li are randomized within the feasible ranges of �10%. The parameters having main effect, Si > 10% on output QoI are not shown in the Figure 3. Each row in Figure 3 represents the ranking of influential (bifurcation) parameters. For convenience, the electrical parameters Ri;Ci ð Þ ; Li , i ¼ 1, 2, 3, …, 15 that have impact greater than 10% on pressure and flow are considered as bifurcation parameters which further can be used in bifurcation analysis. For example, for pressure

Figure 3. Main effect sensitivity of 45-electrical parameters Ri; Ci ð Þ ; Li on pressure and flow time series pi ; qi at 15-segments ð Þ N1; N2; …; N<sup>15</sup> of the arm arteries with number of simulations run per parameter, N is 4000. In total N(K + 2) = 4000(45 + 2) = 188,000 = 0.188 million of simulations run are required to compute the main and total sensitivity indices. The total time taken to compute the sensitivity indices is approximately 3 hours.

at node-2, L<sup>1</sup> and L<sup>2</sup> are the bifurcation parameters, see in Figure 4 (top). Whereas, for flow at node -2, L<sup>1</sup> and L<sup>2</sup> are considered as bifurcation parameters, see Figure 4 (bottom).

In a similar fashion, each row of Figure 3 represents the ranking of bifurcation parameters which further can be used in bifurcation analysis. The parameters which have main effect Si < 10% can be exempted from the bifurcation analysis. The criteria for factor fixing vary from problem to problem.

#### 5. Conclusion

In this chapter, a 5-step procedure of global sensitivity analysis is presented to identify the bifurcation parameters in a lumped-parameter model of the arm arteries. Moreover, the proposed procedure can be applied on any morphology or structure of the systemic circulation (carotid bifurcation, aorta or complete systemic circulation). The results of sensitivity analysis are useful to identify and rank the bifurcation parameters, as well as help which parameters could be exempted from the bifurcation analysis. In this particular example of the arm arteries, 23 out of 45 parameters can be excluded from the bifurcation analysis. Whereas, 22 identified as bifurcation parameters, which further can be used/studied in the bifurcation analysis.

Figure 4. Ranking of bifurcation parameters Ri;Ci ð Þ ; Li in complete arm arteries for pressure (top) and flow (bottom) at node-2. It can be clearly seen that, L1, L<sup>2</sup> and L1, L<sup>3</sup> are considered as bifurcation parameters for pressure and flow at

Sensitivity Analysis: A Useful Tool for Bifurcation Analysis

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79

node-2 respectively.

at node-2, L<sup>1</sup> and L<sup>2</sup> are the bifurcation parameters, see in Figure 4 (top). Whereas, for flow at

15-segments ð Þ N1; N2; …; N<sup>15</sup> of the arm arteries with number of simulations run per parameter, N is 4000. In total N(K + 2) = 4000(45 + 2) = 188,000 = 0.188 million of simulations run are required to compute the main and total sensitivity

Figure 3. Main effect sensitivity of 45-electrical parameters Ri; Ci ð Þ ; Li on pressure and flow time series pi

; qi at

In a similar fashion, each row of Figure 3 represents the ranking of bifurcation parameters which further can be used in bifurcation analysis. The parameters which have main effect Si < 10% can be exempted from the bifurcation analysis. The criteria for factor fixing vary

In this chapter, a 5-step procedure of global sensitivity analysis is presented to identify the bifurcation parameters in a lumped-parameter model of the arm arteries. Moreover, the proposed procedure can be applied on any morphology or structure of the systemic circulation (carotid bifurcation, aorta or complete systemic circulation). The results of sensitivity analysis are useful to identify and rank the bifurcation parameters, as well as help which parameters could be exempted from the bifurcation analysis. In this particular example of the arm arteries, 23 out of 45 parameters can be excluded from the bifurcation analysis. Whereas, 22 identified as bifurcation parameters, which further can be used/studied in the bifurcation analysis.

node -2, L<sup>1</sup> and L<sup>2</sup> are considered as bifurcation parameters, see Figure 4 (bottom).

indices. The total time taken to compute the sensitivity indices is approximately 3 hours.

78 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

from problem to problem.

5. Conclusion

Figure 4. Ranking of bifurcation parameters Ri;Ci ð Þ ; Li in complete arm arteries for pressure (top) and flow (bottom) at node-2. It can be clearly seen that, L1, L<sup>2</sup> and L1, L<sup>3</sup> are considered as bifurcation parameters for pressure and flow at node-2 respectively.

#### Author details

Raheem Gul<sup>1</sup> \* and Stefan Bernhard<sup>2</sup>

\*Address all correspondence to: gulrehman@ciit.net.pk

1 COMSATS Institute of Information Technology, Abbottabad, Pakistan

2 Department of Electrical Engineering and Information Technology, Pforzheim University of Applied Sciences, Germany

[12] Gul R, Bernhard S. Parametric uncertainty and global sensitivity analysis in a model of the carotid bifurcation: Identification and ranking of most sensitive model parameters.

Sensitivity Analysis: A Useful Tool for Bifurcation Analysis

http://dx.doi.org/10.5772/intechopen.72345

81

[13] Noordergraaf A, Verdouw PD, Boom HBK. The use of an analog computer in a circula-

[14] Westerhof N, Noordergraaf A. Arterial viscoelasticity: A generalized model, effect of input impedance and wave travel in the systemic tree. Journal of Biomechanics. 1970;3:

[15] Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto F, Tarantola S. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index.

[16] Jansen MJW. Analysis of variance designs for model output. Computer Physics Commu-

[17] Jansen MJW, Rossing WAH, Daamen RA. Monte Carlo estimation of uncertainty contributions from several independent multivariate sources. In: Predictability and Nonlinear Modeling in Natural Sciences and Economics. Dordrecht: Kluwer Academic Publishers;

[18] Yang J. Convergence and uncertainty analyses in Monte-Carlo based analysis. Environ-

tion model. Progress in Cardiovascular Diseases. 1963;5(5):419-439

Computer Physics Communications. 2010;181:259-270

mental Modeling and Software. 2011;26:444-457

Mathematical Biosciences. 2015;269:104-116

357-379

nications. 1999;117:35-43

1994. pp. 334-343

#### References


[12] Gul R, Bernhard S. Parametric uncertainty and global sensitivity analysis in a model of the carotid bifurcation: Identification and ranking of most sensitive model parameters. Mathematical Biosciences. 2015;269:104-116

Author details

Applied Sciences, Germany

& Sons Ltd; 2008

Biology. 2011;5(6):336-346

metrics. 1991;33:161-174

medical Engineering. 2013:1-24

\* and Stefan Bernhard<sup>2</sup>

\*Address all correspondence to: gulrehman@ciit.net.pk

80 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Vol. 112. New York: Springer Verlag; 2004

1 COMSATS Institute of Information Technology, Abbottabad, Pakistan

2 Department of Electrical Engineering and Information Technology, Pforzheim University of

[1] Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S. Global Sensitivity Analysis. The Atrium, Chichester, West Sussex, England: John Wiley

[2] Kuznetsov YA. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences.

[3] Zi Z. Sensitivity analysis approaches applied to systems biology models. IET Systems

[4] Gul R, Schütte, Bernhard S. Mathematical modeling and sensitivity analysis of arterial anastomosis in arm arteries. Applied Mathematical Modeling. 2016;40:7724-7738

[5] Gul R. Mathematical modeling and sensitivity analysis of lumped-parameter model of

[6] Saltelli A, Tarantola S, Chan KPS. A quantitative model-independent method for global

[7] Morris MD. Factorial sampling plans for preliminary computational experiments. Techno-

[8] Sobol I. Sensitivity estimates for nonlinear mathematical models. Matematicheskoe

[9] Homma T, Saltelli A. Importance measures in global sensitivity analysis of nonlinear

[10] Chen P, Quarteroni A, Rozza G. Simulation-based uncertainty quantification of human arterial network hemodynamics. International Journal for Numerical Methods in Bio-

[11] Gul R, Bernhard S. Optimal measurement locations for diagnosis of aortic stenoses and aneurysms in a lumped-parameter model of the systemic circulation using sensitivity

the human cardiovascular system. PhD Thesis. Germany: FU Berlin; 2016

sensitivity analysis of model output. Technometrics. 1999;41(1):39-56

Modelirovanie. 1990;2:112-118. In Russian, translated in English

models. Reliability Engineering and System Safety. 1996;52:117

analysis. International Journal of Biomathematics. 2017;10(8):1750116

Raheem Gul<sup>1</sup>

References


**Chapter 5**

**Provisional chapter**

**Biological Hypercomputation and Degrees of Freedom**

This chapter presents the idea of biological hypercomputation (BH) and discusses how and why it entails degrees of freedom. Crossing a biological and computational point of view, the claim is made that living beings cannot be considered as machines in any sense of the word, the arguments are provided, and the consequence is drawn: the complexity of life is the very process by which living beings gain degrees of freedom. The leading thread for the analysis here is the relationship between matter, energy and information.

Biological hypercomputation (BH) is the title that summarizes the fact that living systems are not machines, and hence, do not process information like any machine, no matter what. The leading thread, thereafter, is to understanding the very way in which living systems process information. For a living being processing information is after all a matter of life or death.

A normal understanding of the issue would claim that living systems receive information from the environment and then process it, so that they would be able to learn and adapt to the changing environment. This text claims that it is a wrong take. Instead, it will be argued that there is no information before the very act of processing, and as a consequence, there is no information after the processing. Reality is the outcome of the very process of information

To be sure, ranging from the various Turing Machines (TM)—u-TM, o-TM, d-TM, and many others, up to the various levels of life, say, from bacteria to living cells to plants to animals, etc., there is a growing process by which new and previously unforeseen degrees of freedom are reached. Reality, nature and the world can be seen as an increasing process of degrees of

**Biological Hypercomputation and Degrees of Freedom**

DOI: 10.5772/intechopen.73179

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Carlos Eduardo Maldonado

**Abstract**

**1. Introduction**

processing.

Carlos Eduardo Maldonado

http://dx.doi.org/10.5772/intechopen.73179

Additional information is available at the end of the chapter

**Keywords:** life, computation, freedom, nature, complexity

Additional information is available at the end of the chapter

**Provisional chapter**

## **Biological Hypercomputation and Degrees of Freedom**

**Biological Hypercomputation and Degrees of Freedom**

DOI: 10.5772/intechopen.73179

#### Carlos Eduardo Maldonado Carlos Eduardo Maldonado Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73179

#### **Abstract**

This chapter presents the idea of biological hypercomputation (BH) and discusses how and why it entails degrees of freedom. Crossing a biological and computational point of view, the claim is made that living beings cannot be considered as machines in any sense of the word, the arguments are provided, and the consequence is drawn: the complexity of life is the very process by which living beings gain degrees of freedom. The leading thread for the analysis here is the relationship between matter, energy and information.

**Keywords:** life, computation, freedom, nature, complexity

#### **1. Introduction**

Biological hypercomputation (BH) is the title that summarizes the fact that living systems are not machines, and hence, do not process information like any machine, no matter what. The leading thread, thereafter, is to understanding the very way in which living systems process information. For a living being processing information is after all a matter of life or death.

A normal understanding of the issue would claim that living systems receive information from the environment and then process it, so that they would be able to learn and adapt to the changing environment. This text claims that it is a wrong take. Instead, it will be argued that there is no information before the very act of processing, and as a consequence, there is no information after the processing. Reality is the outcome of the very process of information processing.

To be sure, ranging from the various Turing Machines (TM)—u-TM, o-TM, d-TM, and many others, up to the various levels of life, say, from bacteria to living cells to plants to animals, etc., there is a growing process by which new and previously unforeseen degrees of freedom are reached. Reality, nature and the world can be seen as an increasing process of degrees of

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

freedom. This view, however, should not be understood hierarchically. Rather, the architecture is to be seen as a fractal organization. As the growing architecture of life increases, an amplification of previous lower layers is produced, that does not obliterate or supersedes the lower levels.

Matter ∊ Energy ∊ Information (1)

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 85

Meaning that energy explains better and deeper what matter is supposed to explain, and that information explains still much better and deeper what energy was supposed to explain. Thus, the story from classical mechanics to thermodynamics to information theory is the very story by which a higher and more achieved understanding of reality is accomplished. This, however should not imply by any means a linear and necessarily accumulative process, but a story of improving, gaining more knowledge and wisdom about the universe, nature, and society. The story of knowledge can very appropriately be grasped in terms of

The amount of information of a system depends on how probable an event is. In other words, the higher the probability that an event takes place, the lower the amount—and the quality—of the system. And vice versa: the lower the probability that an event happens, the higher—and better—the information of that event. Briefly said, the information is inversely proportional to the probability—of an event. This means that information is a measure of how

As it can be seen, the characterization I have provided about information does not necessarily coincide with the standard view—say, Shannon- that identifies information and entropy. Rather, entropy can be adequately grasped as the quantification of randomness—of a system. Therefore, it can be easily stated that the amount of information in the universe as a whole can only increase, which means that what was considered as an expanding universe in terms of the transition from energy to matter is really an expansion of information [4]. This coincides with Formula (1) presented above. Information is the concept or reality that encompasses and explains energy and matter (mass). Whereas matter presents a view of the world rather disconnected (indeed, it is the laws that gather and unify matter, according to Newton), and energy unfolds and transforms as a set of expressions and processes explained in terms of Fourier's and Boltzmann's laws, information set out a convergent, yet in-process, reality that allows for evolution [5]. Formula (2) expresses the unity of matter, energy, and information, showing hence that there are not three things but only one—read thereafter in terms of infor-

Matter − Energy − Information (2)

Formula (2) serves as a synthesis that claims that matter, energy, and information are one and the same thing that were historically discovered or brought out by stages. That story is exactly the story that leads from classical mechanics to thermodynamics to information theory.

Now, summarized, information theory is cryptography, quantum information, entanglement, and teleportation [6]. These are three different faces of one and the same development—an achievement that crosses the most conspicuous technologies nowadays, as well as spearhead science in any domain you wish. By and large information pervades both science and culture,

bifurcations [2].

surprising something is [3].

mation, according to Formula (1).

currently.

To be sure, the process by which the living systems gain degrees of freedom coincides with the entire weave of networks, fractal architectures, and interdependencies among layers, planes, and contexts. As a result, life and nature can be seen as a fantastic interweaving of times, layers, and interdependencies.

A number of works have been done that permit the approach of this text. The various authors, takes, and approaches will be discussed along the text, which brings a fresh state-of-the arte about biological hypercomputation and the weave of information processing in nature.

Throughout this chapter the basic argument is that life can be seen as a large cooperative game by which the increasing complexity of life lies on the very increasing processes of information processing.

## **2. A serious problem: what is information (one more time!)**

In order to explain nature band the universe, the eighteenth century invented or created a physical concept, namely mas or matter. Thanks to it, the entire universe could be explained in terms of three basic but elegant laws. This was I. Newton's achievement. Anything that is or happens is the outcome of what happens to mass, thus: relationships of action-and-reaction, the relationship between two bodies based on a force equal in magnitude and opposite in direction with each other, and the heaviness or mass of a body so that the heavier body will always attract the lighter body to itself, i.e. gravity.

In the framework of the study of heat a brand new science originating both in physics and chemistry arose in the nineteenth century that introduced a quite different new concept to understand and explain nature and reality, namely energy. Whereas mass or matter is a univocal concept, energy shows a manifold of ways of existence: caloric energy, kinetic energy, potential energy, and others. The contributions of scientists such as Fourier, Carnot, Boltzmann, Lord Kelvin, and others, had as a result the identification of three laws of thermodynamics. Those three laws were sufficient to explain not only the dynamics of the universe and nature, but also the arrow of time and the challenges it arose.

Now, the twentieth century discovered still another physical concept highly more complex and different form the two original ones. This was the concept of information, originally brought out by Shannon and Weaver, which, however, was to know still further developments manly due to the research on cryptography, quantum theory, quantum computation and quantum science, mainly. The crux here is that information is a physical and yet nontangential, non-material concept—in contrast with mass and energy [1].

Now, the relation among the three physical concepts can be adequately identified as follows:

$$\text{Matter } \mathfrak{e} \text{ Energy } \mathfrak{e} \text{ Information} \tag{1}$$

Meaning that energy explains better and deeper what matter is supposed to explain, and that information explains still much better and deeper what energy was supposed to explain. Thus, the story from classical mechanics to thermodynamics to information theory is the very story by which a higher and more achieved understanding of reality is accomplished. This, however should not imply by any means a linear and necessarily accumulative process, but a story of improving, gaining more knowledge and wisdom about the universe, nature, and society. The story of knowledge can very appropriately be grasped in terms of bifurcations [2].

freedom. This view, however, should not be understood hierarchically. Rather, the architecture is to be seen as a fractal organization. As the growing architecture of life increases, an amplification of previous lower layers is produced, that does not obliterate or supersedes the

To be sure, the process by which the living systems gain degrees of freedom coincides with the entire weave of networks, fractal architectures, and interdependencies among layers, planes, and contexts. As a result, life and nature can be seen as a fantastic interweaving of

A number of works have been done that permit the approach of this text. The various authors, takes, and approaches will be discussed along the text, which brings a fresh state-of-the arte about biological hypercomputation and the weave of information processing in nature.

Throughout this chapter the basic argument is that life can be seen as a large cooperative game by which the increasing complexity of life lies on the very increasing processes of infor-

In order to explain nature band the universe, the eighteenth century invented or created a physical concept, namely mas or matter. Thanks to it, the entire universe could be explained in terms of three basic but elegant laws. This was I. Newton's achievement. Anything that is or happens is the outcome of what happens to mass, thus: relationships of action-and-reaction, the relationship between two bodies based on a force equal in magnitude and opposite in direction with each other, and the heaviness or mass of a body so that the heavier body will

In the framework of the study of heat a brand new science originating both in physics and chemistry arose in the nineteenth century that introduced a quite different new concept to understand and explain nature and reality, namely energy. Whereas mass or matter is a univocal concept, energy shows a manifold of ways of existence: caloric energy, kinetic energy, potential energy, and others. The contributions of scientists such as Fourier, Carnot, Boltzmann, Lord Kelvin, and others, had as a result the identification of three laws of thermodynamics. Those three laws were sufficient to explain not only the dynamics of the universe

Now, the twentieth century discovered still another physical concept highly more complex and different form the two original ones. This was the concept of information, originally brought out by Shannon and Weaver, which, however, was to know still further developments manly due to the research on cryptography, quantum theory, quantum computation and quantum science, mainly. The crux here is that information is a physical and yet non-

Now, the relation among the three physical concepts can be adequately identified as follows:

**2. A serious problem: what is information (one more time!)**

always attract the lighter body to itself, i.e. gravity.

and nature, but also the arrow of time and the challenges it arose.

tangential, non-material concept—in contrast with mass and energy [1].

lower levels.

mation processing.

times, layers, and interdependencies.

84 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

The amount of information of a system depends on how probable an event is. In other words, the higher the probability that an event takes place, the lower the amount—and the quality—of the system. And vice versa: the lower the probability that an event happens, the higher—and better—the information of that event. Briefly said, the information is inversely proportional to the probability—of an event. This means that information is a measure of how surprising something is [3].

As it can be seen, the characterization I have provided about information does not necessarily coincide with the standard view—say, Shannon- that identifies information and entropy. Rather, entropy can be adequately grasped as the quantification of randomness—of a system.

Therefore, it can be easily stated that the amount of information in the universe as a whole can only increase, which means that what was considered as an expanding universe in terms of the transition from energy to matter is really an expansion of information [4]. This coincides with Formula (1) presented above. Information is the concept or reality that encompasses and explains energy and matter (mass). Whereas matter presents a view of the world rather disconnected (indeed, it is the laws that gather and unify matter, according to Newton), and energy unfolds and transforms as a set of expressions and processes explained in terms of Fourier's and Boltzmann's laws, information set out a convergent, yet in-process, reality that allows for evolution [5]. Formula (2) expresses the unity of matter, energy, and information, showing hence that there are not three things but only one—read thereafter in terms of information, according to Formula (1).

$$\text{Matter} - \text{Energy} - \text{Information} \tag{2}$$

Formula (2) serves as a synthesis that claims that matter, energy, and information are one and the same thing that were historically discovered or brought out by stages. That story is exactly the story that leads from classical mechanics to thermodynamics to information theory.

Now, summarized, information theory is cryptography, quantum information, entanglement, and teleportation [6]. These are three different faces of one and the same development—an achievement that crosses the most conspicuous technologies nowadays, as well as spearhead science in any domain you wish. By and large information pervades both science and culture, currently.

A short survey to the history of computation may be very helpful here. Indeed, information processing encounters five main levels of development, thus:

of entities but of information [7]. Exactly in this sense, too, teleportation is not so much the transportation of a physical entity, what of what makes that entity as such, namely information. It is information, indeed, what is teleported, not the thing as such, in its materiality.

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 87

Rightly understood, we can safely say that there is information processing in nature. Briefly

Living beings do not just read the environment; moreover, they accordingly unceasingly create brand new information that was non-existing before. In other words, living beings read the environment but they also, at the same time, write on the environment they are reading

Whereas any Turing Machine (TM) processes information top-down, sequentially, linearly, and mechanistically, according to the prevailing Von Neumann architecture of TM-computers, living beings, to say the least, process information in-non classical way if the framework is the

This text argues that the processing of information among living beings consists not only and not so much—in rightly reading and interpreting the environment and the surroundings, but also—and mainly—in introducing or creating brand new information into the world,

Thus, over against a popular understanding of the issue, information processing does not have anything to do with things such as "following a thread," "analyzing," nor even "understanding." Processing information can rather be grasped in biological or medical terms as metabolizing, i.e., changing one thing—say A—into another, B, for instance -. Good metabolizing, hence, transforms a nice meal into a poem or a scientific paper, or also a nice rest in the

For the living beings, processing information is a matter of life or death—in that a bad information processing may entail danger, peril or death. Biologically or culturally speaking, information processing might entail identifying a good male or female a god territory, the presence of a friend or an enemy, who truly loves you and who just pretends to love you, which food is healthy and which poisonous, for instance. Living beings that process information rightly may encounter better circumstances for adaptation and learning, and be literally the fittest. Evolution and adaptation are after all, it appears, a matter of good information processing.

Yet, life is not just an emergent property of the universe. A number of authors have claimed that the universe herself is alive and conscious [8–10]. The claim is hence about consciousness not as an epiphenomenon of the universe, and life as an emergent stance. In this take, life is an essential feature of the universe that can be traced back to the very processing of information, precisely. For the sake of brevity I shall put such a claim here into brackets and focus on the

**3. Living beings process information: a complexity understanding**

Matter is information, which brings us back to Formulas (1) and (2).

said, nature can be understood in terms of information.

on—if the analogy is permitted here.

evening into insight and strength, for example.

way in which living systems do process information.

Turing-Church thesis.

correspondingly.


It is clear that throughout the four mentioned layers, the very technological capabilities are developed and enhanced, for example via the so-called Moore law (after G. Moore, then president of Intel), that deals with the speed of change of the processors in computers. Undoubtedly, information processing has been changing the very understanding of computing and information, as well as it has been the subject—ultimately of software engineering as well as from hardware engineering—the two basic domains of computation, roughly said.

From the standpoint of quantum theory, information sets out the ground to unify epistemology and ontology. The link that allows such a connection is the wave function. Quantum science does not allow for the distinction, and even less the hierarchy, between ontology and epistemology, on either side. Instead, physics is about what we can say about nature—not about what nature is, any longer.

To be sure, the concept-behavior that allows for the unity of ontology and epistemology is the concept of entanglement, a concept firstly introduced by Schrödinger (*Vershränkung*), but truly implemented by J. Bell. When entangled, the unity of two or more particles becomes more important than the particles individually considered. Entanglement is not really entanglement of entities but of information [7]. Exactly in this sense, too, teleportation is not so much the transportation of a physical entity, what of what makes that entity as such, namely information. It is information, indeed, what is teleported, not the thing as such, in its materiality. Matter is information, which brings us back to Formulas (1) and (2).

Rightly understood, we can safely say that there is information processing in nature. Briefly said, nature can be understood in terms of information.

## **3. Living beings process information: a complexity understanding**

A short survey to the history of computation may be very helpful here. Indeed, information

• Linear, sequential, top-down information processing. Amply, this is the largely predominant paradigm in computation and computing science. Ever since the first computers, Eniac I, Eniac II, etc., on until now, computers and computations are defined by the Von Neumann architecture, and the Church-Turing thesis. A programmer programs a program and the computer follows the indications and directions. Design, parametrization, control are the key operations and ways of working and dealing with computers and computation. • Dynamic and evolving information processing. Genetic algorithms, time series, and the first steps in artificial intelligence (AI) such as the works by Ray, Conway, and neural networks, can be said to define the second layer of information processing. The shift toward bottom-up concerns and interests is produced that largely enhance computation as a dynamic process. Not eventually modeling and simulation come along with literal explosion

• Biologically inspired information processing. More recently bio-inspired computation takes as model or guide the behavior of living systems—say, DNA, proteins, ant colonies, etc.—and develops programs that come closer to the way living beings behave. This kind of information processing has shed so far brand new fresh lights on to the very way in which computing science had being previously conceived. Thus, computation and biology at large come closer and learn from each other making computation much more flexible

• Biological hypercomputation. On the basis of the third layer, trying to understand the way in which living beings process information clearly entails a distance from classical computation, no matter what. Such a computation has been named biological hypercomputation.

It is clear that throughout the four mentioned layers, the very technological capabilities are developed and enhanced, for example via the so-called Moore law (after G. Moore, then president of Intel), that deals with the speed of change of the processors in computers. Undoubtedly, information processing has been changing the very understanding of computing and information, as well as it has been the subject—ultimately of software engineering as well as from hardware engineering—the two basic domains of computation, roughly said.

From the standpoint of quantum theory, information sets out the ground to unify epistemology and ontology. The link that allows such a connection is the wave function. Quantum science does not allow for the distinction, and even less the hierarchy, between ontology and epistemology, on either side. Instead, physics is about what we can say about nature—not

To be sure, the concept-behavior that allows for the unity of ontology and epistemology is the concept of entanglement, a concept firstly introduced by Schrödinger (*Vershränkung*), but truly implemented by J. Bell. When entangled, the unity of two or more particles becomes more important than the particles individually considered. Entanglement is not really entanglement

processing encounters five main levels of development, thus:

86 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

of programming languages aimed at a variety of goals and abilities.

and robust at the same time. Adaptive computation comes to the fore.

This is the core of this text.

about what nature is, any longer.

Living beings do not just read the environment; moreover, they accordingly unceasingly create brand new information that was non-existing before. In other words, living beings read the environment but they also, at the same time, write on the environment they are reading on—if the analogy is permitted here.

Whereas any Turing Machine (TM) processes information top-down, sequentially, linearly, and mechanistically, according to the prevailing Von Neumann architecture of TM-computers, living beings, to say the least, process information in-non classical way if the framework is the Turing-Church thesis.

This text argues that the processing of information among living beings consists not only and not so much—in rightly reading and interpreting the environment and the surroundings, but also—and mainly—in introducing or creating brand new information into the world, correspondingly.

Thus, over against a popular understanding of the issue, information processing does not have anything to do with things such as "following a thread," "analyzing," nor even "understanding." Processing information can rather be grasped in biological or medical terms as metabolizing, i.e., changing one thing—say A—into another, B, for instance -. Good metabolizing, hence, transforms a nice meal into a poem or a scientific paper, or also a nice rest in the evening into insight and strength, for example.

For the living beings, processing information is a matter of life or death—in that a bad information processing may entail danger, peril or death. Biologically or culturally speaking, information processing might entail identifying a good male or female a god territory, the presence of a friend or an enemy, who truly loves you and who just pretends to love you, which food is healthy and which poisonous, for instance. Living beings that process information rightly may encounter better circumstances for adaptation and learning, and be literally the fittest. Evolution and adaptation are after all, it appears, a matter of good information processing.

Yet, life is not just an emergent property of the universe. A number of authors have claimed that the universe herself is alive and conscious [8–10]. The claim is hence about consciousness not as an epiphenomenon of the universe, and life as an emergent stance. In this take, life is an essential feature of the universe that can be traced back to the very processing of information, precisely. For the sake of brevity I shall put such a claim here into brackets and focus on the way in which living systems do process information.

Whereas at the same time living beings—from bacteria to cells, from organisms to biomes, for example—continuously read the environment and the surroundings they create brand new information that is brought into the world in a variety of forms, thus: as acts and actions, as behaviors, as creation in same cases of tools and rods, as the use of tools as devices aimed for a certain goal, definitely as language and communication, and as forms of organization, whether individual or collective [11]. It is the very creation of new information into the world, which is usually grasped as adaptation. Adaptation is the biological concept that originally is the outcome of information processing.

or even symbolic epigenetics, becomes central here, for it most adequately expresses the idea that very much as there is no difference between software and hardware, in the same tenure, there is no difference between nature and culture. They are closely intertwined and cannot be

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 89

As a consequence, computing (= BH) is for life one and the same thing with evolution. Evolving and computing are two faces of one and the same token. Evolution is the process through

Information processing among living beings takes place as an unceasing creation of information that is introduced into the world. Thus, information processing is not just reading the environment and adequately interpreting it, but, moreover, correspondingly, bringing new information into the world in the form of actions, processes, behaviors, or achievements.

There is no information before the information processing by living beings, very much as there is no information after the information processing by living beings. Information exists in so far as it is processed, that is, both created and changed. In other words, there is no reality previous to the processing of information, but neither is there any reality "outside" the information processing. Yet, the universe becomes increasingly complex precisely thanks to the unceasing processing of information, i.e. the reduction of entropy in the universe. Life is

As a consequence, the universe is alive it appears. A number of interpretations in this sense can be mentioned that challenge the traditional view of the universe that goes as follows:

PhysicsInorganic ChemistryOrganic ChemistryBiologyCulture (3)

I argue that Formula (3) is erroneous because it is set on the physical predominance of physics over biochemistry or biology, for example. Formula (3) is really a translation of Formula (1), and thus, it sets out the ground for a reductionist view of nature and the universe. A clear

The increasing complexity of reality and the universe is one and the same process as the increasing complexity in the information processes [13, 14]. We have been gaining an enormous field in understanding what life does and how it is made possible. Thus, for instance, regarding the realm of plants, Refs. [15, 16] have shed solid lights about how they think, literally. Down the scale, [17] did steadily study the way in which information processing takes place among bacteria. Furthermore, information processing has been studied additionally among social insect networks [18]. The range of studies and the cope is always wider and deeper, encompassing animals at large [19], until of course we reach the level of human

that system that creates order and exhibits the best form of order possible.

reductionist approach serves as ground for that formula.

beings, and computers and computation.

divided or split.

which living beings gain degrees of freedom.

**4. Living beings gain degrees of freedom**

As it can be easily seen, living beings compute. Moreover, life without computation is inconceivable.

Computation is therefore a concept that truly means interaction—with the environment, and with other living beings. Natural computation is much more than a metaphor, but the distinguishing feature of living beings. Life is a large weave of computation that takes place in a manifold of ways simultaneously, as follows:

There is classical computation in nature in either form of a Turing Machine. At the same time there is parallel computation, and multilevel computation, non-local computation as well as distributed computation, fuzzy and random computation very much as also quantum computation and emergent computation, not to mention interactive computation. A typology of computation and computational models can be seen in [12]. A metaphor can be introduced here, namely a variety of computations corresponds to the diversity of life and living beings. It appears that ecology and biology go hand in hand with information theory and computing science, even though the two latter are more recent that the two before [13].

To be sure, biological hypercomputation is the way in which living beings process information—is anything but machines. However, there are a number of explanations within the health sciences and the living sciences that still operate mechanically. Thus, for instance, the functioning of the brain many times is explained in terms of "on" and "off" switches or operations, and the very functioning of the heart is usually conceived as a pumping machine, period. Many other cases can be introduced here. The crux is that those explanations still owe a big deal to the past and remain very short vis-à-vis more contemporary and spearhead explanations. Computing sciences is such one of those frontline explanations, under the proviso that it be not a Turing Machine one, in any concern.

Biological hypercomputation (BH) has been introduced [15] to mean that life and the living processes cannot be understood from lower stances: in other words, life is to be explained as it happens: life is an uncompressible "program." The most crucial and basic problems of living beings are certainly not tackled in terms of mathematical functions, whatsoever. Instead, the problems living beings faced are normally solved in terms of non-classical logics, a field that has not deserved as much attention as its own value.

More fundamentally still, computationally speaking, the difference between software and hardware is irrelevant. More exactly, it does not exist. The phenotype expresses the genotype according to evolution and the environment. Epigenetics, for example behavioral epigenetics or even symbolic epigenetics, becomes central here, for it most adequately expresses the idea that very much as there is no difference between software and hardware, in the same tenure, there is no difference between nature and culture. They are closely intertwined and cannot be divided or split.

As a consequence, computing (= BH) is for life one and the same thing with evolution. Evolving and computing are two faces of one and the same token. Evolution is the process through which living beings gain degrees of freedom.

## **4. Living beings gain degrees of freedom**

Whereas at the same time living beings—from bacteria to cells, from organisms to biomes, for example—continuously read the environment and the surroundings they create brand new information that is brought into the world in a variety of forms, thus: as acts and actions, as behaviors, as creation in same cases of tools and rods, as the use of tools as devices aimed for a certain goal, definitely as language and communication, and as forms of organization, whether individual or collective [11]. It is the very creation of new information into the world, which is usually grasped as adaptation. Adaptation is the biological concept that originally is

As it can be easily seen, living beings compute. Moreover, life without computation is

Computation is therefore a concept that truly means interaction—with the environment, and with other living beings. Natural computation is much more than a metaphor, but the distinguishing feature of living beings. Life is a large weave of computation that takes place in a

There is classical computation in nature in either form of a Turing Machine. At the same time there is parallel computation, and multilevel computation, non-local computation as well as distributed computation, fuzzy and random computation very much as also quantum computation and emergent computation, not to mention interactive computation. A typology of computation and computational models can be seen in [12]. A metaphor can be introduced here, namely a variety of computations corresponds to the diversity of life and living beings. It appears that ecology and biology go hand in hand with information theory and computing

To be sure, biological hypercomputation is the way in which living beings process information—is anything but machines. However, there are a number of explanations within the health sciences and the living sciences that still operate mechanically. Thus, for instance, the functioning of the brain many times is explained in terms of "on" and "off" switches or operations, and the very functioning of the heart is usually conceived as a pumping machine, period. Many other cases can be introduced here. The crux is that those explanations still owe a big deal to the past and remain very short vis-à-vis more contemporary and spearhead explanations. Computing sciences is such one of those frontline explanations, under the pro-

Biological hypercomputation (BH) has been introduced [15] to mean that life and the living processes cannot be understood from lower stances: in other words, life is to be explained as it happens: life is an uncompressible "program." The most crucial and basic problems of living beings are certainly not tackled in terms of mathematical functions, whatsoever. Instead, the problems living beings faced are normally solved in terms of non-classical logics, a field that

More fundamentally still, computationally speaking, the difference between software and hardware is irrelevant. More exactly, it does not exist. The phenotype expresses the genotype according to evolution and the environment. Epigenetics, for example behavioral epigenetics

science, even though the two latter are more recent that the two before [13].

viso that it be not a Turing Machine one, in any concern.

has not deserved as much attention as its own value.

the outcome of information processing.

88 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

manifold of ways simultaneously, as follows:

inconceivable.

Information processing among living beings takes place as an unceasing creation of information that is introduced into the world. Thus, information processing is not just reading the environment and adequately interpreting it, but, moreover, correspondingly, bringing new information into the world in the form of actions, processes, behaviors, or achievements.

There is no information before the information processing by living beings, very much as there is no information after the information processing by living beings. Information exists in so far as it is processed, that is, both created and changed. In other words, there is no reality previous to the processing of information, but neither is there any reality "outside" the information processing. Yet, the universe becomes increasingly complex precisely thanks to the unceasing processing of information, i.e. the reduction of entropy in the universe. Life is that system that creates order and exhibits the best form of order possible.

As a consequence, the universe is alive it appears. A number of interpretations in this sense can be mentioned that challenge the traditional view of the universe that goes as follows:

PhysicsInorganic ChemistryOrganic ChemistryBiologyCulture (3)

I argue that Formula (3) is erroneous because it is set on the physical predominance of physics over biochemistry or biology, for example. Formula (3) is really a translation of Formula (1), and thus, it sets out the ground for a reductionist view of nature and the universe. A clear reductionist approach serves as ground for that formula.

The increasing complexity of reality and the universe is one and the same process as the increasing complexity in the information processes [13, 14]. We have been gaining an enormous field in understanding what life does and how it is made possible. Thus, for instance, regarding the realm of plants, Refs. [15, 16] have shed solid lights about how they think, literally. Down the scale, [17] did steadily study the way in which information processing takes place among bacteria. Furthermore, information processing has been studied additionally among social insect networks [18]. The range of studies and the cope is always wider and deeper, encompassing animals at large [19], until of course we reach the level of human beings, and computers and computation.

The concept "degrees of freedom" can be understood from a wide take that says that it consists in the number of parameters a system exhibits or has that may vary independently (Wikipedia) until the calculation of the various scenarios in a system or the number of dimensions that are needed to determine a full vector. In complexity science or theory the concept is pivotal and allows for crossing different approaches, sciences, discipline, and interpretations aiming at making explicit what complexity is all about.

We can safely say that the higher the degrees of freedom a system has the more complex it is. And vice versa, the lower the degrees of freedom, the less complex it is. Formula (4) synthetizes this:

$$\text{\textquotedblleft Degrees of Freedom \textquotedblright} \text{Complexity} \tag{4}$$

**5. The weave of life**

versa.

on or very near the equator.

that *telos* are necessary.

story of learning, adaptation and coevolution.

Information is a continuous process that emerges as a result of intricate non-linear interactions of living beings among themselves and with their environment that both creates nature and the world as it happens, and at the same time changes it unceasingly. This is exactly the

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 91

As it is well known, living beings evolve in rugged adaptive landscapes—a concept originally introduced by Ch. Darwin in 1859, originally named as fitness landscape. Life is a wonderful complex weave that has no center, but many bubs, clusters and nodes, continuously changing on the ever-changing environment on earth. Only that the timescales in geology are vastly deeper than those of the living beings, not to mention human beings. Indeed, whereas the basic time-scale of life is the decade or even the month for some species, the basic time-scale in geology is 1000.000 years. Against all odds, this is the ultimate time-scale of life on earth.

Life is a large and robust weave of cooperation, commensalism, and mutualism, rather than competition and predation. [20] has made on this subject a great contribution, namely the importance of symbiogenesis: a large network (instead than a chain) of mutual interdependency among living beings so that each organism and species benefits from others, and vice

The story of life is a story of increasing complexity, i.e. biodiversity. Such diversity is said to be at three levels, thus: genetic, natural, and cultural. The countries that have the three kinds of biodiversity are called "megadiverse" and they are 17 countries, to-date—all of them place

The story of life pivots around six fantastic moments of increasing diversity followed by six massive extinctions. At each stage, life has made of herself a wonderful asset of possibilities, forms, shapes, structures, behaviors, and characteristics that correspond with dynamics through which, as a whole, amazing degrees of freedom are been attained or reached. And yet, that story is non-teleological. In other words, life has no ends or goals, and evolution has no purpose whatsoever. Such was indeed the scandal that Darwin's *The origins of species by means of natural selection* (1859) meant particularly vis-à-vis the cornerstone of the western civilization, namely teleology [21]. The western world needs believe that there are *telos*, and

To the discomfort of those who believe in goals end, whether their own or imposed or suggested, the theory of evolution—namely the best theory ever developed to think about change and transformation—introduces the idea of the absence of goals or ends (*telos*). Well, the theory of symbiogenesis comes to complete, so to speak, the theory of evolution by showing that the various stages of complexity of life, firstly, and secondly that the whole weave of life naturally tends toward increasing complexity and hence to cooperation and interdependency.

As a consequence, living beings are symbiont: not only it is a cooperative interaction but also a prolonged one among different species and/or organisms. Mutualism and not competition

is the rule in nature, it appears. Nature is the realm of freedom par excellence.

Now, living beings are what they do. And the best they do is information processing, exactly. The better a living system processes information, the fittest it is. As a consequence, the more degrees of freedom it gains. Hence, living is a matter of gaining degrees of freedom. Death, it seems, is the complete loss of degrees of freedom. All possibilities are then closed.

Therefore the claim can be made that information plays an "active" role in the living systems, and a "passive" role in classical physics. Analogously as it happens with any Turing Machine, namely information is something that "happens" in the processor that is conceived as a black box. The audience or the researcher knows what enters (input) and knows what comes out (output), but ignores what and how happens in the black box. Classical computation seems to reduce the degrees of freedom by accepting or assessing the processing of information as a "black box." On the contrary, (BH) is the process by which new degrees of freedom are gained in so far as new adaptations, networks and behaviors are made possible, for instance?

Information processing does not stand as a condition for evolution and adaptation. On the contrary, it is the very process of evolution both of the living beings and the environment. Coevolution is the name that best suits both stances in their entanglement. Living is nothing else that coping with the information of the surroundings and accordingly changing that information in terms of brand new information brought into the world.

The better the living system processes information, the better it copes with challenges, solves problems, adapts to a continuously changing environment. Life ends when it "overdoses" of information and the living being is unable to process it. A saturation point is reached and the system can no longer process any further information [20].

Summarizing, life consists in the process by which information processing is carried out and hence unceasingly new degrees of freedom are reached. The system can be said to be "young." Contrarily, death happens when the processing of information is not possible any longer—it becomes slow, saturated, new information cannot be processed accordingly, and the "screen" slows down in bringing out the information required.

Life is a matter of "controlling" the saturation of information, and thereafter, of information processing. Software and hardware are one and the same thing.

## **5. The weave of life**

The concept "degrees of freedom" can be understood from a wide take that says that it consists in the number of parameters a system exhibits or has that may vary independently (Wikipedia) until the calculation of the various scenarios in a system or the number of dimensions that are needed to determine a full vector. In complexity science or theory the concept is pivotal and allows for crossing different approaches, sciences, discipline, and interpretations

We can safely say that the higher the degrees of freedom a system has the more complex it is. And vice versa, the lower the degrees of freedom, the less complex it is. Formula (4) synthe-

> Degrees of Freedom > Complexity (4)

Now, living beings are what they do. And the best they do is information processing, exactly. The better a living system processes information, the fittest it is. As a consequence, the more degrees of freedom it gains. Hence, living is a matter of gaining degrees of freedom. Death, it

Therefore the claim can be made that information plays an "active" role in the living systems, and a "passive" role in classical physics. Analogously as it happens with any Turing Machine, namely information is something that "happens" in the processor that is conceived as a black box. The audience or the researcher knows what enters (input) and knows what comes out (output), but ignores what and how happens in the black box. Classical computation seems to reduce the degrees of freedom by accepting or assessing the processing of information as a "black box." On the contrary, (BH) is the process by which new degrees of freedom are gained

seems, is the complete loss of degrees of freedom. All possibilities are then closed.

in so far as new adaptations, networks and behaviors are made possible, for instance?

information in terms of brand new information brought into the world.

system can no longer process any further information [20].

the "screen" slows down in bringing out the information required.

processing. Software and hardware are one and the same thing.

Information processing does not stand as a condition for evolution and adaptation. On the contrary, it is the very process of evolution both of the living beings and the environment. Coevolution is the name that best suits both stances in their entanglement. Living is nothing else that coping with the information of the surroundings and accordingly changing that

The better the living system processes information, the better it copes with challenges, solves problems, adapts to a continuously changing environment. Life ends when it "overdoses" of information and the living being is unable to process it. A saturation point is reached and the

Summarizing, life consists in the process by which information processing is carried out and hence unceasingly new degrees of freedom are reached. The system can be said to be "young." Contrarily, death happens when the processing of information is not possible any longer—it becomes slow, saturated, new information cannot be processed accordingly, and

Life is a matter of "controlling" the saturation of information, and thereafter, of information

aiming at making explicit what complexity is all about.

90 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

tizes this:

Information is a continuous process that emerges as a result of intricate non-linear interactions of living beings among themselves and with their environment that both creates nature and the world as it happens, and at the same time changes it unceasingly. This is exactly the story of learning, adaptation and coevolution.

As it is well known, living beings evolve in rugged adaptive landscapes—a concept originally introduced by Ch. Darwin in 1859, originally named as fitness landscape. Life is a wonderful complex weave that has no center, but many bubs, clusters and nodes, continuously changing on the ever-changing environment on earth. Only that the timescales in geology are vastly deeper than those of the living beings, not to mention human beings. Indeed, whereas the basic time-scale of life is the decade or even the month for some species, the basic time-scale in geology is 1000.000 years. Against all odds, this is the ultimate time-scale of life on earth.

Life is a large and robust weave of cooperation, commensalism, and mutualism, rather than competition and predation. [20] has made on this subject a great contribution, namely the importance of symbiogenesis: a large network (instead than a chain) of mutual interdependency among living beings so that each organism and species benefits from others, and vice versa.

The story of life is a story of increasing complexity, i.e. biodiversity. Such diversity is said to be at three levels, thus: genetic, natural, and cultural. The countries that have the three kinds of biodiversity are called "megadiverse" and they are 17 countries, to-date—all of them place on or very near the equator.

The story of life pivots around six fantastic moments of increasing diversity followed by six massive extinctions. At each stage, life has made of herself a wonderful asset of possibilities, forms, shapes, structures, behaviors, and characteristics that correspond with dynamics through which, as a whole, amazing degrees of freedom are been attained or reached. And yet, that story is non-teleological. In other words, life has no ends or goals, and evolution has no purpose whatsoever. Such was indeed the scandal that Darwin's *The origins of species by means of natural selection* (1859) meant particularly vis-à-vis the cornerstone of the western civilization, namely teleology [21]. The western world needs believe that there are *telos*, and that *telos* are necessary.

To the discomfort of those who believe in goals end, whether their own or imposed or suggested, the theory of evolution—namely the best theory ever developed to think about change and transformation—introduces the idea of the absence of goals or ends (*telos*). Well, the theory of symbiogenesis comes to complete, so to speak, the theory of evolution by showing that the various stages of complexity of life, firstly, and secondly that the whole weave of life naturally tends toward increasing complexity and hence to cooperation and interdependency.

As a consequence, living beings are symbiont: not only it is a cooperative interaction but also a prolonged one among different species and/or organisms. Mutualism and not competition is the rule in nature, it appears. Nature is the realm of freedom par excellence.

#### **6. Conclusions**

We have three concepts: matter, energy and information. However, a right understanding of them brings to the fore the fact that they are not three, but one and the same concept: matter-energy-information. Moreover, information can be rightly understood as fundamental (= grounding) energy and mass.

What happens to matter is truly a matter of energy and relations among various types of energy. And still the very processes, dynamics and configurations of energy are dynamics in and of information. Formula (4) expresses this understanding.

$$\text{Information} \le \text{Energy} \le \text{Matter (Mass)}\tag{5}$$

[4] Roederer JG. Information and Its Role in Nature. Springer Verlag; 2005

University Press; 2001

Vintage Books; 2008

pe=pdf

2010

2012;**7**(7):e40337

Vintage Books; 2003

Farrar, Straus and Giroux; 2010

Theory. Springer Verlag; 1990

[5] Chaisson E. Cosmic Evolution. The Rise of Complexity in Nature. Harvard: Harvard

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[6] Zeilinger A. Dance of the Photons. From Einstein to Quantum Teleportation. New York:

[7] Gilder L. The Age of Entanglement. When Quantum Physics was Reborn. New York:

[8] Kafatos M, Nadeau R. The Conscious Universe. Part and Whole in Modern Physical

[9] Gribbin J. In the Beginning: The Birth of the Living Universe. Little Brown & Co.; 1995

[10] Kauffman S. Humanity in a Creative Universe. Oxford: Oxford University Press; 2016

Ciencias Exactas, Físicas y Naturales; 2017. pp. 109-124

Cambridge: Cambridge University Press; 2013

New York Academy of Sciences. Oct 2009;**1178**:78-90

Animal Studies. New York: Columbia University Press; 2012

Intelligence. Island Press; 2015

[11] Maldonado CE. Hipercomputación biológica y comunicación entre los seres vivos. In: Arboleda LC, editor. Un Festschrift para José Luis Villaveces. Bogotá: Academia de

[12] Burgin M, Dodig-Crnkovic G. Typologies of Computation and Computational Models; 2013. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.446.5721&rep=rep1&ty

[13] Linewaever CH, Davies PCW, Ruse M, editors. Complexity and the Arrow of Time.

[14] Barrow JD, Davies PCW, Harper CL Jr, editors. Science and Ultimate Reality. Quantum Theory, Cosmology and Complexity. Cambridge: Cambridge University Press; 2005

[15] Maldonado CE, Gómez-Cruz N. Biological hypercomputation: A new research problem

[16] Mancuso S, Viola A. Brilliant Green. The Surprising History and Science of Plant

[17] Baluska F. Signaling in Plants (Signaling and Communication in Plants). Springer Verlag;

[18] Ben-Jacob E. Learning from bacteria about natural information processing. Annals of the

[19] Waters JS, Fewell JH. Information processing in social insect networks. PLoS One.

[20] Gross A, Vallely A, editors. Animals and the Human Imagination. A Companion to

[21] Margulis L, Sagan D. Acquiring Genomes. A Theory of the Origins of Species. New York:

in complexity theory. Complexity. 2015;**20**(4):8-18. ISSN 1099-0526

Formula (5) simply means that matter explains less (and worst) what energy does explain, and furthermore, information allows for a deeper and better understanding of what energy means to explain.

Now, when looked under the light of life and how life is possible, the three concepts are transformed into one process, namely biological hypercomputation. In the processing of information among the living beings matter, energy and information become one single unity: life. Thus, life allows for the overcoming three different layers of reality (physics, thermodynamics, and information); such overcoming is also the non-differentiation between software and hardware. Processing information for life consists in gaining degrees of freedom.

Living beings process information much more than (just) matter and energy, which they definitely do. Evolution can be seen as the processing of information processing by living beings through which the world and nature are complexified. Information allows for the rejection of a reductionist approach of life and nature.

#### **Author details**

Carlos Eduardo Maldonado

Address all correspondence to: maldonado.carloseduardo@gmail.com

School of Medicine, Universidad El Bosque, Bogotá, Colombia

#### **References**


[4] Roederer JG. Information and Its Role in Nature. Springer Verlag; 2005

**6. Conclusions**

means to explain.

**Author details**

**References**

Carlos Eduardo Maldonado

University Press; 2010

(= grounding) energy and mass.

a reductionist approach of life and nature.

We have three concepts: matter, energy and information. However, a right understanding of them brings to the fore the fact that they are not three, but one and the same concept: matter-energy-information. Moreover, information can be rightly understood as fundamental

What happens to matter is truly a matter of energy and relations among various types of energy. And still the very processes, dynamics and configurations of energy are dynamics in

Information ≤ Energy ≤ Matter (Mass) (5)

Formula (5) simply means that matter explains less (and worst) what energy does explain, and furthermore, information allows for a deeper and better understanding of what energy

Now, when looked under the light of life and how life is possible, the three concepts are transformed into one process, namely biological hypercomputation. In the processing of information among the living beings matter, energy and information become one single unity: life. Thus, life allows for the overcoming three different layers of reality (physics, thermodynamics, and information); such overcoming is also the non-differentiation between software and

Living beings process information much more than (just) matter and energy, which they definitely do. Evolution can be seen as the processing of information processing by living beings through which the world and nature are complexified. Information allows for the rejection of

[1] Gleick J. The Information: A History, a Theory, a Flood. New York: Vintage Books; 2012

[3] Vedral V. Decoding Reality. The Universe as Quantum Information. Oxford: Oxford

hardware. Processing information for life consists in gaining degrees of freedom.

Address all correspondence to: maldonado.carloseduardo@gmail.com

School of Medicine, Universidad El Bosque, Bogotá, Colombia

[2] Serres M. Éléments d'histoire des sciences. Bordas; 1993

and of information. Formula (4) expresses this understanding.

92 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals


**Section 2**

**Complexity in Physical Systems**

**Complexity in Physical Systems**

**Chapter 6**

Provisional chapter

**Self-Organization, Coherence and Turbulence in Laser**

DOI: 10.5772/intechopen.71478

In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubicquintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent "internal modes" that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure,

Keywords: optical turbulence, dissipative solitons, chaos in nonlinear optical systems, generalized cubic-quintic nonlinear Ginzburg-Landau equation,

Coherent and partially coherent structures emerging in nonlinear systems far from the thermodynamic equilibrium play an important role in different research areas ranging from hydrodynamics and plasma physics to biophysics and sociology. Nontrivial dynamics of such

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Self-Organization, Coherence and Turbulence

Vladimir L. Kalashnikov and Evgeni Sorokin

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Vladimir L. Kalashnikov and Evgeni Sorokin

http://dx.doi.org/10.5772/intechopen.71478

namely, a dissipative Raman soliton.

dissipative Raman soliton

1. Introduction

**Optics**

in Laser Optics

Abstract

Provisional chapter

#### **Self-Organization, Coherence and Turbulence in Laser Optics** Self-Organization, Coherence and Turbulence

DOI: 10.5772/intechopen.71478

in Laser Optics

Vladimir L. Kalashnikov and Evgeni Sorokin

Additional information is available at the end of the chapter Vladimir L. Kalashnikov and Evgeni Sorokin

http://dx.doi.org/10.5772/intechopen.71478 Additional information is available at the end of the chapter

#### Abstract

In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubicquintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent "internal modes" that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure, namely, a dissipative Raman soliton.

Keywords: optical turbulence, dissipative solitons, chaos in nonlinear optical systems, generalized cubic-quintic nonlinear Ginzburg-Landau equation, dissipative Raman soliton

#### 1. Introduction

Coherent and partially coherent structures emerging in nonlinear systems far from the thermodynamic equilibrium play an important role in different research areas ranging from hydrodynamics and plasma physics to biophysics and sociology. Nontrivial dynamics of such

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

structures including chaos and turbulence is a challenge for modern nonlinear science and one may assume that "the problem of turbulence is one of the central problems in theoretical physics" [1]. The reasonable approach to this issue, which can translate some contra-intuitive and obscure ideas in this area into explicit and verifiable concepts, is a realization of simpler dynamics in quite different material context. Such an approach can be named metaphorical or analog modeling [2], and a rapid progress of modern laser technology provides an ideal playground for such enterprise due to high controllability, relative simplicity, and unique potential of statistic gathering. Such progress was marked by the development of the concept of a dissipative soliton (DS) [3]. The existence of DS under nonequilibrium conditions requires a well-organized energy exchange with an environment so that this energy flow forms a nontrivial internal structure of DS, which provides the energy redistribution inside it and can distort the soliton coherence. Such a DS with nontrivial internal structure can develop in lasers, and the DS dynamics can become chaotic and turbulent [3–5]. For instance, such emergent structures can be considered as a classical analog of Bose-Einstein condensate in low dissipative limit and, contrariwise, as a primitive analog of cell in the case of extensive and wellstructured energy exchange with an environment. Formally, these inherently nonHamiltonian entities mimic some features of Hamiltonian systems that remain an obscure and insufficiently explored topic regarding the fundamental properties of coherent dissipative structures. The range of turbulence, noise, and rogue wave phenomena emulated by the optical DS is so broad that it turns them into a universal testbed for studies in the fields of nonlinear dynamical systems and nonequilibrium thermodynamics.

describing an extremely broad class of phenomena. It is possible that the most known one is the famous nonlinear Schrödinger equation (NSE) which describes an evolution of slowly varying wave in a nonlinear medium and can be considered as a "metaphoric" simulation tool

> ∂<sup>2</sup>ω ∂kj∂kl

The dimensionality of this equation is relative: the evolutional coordinate can be a time T or a propagation distance z (T\$ z), the transverse coordinate can be transverse multidimensional spatial xj (j = 1…d) one or a local time t (x\$ t, d = 1). The Fourier representations of a "field" slowly varying envelope Ψ are interchangeable between frequency and momentum domains (ω\$ k, d = 1). Eq. (1) may describe the propagation of optical pulses in a nonlinear medium (then Ψ is a complex field amplitude and |Ψ|2 is proportional to a field power), the capillary waves on a fluid surface, the Langmuir waves in plasma, or the weakly nonlinear Bose-gas in classic limit (in the last case Eq. (1) represents the famous time-dependent Gross-Pitaevskii equation [16]).

term in Eq. (1) can have the different forms; in particular, a nonlinear response can be non-

The notion of turbulence is fuzzy in some sense. Here, the turbulence will be treated as a phenomenon related to the excitation of a sufficiently large number of degrees of freedom that causes a loss of their mutual phase information [15]. As a consequence, a wave package decouples into a set of individual modes ("particles") which interaction can be described in the framework of kinetic theory as many-particle collisions in Bose-gas. In other words, as some degrees of freedom become very large for sufficiently large energies, phase information becomes irrelevant, and the waves decohere [8, 15]. Thus, a wave can be considered as a set of

Thus, we come to a "kinetic" theory of turbulence, for example, to a model of four-boson

Such an equation becomes nontrivial in a dissipative environment [17, 18]. A simple generalization of NSE (1) taking into account the dissipative effects includes a saturable gain (energy

"source") σ, dissipative nonlinearity (self-amplitude modulation, SAM) Ϝ(|Ψ|2

dissipation (spectral in the sense of dissipation in the Fourier space) P<sup>d</sup>

δ ωð Þ þ ω<sup>1</sup> � ω<sup>2</sup> � ω<sup>3</sup> d k<sup>1</sup>

nk1nk2nk<sup>3</sup> þ nknk2nk<sup>3</sup> � nknk1nk<sup>2</sup> � nknk1nk<sup>3</sup> ð Þ

Here Ψ(x, t) is a slowly varying amplitude of wave propagating in dispersive (P<sup>d</sup>

<sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> � �j j <sup>Ψ</sup> <sup>2</sup>

) and nonlinear ( <sup>∂</sup><sup>ω</sup>

decoupled "modes" nk in a spectral (or wave-number) space:

interaction described by the nonlinear Schrödinger equation:

� � !

∂<sup>2</sup>Ψ ∂xj∂xl <sup>þ</sup> <sup>i</sup> <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> !

<sup>Ψ</sup> - term; let us <sup>γ</sup> � <sup>∂</sup><sup>ω</sup>

Ψð Þk Ψ k<sup>0</sup> h i ð Þ ¼ nkδ k � k<sup>0</sup> ð Þ: (2)

! d k<sup>2</sup> ! d k<sup>3</sup> ! :

j j <sup>Ψ</sup> <sup>2</sup>

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

Ψ ¼ 0: (1)

99

j,l¼1 ∂2 ω ∂kj∂kl ∂2Ψ ∂xj∂xl -

(3)

), and spectral

Ψ ∂xj∂xl :

j,l¼<sup>1</sup> <sup>α</sup> xj; xl � � <sup>∂</sup><sup>2</sup>

<sup>∂</sup>j j <sup>Ψ</sup> 2) medium. The nonlinear

for a study of nonlinear phenomena far from equilibrium [14, 15]:

j¼1 ∂ω ∂kj ∂Ψ ∂xj � i 2 X<sup>d</sup> j,l¼1

∂Ψ <sup>∂</sup><sup>T</sup> <sup>þ</sup>X<sup>d</sup>

term; let us <sup>β</sup> � <sup>∂</sup>2<sup>ω</sup>

instantaneous.

∂kj∂kl

∂nk <sup>∂</sup><sup>t</sup> <sup>∝</sup> ð

> � δ k ! þ k<sup>1</sup> ! � k<sup>2</sup> ! � k<sup>3</sup>

In this work, we conjecture a spectacular analogy between the spectral structures of DS and strong Langmuir turbulence. Such close relation leads to chaotization of DS dynamics with the energy growth. This analogy is deepened by analysis of energy flows inside DS so that a DS can be represented as a "glass of boiling water" or, mathematically, as an ensemble of interacting quasi-particles or "nonlinear modes." The phase decoupling of these "modes" leads to turbulence or DS dissolving. Such a representation open the door for building the kinetic theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment. Moreover, our preliminary investigations demonstrated a mechanism of turbulence control provided by noninstantaneous nonlinearity (stimulated Raman scattering in optical case) [6]. This phenomenon is especially interesting because an inherently noisy process (Raman scattering) suppresses a turbulence under some conditions that is the manifestation of stochastic resonance, which can be significant for a dissipation control in coherent quantum systems (particularly, a quantum computer and a quantum cryptography device).

#### 2. Analogy between DS and turbulence

The phenomenon of turbulence appears in many areas of our experience ranging from atmospheric and oceanic rogue events, aero- and hydrodynamics, optics to cardiology and neurophysiology [1, 5, 7–13]. Such a broad class of phenomena cannot be grasped by some single and simple model. However, there are some comparatively simple equations which allow describing an extremely broad class of phenomena. It is possible that the most known one is the famous nonlinear Schrödinger equation (NSE) which describes an evolution of slowly varying wave in a nonlinear medium and can be considered as a "metaphoric" simulation tool for a study of nonlinear phenomena far from equilibrium [14, 15]:

structures including chaos and turbulence is a challenge for modern nonlinear science and one may assume that "the problem of turbulence is one of the central problems in theoretical physics" [1]. The reasonable approach to this issue, which can translate some contra-intuitive and obscure ideas in this area into explicit and verifiable concepts, is a realization of simpler dynamics in quite different material context. Such an approach can be named metaphorical or analog modeling [2], and a rapid progress of modern laser technology provides an ideal playground for such enterprise due to high controllability, relative simplicity, and unique potential of statistic gathering. Such progress was marked by the development of the concept of a dissipative soliton (DS) [3]. The existence of DS under nonequilibrium conditions requires a well-organized energy exchange with an environment so that this energy flow forms a nontrivial internal structure of DS, which provides the energy redistribution inside it and can distort the soliton coherence. Such a DS with nontrivial internal structure can develop in lasers, and the DS dynamics can become chaotic and turbulent [3–5]. For instance, such emergent structures can be considered as a classical analog of Bose-Einstein condensate in low dissipative limit and, contrariwise, as a primitive analog of cell in the case of extensive and wellstructured energy exchange with an environment. Formally, these inherently nonHamiltonian entities mimic some features of Hamiltonian systems that remain an obscure and insufficiently explored topic regarding the fundamental properties of coherent dissipative structures. The range of turbulence, noise, and rogue wave phenomena emulated by the optical DS is so broad that it turns them into a universal testbed for studies in the fields of nonlinear dynamical

98 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

In this work, we conjecture a spectacular analogy between the spectral structures of DS and strong Langmuir turbulence. Such close relation leads to chaotization of DS dynamics with the energy growth. This analogy is deepened by analysis of energy flows inside DS so that a DS can be represented as a "glass of boiling water" or, mathematically, as an ensemble of interacting quasi-particles or "nonlinear modes." The phase decoupling of these "modes" leads to turbulence or DS dissolving. Such a representation open the door for building the kinetic theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment. Moreover, our preliminary investigations demonstrated a mechanism of turbulence control provided by noninstantaneous nonlinearity (stimulated Raman scattering in optical case) [6]. This phenomenon is especially interesting because an inherently noisy process (Raman scattering) suppresses a turbulence under some conditions that is the manifestation of stochastic resonance, which can be significant for a dissipation control in coherent quantum systems (particularly, a quantum computer and a

The phenomenon of turbulence appears in many areas of our experience ranging from atmospheric and oceanic rogue events, aero- and hydrodynamics, optics to cardiology and neurophysiology [1, 5, 7–13]. Such a broad class of phenomena cannot be grasped by some single and simple model. However, there are some comparatively simple equations which allow

systems and nonequilibrium thermodynamics.

quantum cryptography device).

2. Analogy between DS and turbulence

$$\frac{\partial \Psi}{\partial T} + \sum\_{j=1}^{d} \frac{\partial \omega}{\partial k\_{j}} \frac{\partial \Psi}{\partial x\_{j}} - \frac{i}{2} \sum\_{j,l=1}^{d} \frac{\partial^{2} \omega}{\partial k\_{j} \partial k\_{l}} \frac{\partial^{2} \Psi}{\partial x\_{j} \partial x\_{l}} + i \left(\frac{\partial \omega}{\partial |\Psi|^{2}}\right) |\Psi|^{2} \Psi = 0. \tag{1}$$

The dimensionality of this equation is relative: the evolutional coordinate can be a time T or a propagation distance z (T\$ z), the transverse coordinate can be transverse multidimensional spatial xj (j = 1…d) one or a local time t (x\$ t, d = 1). The Fourier representations of a "field" slowly varying envelope Ψ are interchangeable between frequency and momentum domains (ω\$ k, d = 1). Eq. (1) may describe the propagation of optical pulses in a nonlinear medium (then Ψ is a complex field amplitude and |Ψ|2 is proportional to a field power), the capillary waves on a fluid surface, the Langmuir waves in plasma, or the weakly nonlinear Bose-gas in classic limit (in the last case Eq. (1) represents the famous time-dependent Gross-Pitaevskii equation [16]).

Here Ψ(x, t) is a slowly varying amplitude of wave propagating in dispersive (P<sup>d</sup> j,l¼1 ∂2 ω ∂kj∂kl ∂2Ψ ∂xj∂xl term; let us <sup>β</sup> � <sup>∂</sup>2<sup>ω</sup> ∂kj∂kl ) and nonlinear ( <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> � �j j <sup>Ψ</sup> <sup>2</sup> <sup>Ψ</sup> - term; let us <sup>γ</sup> � <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> 2) medium. The nonlinear term in Eq. (1) can have the different forms; in particular, a nonlinear response can be noninstantaneous.

The notion of turbulence is fuzzy in some sense. Here, the turbulence will be treated as a phenomenon related to the excitation of a sufficiently large number of degrees of freedom that causes a loss of their mutual phase information [15]. As a consequence, a wave package decouples into a set of individual modes ("particles") which interaction can be described in the framework of kinetic theory as many-particle collisions in Bose-gas. In other words, as some degrees of freedom become very large for sufficiently large energies, phase information becomes irrelevant, and the waves decohere [8, 15]. Thus, a wave can be considered as a set of decoupled "modes" nk in a spectral (or wave-number) space:

$$
\langle \Psi(k)\Psi(k')\rangle = n\_k \delta(k-k').\tag{2}
$$

Thus, we come to a "kinetic" theory of turbulence, for example, to a model of four-boson interaction described by the nonlinear Schrödinger equation:

$$\begin{split} \frac{\partial n\_k}{\partial t} & \propto \left[ (n\_{k\_1} n\_{k\_2} n\_{k\_3} + n\_k n\_{k\_2} n\_{k\_3} - n\_k n\_{k\_1} n\_{k\_2} - n\_k n\_{k\_1} n\_{k\_3}) \right. \\ & \times \delta \left( \overrightarrow{k} + \overrightarrow{k\_1} - \overrightarrow{k\_2} - \overrightarrow{k\_3} \right) \delta (\omega + \omega\_1 - \omega\_2 - \omega\_3) d \overset{\rightarrow}{d} \overrightarrow{k\_1} \; d \; \overrightarrow{k\_2} \; d \; \overrightarrow{k\_3}. \end{split} \tag{3}$$

Such an equation becomes nontrivial in a dissipative environment [17, 18]. A simple generalization of NSE (1) taking into account the dissipative effects includes a saturable gain (energy "source") σ, dissipative nonlinearity (self-amplitude modulation, SAM) Ϝ(|Ψ|2 ), and spectral dissipation (spectral in the sense of dissipation in the Fourier space) P<sup>d</sup> j,l¼<sup>1</sup> <sup>α</sup> xj; xl � � <sup>∂</sup><sup>2</sup> Ψ ∂xj∂xl :

$$\text{RHS of Eq.}(1) = \sigma \Psi + \operatorname{\mathcal{F}}\left(|\Psi|^2\right)\Psi + \sum\_{j\_l=1}^d \alpha(\mathbf{x}\_j, \mathbf{x}\_l) \frac{\partial^2 \Psi}{\partial \mathbf{x}\_j \partial \mathbf{x}\_l}.\tag{4}$$

Eqs. (1) and (4) called the generalized complex nonlinear Ginzburg-Landau equation have the strongly localized (in x-space) steady-state (in T-space) solutions which are named dissipative solitons (DS) [3]. A classical (nondissipative) soliton, which possesses the quite specific mathematical properties [18–20], develops due to mutual compensation of dispersive spreading and self-compression caused by the phase nonlinearity under the condition of <sup>∂</sup>2<sup>ω</sup> ∂kj∂kl � <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> <sup>&</sup>gt; 0 and is stable in a (1 + 1)-dimensional (i.e., T plus d = 1 in Eq. (1)) case.<sup>1</sup> The parameters of such soliton are not fixed but only interrelated. One may say that a soliton "lives in solitude" ("pratyekabuddha," Figure 1).

Dissipation adds new bounds on the soliton parameters and fixes them so that one may say that the DS lives in "the heart of nonlinear world" ("bodhisattva," Figure 2).

The mutual balance of dispersion and phase nonlinearity remains a crucial factor for DS formation, but its physical meaning differs substantially from that for nondissipative soliton. The crucial factor here is a resonance between dispersive (linear) waves and DS: equality of their wave-numbers defines the frequency window where DS can exist. Indeed, a wavenumber of DS is <sup>q</sup> <sup>=</sup> <sup>γ</sup>P<sup>0</sup> (P<sup>0</sup> � max(|Ψ|<sup>2</sup> )) [23]. The dispersion relation providing the resonance with linear waves is k(ω) = βω<sup>2</sup> /2. To be stable (i.e., nonradiating), the DS spectrum has to be localized within a frequency window �Δ: <sup>k</sup>(�Δ) = <sup>q</sup>, where <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2γP0=β p (Figure 3).<sup>2</sup> Formation of these "domain walls" [24–26] due to phase effects in a dissipative system results in natural frequency cut-off, which is essential for inherent analogy between DS and a turbulent entity.

However, sole dispersive balance is not sufficient for the DS stability. The spectral dissipation �αΔ<sup>2</sup> plays a crucial role cutting the spectrum and defining the DS width (Figure 4). As

will be seen, this factor is crucial for dissipative soliton turbulence. A spectral dissipation

Figure 4. Wigner representation of DS (a) and cut-off due to spectral dissipation defining the DS width (b).

Figure 3. Resonance conditions for DS and linear waves and DS spectra in dependence on DS energy (a); and the Wigner

the first term is leading) that results in the additional relation between soliton spectral width

condition for the soliton existence which combines the dissipative and nondissipative fac-

The key feature of DS is its nontrivial internal structure revealing itself in the phase inhomo-

∂t <sup>2</sup> .

βκ <sup>≤</sup> <sup>1</sup>=2.<sup>3</sup> One has to note, that both considered mechanisms of DS formation act in the spectral domain and, as was shown in [28], a transition to spectral domain is fruitful for

κP0=α p . In combination with the dispersive relation, it gives the

βκ ≤

1=3 if E ! ∞ 1=2 if E ! 0

) <sup>≈</sup> <sup>κ</sup>|Ψ|<sup>2</sup> � <sup>ζ</sup>|Ψ|<sup>4</sup> <sup>+</sup> …, where

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

101

, where E is a DS energy.

must be balanced by a nonlinear gain �κP<sup>0</sup> (we assume <sup>Ϝ</sup>(|Ψ|<sup>2</sup>

geneity<sup>4</sup> and the internal energy redistribution (E is an energy flow):

More precise analysis [27] gives the conditions of asymptotical stability: αγ

The measure of this inhomogeneity is a so-called chirp <sup>Θ</sup> <sup>∝</sup> <sup>∂</sup><sup>2</sup> argð Þ <sup>Ψ</sup>

and its peak power: <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

(time-spectral) diagram of DS (b) [23].

developing a DS theory.

tors: αγ

3

4

Figure 1. Soliton exists under a balance between phase nonlinearity and dispersion [21, 22].

$$\begin{array}{c} \textbf{phase\ nonlinearity} \longrightarrow \textbf{O} \xleftarrow{\text{losses}} \text{ dispersion} \\ \textbf{!} \text{ gain} \end{array}$$

Figure 2. DS parameters are fixed by both nondissipative and dissipative factors [21, 22].

<sup>1</sup> Further, namely one-dimensional (d = 1) systems will be under consideration that is a quite precise approximation for solid-state and fiber laser dynamics [21].

<sup>2</sup> One has to remind the x \$ t and k\$ ω dualities in Eq. (1).

RHS of Eq:ð Þ¼ <sup>1</sup> <sup>σ</sup><sup>Ψ</sup> <sup>þ</sup> <sup>Ϝ</sup> j j <sup>Ψ</sup> <sup>2</sup> � �

100 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

self-compression caused by the phase nonlinearity under the condition of <sup>∂</sup>2<sup>ω</sup>

that the DS lives in "the heart of nonlinear world" ("bodhisattva," Figure 2).

to be localized within a frequency window �Δ: <sup>k</sup>(�Δ) = <sup>q</sup>, where <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Figure 1. Soliton exists under a balance between phase nonlinearity and dispersion [21, 22].

Figure 2. DS parameters are fixed by both nondissipative and dissipative factors [21, 22].

("pratyekabuddha," Figure 1).

number of DS is <sup>q</sup> <sup>=</sup> <sup>γ</sup>P<sup>0</sup> (P<sup>0</sup> � max(|Ψ|<sup>2</sup>

nance with linear waves is k(ω) = βω<sup>2</sup>

a turbulent entity.

1

2

solid-state and fiber laser dynamics [21].

One has to remind the x \$ t and k\$ ω dualities in Eq. (1).

<sup>Ψ</sup> <sup>þ</sup> <sup>X</sup> d

Eqs. (1) and (4) called the generalized complex nonlinear Ginzburg-Landau equation have the strongly localized (in x-space) steady-state (in T-space) solutions which are named dissipative solitons (DS) [3]. A classical (nondissipative) soliton, which possesses the quite specific mathematical properties [18–20], develops due to mutual compensation of dispersive spreading and

stable in a (1 + 1)-dimensional (i.e., T plus d = 1 in Eq. (1)) case.<sup>1</sup> The parameters of such soliton are not fixed but only interrelated. One may say that a soliton "lives in solitude"

Dissipation adds new bounds on the soliton parameters and fixes them so that one may say

The mutual balance of dispersion and phase nonlinearity remains a crucial factor for DS formation, but its physical meaning differs substantially from that for nondissipative soliton. The crucial factor here is a resonance between dispersive (linear) waves and DS: equality of their wave-numbers defines the frequency window where DS can exist. Indeed, a wave-

Formation of these "domain walls" [24–26] due to phase effects in a dissipative system results in natural frequency cut-off, which is essential for inherent analogy between DS and

However, sole dispersive balance is not sufficient for the DS stability. The spectral dissipation �αΔ<sup>2</sup> plays a crucial role cutting the spectrum and defining the DS width (Figure 4). As

Further, namely one-dimensional (d = 1) systems will be under consideration that is a quite precise approximation for

j, <sup>l</sup>¼<sup>1</sup>

α xj; xl � � <sup>∂</sup><sup>2</sup><sup>Ψ</sup> ∂xj∂xl

: (4)

∂kj∂kl

)) [23]. The dispersion relation providing the reso-

/2. To be stable (i.e., nonradiating), the DS spectrum has

� <sup>∂</sup><sup>ω</sup>

2γP0=β p (Figure 3).<sup>2</sup>

<sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> <sup>&</sup>gt; 0 and is

Figure 3. Resonance conditions for DS and linear waves and DS spectra in dependence on DS energy (a); and the Wigner (time-spectral) diagram of DS (b) [23].

Figure 4. Wigner representation of DS (a) and cut-off due to spectral dissipation defining the DS width (b).

will be seen, this factor is crucial for dissipative soliton turbulence. A spectral dissipation must be balanced by a nonlinear gain �κP<sup>0</sup> (we assume <sup>Ϝ</sup>(|Ψ|<sup>2</sup> ) <sup>≈</sup> <sup>κ</sup>|Ψ|<sup>2</sup> � <sup>ζ</sup>|Ψ|<sup>4</sup> <sup>+</sup> …, where the first term is leading) that results in the additional relation between soliton spectral width and its peak power: <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi κP0=α p . In combination with the dispersive relation, it gives the condition for the soliton existence which combines the dissipative and nondissipative factors: αγ βκ <sup>≤</sup> <sup>1</sup>=2.<sup>3</sup> One has to note, that both considered mechanisms of DS formation act in the spectral domain and, as was shown in [28], a transition to spectral domain is fruitful for developing a DS theory.

The key feature of DS is its nontrivial internal structure revealing itself in the phase inhomogeneity<sup>4</sup> and the internal energy redistribution (E is an energy flow):

<sup>3</sup> More precise analysis [27] gives the conditions of asymptotical stability: αγ βκ ≤ 1=3 if E ! ∞ 1=2 if E ! 0 , where E is a DS energy.

<sup>4</sup> The measure of this inhomogeneity is a so-called chirp <sup>Θ</sup> <sup>∝</sup> <sup>∂</sup><sup>2</sup> argð Þ <sup>Ψ</sup> ∂t <sup>2</sup> .

$$E \equiv \frac{i}{2} \frac{\partial}{\partial t} \left( \Psi \frac{\partial \Psi^\*}{\partial t} - \Psi^\* \frac{\partial \Psi}{\partial t} \right) = 2\sigma |\Psi|^2 + 2\kappa |\Psi|^4 - 2\alpha \left| \frac{\partial \Psi}{\partial t} \right| + \alpha \frac{\partial^2 |\Psi|^2}{\partial t^2} . \tag{5}$$

The third term in RHS of Eq. (5) is phase-sensitive and, thus, there is an energy flow from DS center, where spectral components with minimal relative frequencies are located, to the DS wings, where frequency components with maximum relative frequencies are located (Figure 5). Here, energy dissipates. Such nontrivial internal "life" of DS intensifies with the growth of phase inhomogeneity Θ. Simultaneously, DS becomes an energy scalable coherent concentrate with the energy ("concentrate mass") ∝Θ [5].

As a result of phase inhomogeneity and intensive internal energy flows, the internal coherence of DS can become partially broken. Then, DS splits into partially coherent "internal modes" which interact with each other as the independent "sub-solitons." [29–31] Thus, DS becomes a strongly localized "cloud" of interacting "quasi-particles" or "glass of boiling water" (Figure 6).

These figures demonstrate an affinity between the structures of DS and turbulence [8, 23]. Both spectral structures are defined by dispersion relations: between soliton and dispersive waves for the former and Langmuir dispersion relation for the latter (Figure 7). Secondly, both highenergy DS and turbulence are characterized by spectral condensation at zero frequency (wavenumber) with subsequent scattering to higher frequencies confined by cut-off at �Δ.

Such an analogy between DS and turbulence opens a door for building the kinetic and quantum [5, 32–34] theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment.

3. Transition to a DS turbulence

The value ∝1/Ξ can be treated as a measure of "long-range" correlation scale.

The value ∝1/Δ can be treated as a measure of "short-range" correlation scale.

dissipation (adapted from [8]).

<sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

5

6

The mechanism of transition to turbulence for DS can be associated with the time/spectral duality (Figure 8). When the energy increases (i.e., E ! ∞ that corresponds to a system with "infinite capacity" [36]), the spectrum condensates around ω = 0 within a diapason of Ξ ! 0 (Figure 7). Simultaneously, DS broadens in time domain ∝1/Ξ by analogy with a growth of Bose-Einstein condensate "mass."<sup>5</sup> The DS peak power tends to some constant value P0∝1/ζ defined by a saturation of dissipative nonlinearity (see above), and, thereby, the cut-off frequency

Figure 7. DS spectrum (central red curve in the left picture) where the cut-off frequency Δ is defined by the resonance condition between linear waves with a wave-number k and DS with a wave-number q (parabolic black curve in the left picture; P<sup>0</sup> is a DS peak power) [23]. The turbulence in the wave-number space is defined by the Langmuir dispersion relation (parabolic black curve in the right picture). Spectral condensation at k = 0 is illustrated by shading and forms a characteristic turbulence spectrum (central red curve; right picture) with the cut-off wave-number �kdiss defined by a

Figure 6. DS spectrum as a "glass of boiling water" (left) [30] demonstrating the dynamics of internal perturbation modes (right) [31]. The last picture is obtained by a perturbation analysis in spectral domain with the Neumann series expansion.

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κP0=α p tends to be constant. The last value defines the coherence scale ∝1/Δ (few picoseconds for a typical DS).6 As a result, DS becomes "decoupled," and even small perturbations can

Figure 5. Energy flows (left column) and corresponding spectral profiles (right column, ℱ is a Fourier image of Ψ) in dependence on dispersion β and chirp Θ for a DS with the profile Ψ ∝sech(t) 1 + <sup>i</sup><sup>Θ</sup> [4].

Figure 6. DS spectrum as a "glass of boiling water" (left) [30] demonstrating the dynamics of internal perturbation modes (right) [31]. The last picture is obtained by a perturbation analysis in spectral domain with the Neumann series expansion.

Figure 7. DS spectrum (central red curve in the left picture) where the cut-off frequency Δ is defined by the resonance condition between linear waves with a wave-number k and DS with a wave-number q (parabolic black curve in the left picture; P<sup>0</sup> is a DS peak power) [23]. The turbulence in the wave-number space is defined by the Langmuir dispersion relation (parabolic black curve in the right picture). Spectral condensation at k = 0 is illustrated by shading and forms a characteristic turbulence spectrum (central red curve; right picture) with the cut-off wave-number �kdiss defined by a dissipation (adapted from [8]).

#### 3. Transition to a DS turbulence

<sup>E</sup> � <sup>i</sup> 2 ∂ ∂t <sup>Ψ</sup> <sup>∂</sup>Ψ<sup>∗</sup>

with the energy ("concentrate mass") ∝Θ [5].

<sup>∂</sup><sup>t</sup> � <sup>Ψ</sup><sup>∗</sup> <sup>∂</sup><sup>Ψ</sup> ∂t

<sup>¼</sup> <sup>2</sup>σj j <sup>Ψ</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>κj j <sup>Ψ</sup> <sup>4</sup> � <sup>2</sup><sup>α</sup>

The third term in RHS of Eq. (5) is phase-sensitive and, thus, there is an energy flow from DS center, where spectral components with minimal relative frequencies are located, to the DS wings, where frequency components with maximum relative frequencies are located (Figure 5). Here, energy dissipates. Such nontrivial internal "life" of DS intensifies with the growth of phase inhomogeneity Θ. Simultaneously, DS becomes an energy scalable coherent concentrate

As a result of phase inhomogeneity and intensive internal energy flows, the internal coherence of DS can become partially broken. Then, DS splits into partially coherent "internal modes" which interact with each other as the independent "sub-solitons." [29–31] Thus, DS becomes a strongly

These figures demonstrate an affinity between the structures of DS and turbulence [8, 23]. Both spectral structures are defined by dispersion relations: between soliton and dispersive waves for the former and Langmuir dispersion relation for the latter (Figure 7). Secondly, both highenergy DS and turbulence are characterized by spectral condensation at zero frequency (wavenumber) with subsequent scattering to higher frequencies confined by cut-off at �Δ. Such an analogy between DS and turbulence opens a door for building the kinetic and quantum [5, 32–34] theory of open (dissipative) semi-coherent structures which mimics, in

Figure 5. Energy flows (left column) and corresponding spectral profiles (right column, ℱ is a Fourier image of Ψ) in

1 + <sup>i</sup><sup>Θ</sup> [4].

dependence on dispersion β and chirp Θ for a DS with the profile Ψ ∝sech(t)

localized "cloud" of interacting "quasi-particles" or "glass of boiling water" (Figure 6).

particular, a quantum Bose-Einstein condensate in a dissipative environment.

∂Ψ ∂t 

 þ α <sup>∂</sup><sup>2</sup>j j <sup>Ψ</sup> <sup>2</sup> ∂t

<sup>2</sup> : (5)

102 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

The mechanism of transition to turbulence for DS can be associated with the time/spectral duality (Figure 8). When the energy increases (i.e., E ! ∞ that corresponds to a system with "infinite capacity" [36]), the spectrum condensates around ω = 0 within a diapason of Ξ ! 0 (Figure 7). Simultaneously, DS broadens in time domain ∝1/Ξ by analogy with a growth of Bose-Einstein condensate "mass."<sup>5</sup> The DS peak power tends to some constant value P0∝1/ζ defined by a saturation of dissipative nonlinearity (see above), and, thereby, the cut-off frequency <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi κP0=α p tends to be constant. The last value defines the coherence scale ∝1/Δ (few picoseconds for a typical DS).6 As a result, DS becomes "decoupled," and even small perturbations can

<sup>5</sup> The value ∝1/Ξ can be treated as a measure of "long-range" correlation scale.

<sup>6</sup> The value ∝1/Δ can be treated as a measure of "short-range" correlation scale.

Figure 8. DS spectrum (red curves) which cut-off frequency Δ defines the correlation scale ∝1/Δ. It tends to some constant value with the DS energy growth. Simultaneously, DS broadens in time domain ∝1/Ξ (blue curves) in parallel with the concentration of energy around ω = 0 in spectral domain [35].

destroy its internal coherence and split ("nucleate") it into a set of "internal modes" shown in Figure 6.

More close insight into this mechanism can be provided by the adiabatic theory of DS in spectral domain presented in [28]. As was shown, the DS spectrum can be expressed as follows:

$$p(\omega) = \frac{\Upsilon}{\omega^2 + \Xi^2} H(\Delta^2 - \omega^2),\tag{6}$$

characterized by the short-range correlation time about of 1 ps. In the process of evolution, an incoherent DS splits into three almost identical coherent solitons, which widths are lower substantially and the corresponding long-range/short-range correlation times become smaller/larger, respectively. Small long-range correlation time prevents the DSs merging and

Figure 9. Contour plot of the DS power evolution and the corresponding autocorrelation function on the turbulent stage. The evolution coordinate is measured in the laser cavity round-trips, and the transverse coordinate corresponds to the

fused silica nonlinear coefficient and Δt=1 femtosecond is a time discretization step. Other parameters are: κ = 0.1γ,

, where γ corresponds to a

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local time measured in picoseconds. The energy parameter Es normalized on <sup>γ</sup>/Δ<sup>t</sup> is of 3 � 105

An analysis of turbulent DS demonstrates its complicate internal structure which can be interpreted as the complex of strongly interacting bright, dark, and gray DSs on a finite but strongly self-localized background concentrating almost the entire part of the energy.<sup>8</sup> In some sense, an appearance of DS turbulence resembles the laminar-turbulent transition in a fiber laser when a macroscopically coherent field (Ξ!0) becomes chaotically self-modulated [45]. Nevertheless, such a scenario is not unique. The turbulent dynamics can result from the strong interaction between "individual" DSs forming a "soliton gas" or turbulent "soliton cluster" (Figure 10) [1, 5, 47]. Interaction of such cluster with a low-intensity background field can result in permanent radiation or absorption of DSs in the form of so-called "soliton rains" [48, 49].

Separately, one may note the chaotization of DS dynamics caused by resonant interaction with the dispersive waves in the presence of higher-order corrections to the dispersion term in Eq. (1). In this case, the collisions between DS and dispersive wave, which radiates by it, results

One has to distinguish such a structure from the breather-like structures on a continuous-wave background. Such

in a chaotic dynamic preserving, nevertheless the DS integrity (Figure 11) [23].

structures can demonstrate chaotic and rogue waves dynamics, as well (e.g., see [43, 44]).

larger short-range correlation time provides DS coherence.

ζ = 0.05γ, δ = 0.05, and α = 40 nm (Yb-fiber laser) [6].

8

where p(ω) is a DS spectral power, and H is the Heaviside function. Eq. (6) represents the spectra shown in Figures 7 and 8, and can be interpreted by analogy with the Rayleigh-Jeans distribution, so that Ξ<sup>2</sup> plays a role of negative "chemical potential" [8, 36, 37]. The parameter Υ = 6πγ/κζ is an analog of "temperature" and is defined by both dissipative and nondissipative nonlinear parameters.

Since the "chemical potential" Ξ<sup>2</sup> decays with the energy growth (Figure 8), a system tends to the state of "soliton gas" [38] with the characteristic "soliton size" ∝1/Δ. Thereby, a coherent "condensate" with minimum entropy becomes a state of the decomposed "quasi-particles" with the chaotically modulated powers because the required entropy growth is provided by such modulation<sup>7</sup> [40]. Thus, the energy growth (i.e., the growth of "condensate mass" ∝1/Ξ) leads to extra-sensitivity to quantum-level noises [40, 41] that urges the quantum theory of coherent and semi-coherent dissipative structures, which would weave largest and smallest scales in the DS dynamics.

The example of such "DS decomposition" through a turbulence is shown in Figure 9. This figure is obtained by numerical simulation of Eqs. (1) and (4) with taking into account of the gain saturation in the form of σ = δ(1 � E/Es) [27]. Figure 9 demonstrates clearly two stages of DS evolution. The first stage corresponds to an incoherent and strongly turbulent DS, which is

<sup>7</sup> Here, one may draw an analogy with Hamiltonian systems, where the gradient of field is a measure of the amount of fluctuations [39].

Self-Organization, Coherence and Turbulence in Laser Optics http://dx.doi.org/10.5772/intechopen.71478 105

Figure 9. Contour plot of the DS power evolution and the corresponding autocorrelation function on the turbulent stage. The evolution coordinate is measured in the laser cavity round-trips, and the transverse coordinate corresponds to the local time measured in picoseconds. The energy parameter Es normalized on <sup>γ</sup>/Δ<sup>t</sup> is of 3 � 105 , where γ corresponds to a fused silica nonlinear coefficient and Δt=1 femtosecond is a time discretization step. Other parameters are: κ = 0.1γ, ζ = 0.05γ, δ = 0.05, and α = 40 nm (Yb-fiber laser) [6].

destroy its internal coherence and split ("nucleate") it into a set of "internal modes" shown in

Figure 8. DS spectrum (red curves) which cut-off frequency Δ defines the correlation scale ∝1/Δ. It tends to some constant value with the DS energy growth. Simultaneously, DS broadens in time domain ∝1/Ξ (blue curves) in parallel with the

More close insight into this mechanism can be provided by the adiabatic theory of DS in spectral

where p(ω) is a DS spectral power, and H is the Heaviside function. Eq. (6) represents the spectra shown in Figures 7 and 8, and can be interpreted by analogy with the Rayleigh-Jeans distribution, so that Ξ<sup>2</sup> plays a role of negative "chemical potential" [8, 36, 37]. The parameter Υ = 6πγ/κζ is an analog of "temperature" and is defined by both dissipative and nondissipative

Since the "chemical potential" Ξ<sup>2</sup> decays with the energy growth (Figure 8), a system tends to the state of "soliton gas" [38] with the characteristic "soliton size" ∝1/Δ. Thereby, a coherent "condensate" with minimum entropy becomes a state of the decomposed "quasi-particles" with the chaotically modulated powers because the required entropy growth is provided by such modulation<sup>7</sup> [40]. Thus, the energy growth (i.e., the growth of "condensate mass" ∝1/Ξ) leads to extra-sensitivity to quantum-level noises [40, 41] that urges the quantum theory of coherent and semi-coherent dissipative structures, which would weave largest and smallest

The example of such "DS decomposition" through a turbulence is shown in Figure 9. This figure is obtained by numerical simulation of Eqs. (1) and (4) with taking into account of the gain saturation in the form of σ = δ(1 � E/Es) [27]. Figure 9 demonstrates clearly two stages of DS evolution. The first stage corresponds to an incoherent and strongly turbulent DS, which is

Here, one may draw an analogy with Hamiltonian systems, where the gradient of field is a measure of the amount of

<sup>ω</sup><sup>2</sup> <sup>þ</sup> <sup>Ξ</sup><sup>2</sup> <sup>H</sup> <sup>Δ</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup> , (6)

domain presented in [28]. As was shown, the DS spectrum can be expressed as follows:

Υ

pð Þ¼ ω

concentration of energy around ω = 0 in spectral domain [35].

104 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Figure 6.

nonlinear parameters.

scales in the DS dynamics.

7

fluctuations [39].

characterized by the short-range correlation time about of 1 ps. In the process of evolution, an incoherent DS splits into three almost identical coherent solitons, which widths are lower substantially and the corresponding long-range/short-range correlation times become smaller/larger, respectively. Small long-range correlation time prevents the DSs merging and larger short-range correlation time provides DS coherence.

An analysis of turbulent DS demonstrates its complicate internal structure which can be interpreted as the complex of strongly interacting bright, dark, and gray DSs on a finite but strongly self-localized background concentrating almost the entire part of the energy.<sup>8</sup> In some sense, an appearance of DS turbulence resembles the laminar-turbulent transition in a fiber laser when a macroscopically coherent field (Ξ!0) becomes chaotically self-modulated [45].

Nevertheless, such a scenario is not unique. The turbulent dynamics can result from the strong interaction between "individual" DSs forming a "soliton gas" or turbulent "soliton cluster" (Figure 10) [1, 5, 47]. Interaction of such cluster with a low-intensity background field can result in permanent radiation or absorption of DSs in the form of so-called "soliton rains" [48, 49].

Separately, one may note the chaotization of DS dynamics caused by resonant interaction with the dispersive waves in the presence of higher-order corrections to the dispersion term in Eq. (1). In this case, the collisions between DS and dispersive wave, which radiates by it, results in a chaotic dynamic preserving, nevertheless the DS integrity (Figure 11) [23].

<sup>8</sup> One has to distinguish such a structure from the breather-like structures on a continuous-wave background. Such structures can demonstrate chaotic and rogue waves dynamics, as well (e.g., see [43, 44]).

U tðÞ¼ <sup>T</sup><sup>2</sup>

discharging" is like the turbulence decay shown in Figure 9.

Figure 13. Wigner (time-spectral) diagram of a sole DRS developing for large β [6].

nonlinear medium.

power; right) [35].

<sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> 2 T1T<sup>2</sup> 2

exp � <sup>t</sup> T2 sin <sup>t</sup>

Where T<sup>2</sup> and T<sup>1</sup> define the effective relaxation time and resonant frequency of phonons in a

The simulations demonstrate [6, 35] that SRS suppresses the DS turbulence for the sufficiently large dispersions β. The first scenario is formation of uncoupled complex of DS and the dissipative Raman soliton (DRS) [6, 35, 42, 52] (Figure 12). One may assume, that such "energy

DRS is characterized by large chirp Θ and frequency down-shift. The last results from intrapulse SRS which is possible due to a broad spectrum, which is a common characteristic of DS and results from its large Θ. A sole DRS develops with growing β (Figure 13) [6, 35]. It is

Figure 12. Wigner (time-spectral) diagram of the DS + DRS complex (left) and its evolution (contour plot of the field

T1

, (8)

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107

Self-Organization, Coherence and Turbulence in Laser Optics

Figure 10. DS clusters in the form of a "persistent and coherent quasi-soliton" (left) [46] and a "sporadic rogue waves events that emerge from turbulent fluctuations" (right) [41].

Figure 11. Wigner (time-spectral) diagram of the chaotic DS in the presence of third-order dispersion [23]. DS (dark-red region around 2.3 µm) radiates a dispersive wave (blue tail around 2.4 µm). As a result of the difference between their group velocities, DS collides permanently with a dispersive wave that causes chaotization of dynamics and modulation of both DS spectrum and time-profile.

#### 4. Coherence of DS in the presence of nonlinearity with nonlocal/noninstantaneous response

A nonlinearity with the nonlocal/noninstanteneous response, which is of interest in optical context, can be taken into account by inclusion in Eq. (1) of the following term [50]

$$-i\gamma^{\mu}\Psi\int \mathcal{U}(\mathbf{x}-\mathbf{x}')|\Psi|^2(T,\mathbf{x}')d\mathbf{x}'.\tag{7}$$

In the case of nonstationarity (i.e., x !t replacement), this equation describes the stimulated Raman scattering (SRS), for instance. Then, the response function is [6, 51]:

$$\Delta U(t) = \frac{T\_1^2 + T\_2^2}{T\_1 T\_2^2} \exp\left(-\frac{t}{T\_2}\right) \sin\left(\frac{t}{T\_1}\right),\tag{8}$$

Where T<sup>2</sup> and T<sup>1</sup> define the effective relaxation time and resonant frequency of phonons in a nonlinear medium.

The simulations demonstrate [6, 35] that SRS suppresses the DS turbulence for the sufficiently large dispersions β. The first scenario is formation of uncoupled complex of DS and the dissipative Raman soliton (DRS) [6, 35, 42, 52] (Figure 12). One may assume, that such "energy discharging" is like the turbulence decay shown in Figure 9.

DRS is characterized by large chirp Θ and frequency down-shift. The last results from intrapulse SRS which is possible due to a broad spectrum, which is a common characteristic of DS and results from its large Θ. A sole DRS develops with growing β (Figure 13) [6, 35]. It is

Figure 12. Wigner (time-spectral) diagram of the DS + DRS complex (left) and its evolution (contour plot of the field power; right) [35].

Figure 13. Wigner (time-spectral) diagram of a sole DRS developing for large β [6].

4. Coherence of DS in the presence of nonlinearity with

�iγΨ ð

Raman scattering (SRS), for instance. Then, the response function is [6, 51]:

A nonlinearity with the nonlocal/noninstanteneous response, which is of interest in optical

Figure 11. Wigner (time-spectral) diagram of the chaotic DS in the presence of third-order dispersion [23]. DS (dark-red region around 2.3 µm) radiates a dispersive wave (blue tail around 2.4 µm). As a result of the difference between their group velocities, DS collides permanently with a dispersive wave that causes chaotization of dynamics and modulation of

Figure 10. DS clusters in the form of a "persistent and coherent quasi-soliton" (left) [46] and a "sporadic rogue waves

In the case of nonstationarity (i.e., x !t replacement), this equation describes the stimulated

U x � <sup>x</sup><sup>0</sup> ð Þj j <sup>Ψ</sup> <sup>2</sup> <sup>T</sup>; <sup>x</sup><sup>0</sup> ð Þdx<sup>0</sup>

: (7)

context, can be taken into account by inclusion in Eq. (1) of the following term [50]

nonlocal/noninstantaneous response

both DS spectrum and time-profile.

events that emerge from turbulent fluctuations" (right) [41].

106 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

turbulence-free and characterized by perturbed anti-Stokes component, which is clearly visible on the Wigner diagram. Such perturbation induces a chaotic vibration of the DRS power [35]. Nevertheless, DRS exists within the parametric range, where an ordinary DS cannot develop in the turbulence-free regime. One may assume that the DRS stability results from passive negative feed-back based on interplay between nonlinear down-frequency shift due to SRS and spectral dissipation.

Acknowledgements

Author details

References

2013.359

10.1007/b11728

2012.01.004

tific; 2013

Vladimir L. Kalashnikov\* and Evgeni Sorokin

\*Address all correspondence to: vladimir.kalashnikov@tuwien.ac.at

2004;398:1-65. DOI: 10.1016/j.physrep.2004.04.002

Fiber Laser. Rijeka: InTechOpen; 2016. p. 165-210

and Nonlinear Media. New Jersey: World Scientific; 2006

ics. 1997;69:507-574. DOI: 10.1103/RevModPhys.69.507

Chapman & Hall; 2016. p. 587-626

30126. DOI: 10.1364/OE.22.030118

Institute of Photonics, Vienna University of Technology, Vienna, Austria

The author acknowledges the support from Austrian Science Fund (FWF Project No. P24916- N27). Computational results have been achieved using the Vienna Scientific Cluster (VSC).

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[1] Zakharov V, Diasc F, Pushkarev A. One-dimensional wave turbulence. Physics Reports.

[2] Editorial. The power of analogies. Nature Photonics. 2014;8:1. DOI: 10.1038/nphoton.

[3] Akhmediev N, Ankiewicz A, editors. Dissipative Solitons. Berlin: Springer; 2005. DOI:

[4] Kalashnikov VL, Sergeyev SV. Dissipative solitons in fibre lasers. In: Paul MC, editor.

[5] Kalashnikov VL. Optics and chaos: Chaotic, rogue and noisy optical dissipative Solitons. In: Skiadas CH, Skiadas C, editors. Handbook of Applications of Chaos Theory. London:

[6] Kalashnikov VL, Sorokin E. Dissipative Raman solitons. Optics Express. 2014;22:30118-

[7] Shats M, Punzmann H, editors. Turbulence and Coherent Structures in Fluids, Plasmas

[8] Robinson PA. Nonlinear wave collapse and strong turbulence. Reviews of Modern Phys-

[9] Laurie J, Bortolozzo U, Nazarenko S, Residori S. One-dimensional optical wave turbulence: Experiment and theory. Physics Reports. 2012;514:121-175. DOI: 10.1016/j.physrep.

[10] Shrira V, Nazarenko S, editors. Advances in Wave Turbulence. New Jersey: World Scien-

Another spectacular manifestation of the effect of a noninstantaneous nonlinearity on an incoherent field is an appearance of the spectral incoherent solitons (SIS) [53]. The spectrally localized soliton-like structures appear without any time-localization due to the causality property inherent to SRS so that a field cannot reach thermal equilibrium [54]. Formally, the corresponding evolution equation in the Langmuir turbulence limit has soliton-like solutions in the spectral domain [55, 56]. As a result, such structure localized in spectral domain possesses main properties of solitons including the property of elastic scattering.

#### 5. Conclusions

The problem of DS coherence, chaotic, and turbulent dynamics has been outlined. A nontrivial internal structure of DS caused by it intensive energy exchange with dissipative environment allows conjecturing a close analogy with turbulent structure forming far from equilibrium. The existence of long-range correlation scale provides the DS energy scaling (or mass scaling for Bose-Einstein condensate). However, such "macroscopic" scaling is provided by strong phase inhomogeneity (chirp) so that internal coherence of DS defined by short-range correlation scale breaks and DS becomes a "cloud" of interacting "quasi-particles" or "glass of boiling water." Such structure is very sensitive to perturbation of even quantum level. Such extra-sensitivity combines macro- and micro-scales that raises an issue of the quantum theory of the macroscopic coherent, partially, and incoherent dissipative structures.

In the context of this work, such DS "decomposition" leads to turbulent dynamics and DS fragmentation. In particular, interactions inside such "soliton cluster" can result in the rogue waves' formation. An additional source of soliton destabilization is resonant interaction with a dispersive wave that results in chaotization of dynamics and formation of "soliton rains."

Nonlinearity with noninstanteneous response (e.g., SRS) leads to new interesting effect. In particular, SRS suppresses the DS turbulence due to the formation of DS + DRS pairs or sole DRS. Although DRS is turbulence-free within a broad parametric range, it has a perturbed anti-Stokes component, which causes chaotic vibrations of DRS parameters.

Another and spectacular manifestation of the noninstantaneous response of nonlinearity is the formation of SIS. This structure is a soliton in the spectral domain but incoherent and delocalized in the time domain.

The above-considered phenomena and conjectures are of interest in the context of the development of approaches to the self-consistent theory of nonequilibrium dissipative structures in classical and quantum aspects, which would use the optical DSs as a testbed.

#### Acknowledgements

turbulence-free and characterized by perturbed anti-Stokes component, which is clearly visible on the Wigner diagram. Such perturbation induces a chaotic vibration of the DRS power [35]. Nevertheless, DRS exists within the parametric range, where an ordinary DS cannot develop in the turbulence-free regime. One may assume that the DRS stability results from passive negative feed-back based on interplay between nonlinear down-frequency shift due to SRS

Another spectacular manifestation of the effect of a noninstantaneous nonlinearity on an incoherent field is an appearance of the spectral incoherent solitons (SIS) [53]. The spectrally localized soliton-like structures appear without any time-localization due to the causality property inherent to SRS so that a field cannot reach thermal equilibrium [54]. Formally, the corresponding evolution equation in the Langmuir turbulence limit has soliton-like solutions in the spectral domain [55, 56]. As a result, such structure localized in spectral domain pos-

The problem of DS coherence, chaotic, and turbulent dynamics has been outlined. A nontrivial internal structure of DS caused by it intensive energy exchange with dissipative environment allows conjecturing a close analogy with turbulent structure forming far from equilibrium. The existence of long-range correlation scale provides the DS energy scaling (or mass scaling for Bose-Einstein condensate). However, such "macroscopic" scaling is provided by strong phase inhomogeneity (chirp) so that internal coherence of DS defined by short-range correlation scale breaks and DS becomes a "cloud" of interacting "quasi-particles" or "glass of boiling water." Such structure is very sensitive to perturbation of even quantum level. Such extra-sensitivity combines macro- and micro-scales that raises an issue of the quantum theory of the macro-

In the context of this work, such DS "decomposition" leads to turbulent dynamics and DS fragmentation. In particular, interactions inside such "soliton cluster" can result in the rogue waves' formation. An additional source of soliton destabilization is resonant interaction with a dispersive wave that results in chaotization of dynamics and formation of "soliton rains."

Nonlinearity with noninstanteneous response (e.g., SRS) leads to new interesting effect. In particular, SRS suppresses the DS turbulence due to the formation of DS + DRS pairs or sole DRS. Although DRS is turbulence-free within a broad parametric range, it has a perturbed

Another and spectacular manifestation of the noninstantaneous response of nonlinearity is the formation of SIS. This structure is a soliton in the spectral domain but incoherent and

The above-considered phenomena and conjectures are of interest in the context of the development of approaches to the self-consistent theory of nonequilibrium dissipative structures in

anti-Stokes component, which causes chaotic vibrations of DRS parameters.

classical and quantum aspects, which would use the optical DSs as a testbed.

sesses main properties of solitons including the property of elastic scattering.

108 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

scopic coherent, partially, and incoherent dissipative structures.

and spectral dissipation.

5. Conclusions

delocalized in the time domain.

The author acknowledges the support from Austrian Science Fund (FWF Project No. P24916- N27). Computational results have been achieved using the Vienna Scientific Cluster (VSC).

#### Author details

Vladimir L. Kalashnikov\* and Evgeni Sorokin

\*Address all correspondence to: vladimir.kalashnikov@tuwien.ac.at

Institute of Photonics, Vienna University of Technology, Vienna, Austria

#### References


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**Chapter 7**

**Provisional chapter**

**Interaction of Solitons with the Electromagnetic Field**

Solitary waves and solitons are briefly discussed in their historical context, especially regarding the motivation to use solitons for particle models in the nonlinear Klein-Gordon (NKG) equation. Conservation equations for charge, energy, and momentum follow from Noether's theorem in the usual way, and the NKG equation can be coupled to Maxwell's equations in the standard, gauge invariant manner. A recently proposed model is summarized in which two NKG equations are coupled to the electromagnetic field. In that model, solitons mimic the dynamical behavior of electrons and protons. A new result is then presented that follows from that model. Although the model is purely classical, it turns out that the arc spectrum of hydrogen is emitted into the electromagnetic field when small oscillations of one Klein-Gordon equation occur in the vicinity of a proton-like soliton. It is perhaps unexpected that a purely classical model can exhibit behavior suggestive of a phenomenon that is generally presumed to occur only in a quantum-mechanical context. Because of the way in which that result occurs, however, it is not clear whether there is any possible relevance to the actual physical phenomenon. **Keywords:** nonlinear systems and models, solitons, nonlinear Klein-Gordon-Maxwell

equations, particle models, hydrogen spectrum, nonlinear field theories

**Interaction of Solitons with the Electromagnetic Field** 

DOI: 10.5772/intechopen.71215

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>ψ</sup>* <sup>+</sup> *<sup>ψ</sup>* <sup>=</sup> <sup>0</sup> (1)

and reproduction in any medium, provided the original work is properly cited.

When a term that acts as a restoring force is added to the usual linear wave equation, the equation

is obtained. Schweber ([1], p. 54) points out "When Schrödinger wrote down the nonrelativistic equation now bearing his name, he also formulated the corresponding relativistic

**in Classical Nonlinear Field Models**

**in Classical Nonlinear Field Models**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71215

Jon C. Luke

**Abstract**

**1. Introduction**

<sup>∂</sup><sup>2</sup> *<sup>ψ</sup>* \_\_\_

Jon C. Luke


**Provisional chapter**

#### **Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models in Classical Nonlinear Field Models**

**Interaction of Solitons with the Electromagnetic Field** 

DOI: 10.5772/intechopen.71215

Jon C. Luke Jon C. Luke Additional information is available at the end of the chapter

[43] Onorato M, Residori S, Baronio F, editors. Rogue and Shock Waves in Nonlinear Disper-

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[47] Hammani K, Kibler B, Finot C, Picozzi A. Emergence of rogue waves from optical

[48] Chouli S, Grelu P. Rains of solitons in a fiber laser. Optics Express. 2009;17:11776-11781 [49] Chouli S, Grelu P. Soliton rains in a fiber laser: An experimental study. Physical Review

[50] Garnier J, Lisak M, Picozzi A. Toward a wave turbulence formulation of statistical nonlinear optics. Journal of the Optical Society of America B: Optical Physics. 2012;29:

[52] Babin SA, Podivilov EV, Kharenko DS, Bednyakova AE, Fedoruk MP, Kalashnikov VL, Apolonski A. Multicolour nonlinearly bound chirped dissipative solitons. Nature Com-

[53] Picozzi A, Pitois S, Millot G. Spectral incoherent solitons: A localized soliton behavior in the frequency domain. Physical Review Letters. 2008;101:093901. DOI: 10.1103/PhysRevLett.

[54] Garnier J, Pikozzi A. Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response. Physical Review A. 2010;81:033831.

[55] Musher SL, Rubenchik AM, Zakharov VE. Weak Langmuir turbulence. Physics Reports.

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sive Media. Switzerland: Springer; 2016. DOI: 10.1007/978-3-319-39214-1

International School of Physics). Amsterdam: IOS Press; 2016. p. 233-262

Nature Photonics. 2013;7:783-786. DOI: 10.1038/NPHOTON.2013.246

turbulence. Physics Letters A. 2010;374:3585-3589

112 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

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munications. 2014;5:4653. DOI: 10.1038/ncomms5653

1995;252:177-274. DOI: 10.1016/0370-1573(94)00071-A

DOI: 10.1103/PhysRevA.81.033831

A. 2010;81:063829

2229-2242

101.093901

83.023806

[46] Zavyalov A, Egorov O, Iliew R, Lederer F. Physical Review A. 2012;85:013828

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71215

#### **Abstract**

Solitary waves and solitons are briefly discussed in their historical context, especially regarding the motivation to use solitons for particle models in the nonlinear Klein-Gordon (NKG) equation. Conservation equations for charge, energy, and momentum follow from Noether's theorem in the usual way, and the NKG equation can be coupled to Maxwell's equations in the standard, gauge invariant manner. A recently proposed model is summarized in which two NKG equations are coupled to the electromagnetic field. In that model, solitons mimic the dynamical behavior of electrons and protons. A new result is then presented that follows from that model. Although the model is purely classical, it turns out that the arc spectrum of hydrogen is emitted into the electromagnetic field when small oscillations of one Klein-Gordon equation occur in the vicinity of a proton-like soliton. It is perhaps unexpected that a purely classical model can exhibit behavior suggestive of a phenomenon that is generally presumed to occur only in a quantum-mechanical context. Because of the way in which that result occurs, however, it is not clear whether there is any possible relevance to the actual physical phenomenon.

**Keywords:** nonlinear systems and models, solitons, nonlinear Klein-Gordon-Maxwell equations, particle models, hydrogen spectrum, nonlinear field theories

#### **1. Introduction**

When a term that acts as a restoring force is added to the usual linear wave equation, the equation

$$\frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \psi = 0 \tag{1}$$

is obtained. Schweber ([1], p. 54) points out "When Schrödinger wrote down the nonrelativistic equation now bearing his name, he also formulated the corresponding relativistic

> Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

equation." Within the year of 1926, Klein, Gordon, and at least three others independently proposed the use of Eq. (1) as the relativistic generalization of Schrödinger's equation. Eq. (1) has since become known as the Klein-Gordon equation. Instead of the original time and space variables *t*,̂*x*, ̂*y*, ̂*z* that might be used in a quantum-mechanical context, we have chosen to write ̂ Eq. (1) using dimensionless variables *t*, *x*, *y*, *z* defined as

$$\mathbf{t} = \frac{m}{\hbar} \frac{c^2}{\hbar} \mathbf{\hat{t}}, \quad \mathbf{x} = \frac{mc}{\hbar} \mathbf{\hat{x}}, \quad \mathbf{y} = \frac{mc}{\hbar} \mathbf{\hat{y}}, \quad \mathbf{z} = \frac{mc}{\hbar} \mathbf{\hat{z}}.\tag{2}$$

itself in more detail in Sections 5 and 6. An explicit example of a soliton solution is given in Section 7. In Sections 9–11, we summarize a recently proposed model in which a second NKG equation is introduced so as to model solitons that can be thought of as protons as well as electrons. In that model, the proton and electron-like solitons attract and repel each other in the desired way. Finally, in Section 12, we present new results from the same model where it is shown that small oscillations in the neighborhood of a proton-like soliton emit the wellknown frequencies of the hydrogen spectrum into the electromagnetic field. Since emission of the hydrogen spectrum is a phenomenon that would normally be expected to occur only in a quantum-mechanical setting and not in the present classical field model, we interpret this

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

http://dx.doi.org/10.5772/intechopen.71215

115

The study of solitary waves has a long history [4], which originated with observations by Scott Russell ([5], [6, p. 118]), of an isolated wave moving in a canal. Following along on horseback, he was able to observe that the wave traveled a great distance with little change in form and only gradually died out. The linear theories of water waves available at that time predicted, however, that such waveshapes would necessarily disperse, so for some time Scott Russell's observations were not taken seriously. The full equations for water waves, even in the irrotational, inviscid case, present quite a formidable boundary value problem, so various simple model equations have been examined in much greater detail. Strangely in contrast to the linear theory, a certain nonlinear model of water waves for shallow water predicted that waves of all shapes would steepen and ultimately break. Eventually, various simple, nonlinear model equations such as the equation of Korteweg and deVries (often known as the KdV equation)

> ∂*η*\_\_\_ <sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>ν</sup>*

were formulated, where both the dispersion and steepening effects are included. In these equations, the two effects balance each other and do, in fact, allow an isolated waveshape to

The term soliton was later introduced to emphasize that solitary waves, like particles, tend to maintain their identity. When a nonlinear term is introduced into the usual linear Schrödinger

is obtained. It has soliton solutions and is of particular interest in relation to the ideas of de Broglie and Bohm [7, 8]. Soliton solutions occur in the NKG Eq. (4) and also in the related equation

∂3 *η* \_\_\_

<sup>∂</sup>*x*<sup>3</sup> <sup>=</sup> <sup>0</sup>, (5)

(*ψψ*¯)*ψ* = 0, (6)

<sup>∂</sup>*x*<sup>2</sup> <sup>+</sup> sin*<sup>ψ</sup>* <sup>=</sup> <sup>0</sup> (7)

result in Section 13 and suggest the future direction of the present research.

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> (*c*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*<sup>1</sup> *<sup>η</sup>*)

∂*ψ*\_\_\_

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>ψ</sup>* <sup>+</sup> *<sup>W</sup>*′

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∂</sup><sup>2</sup> *<sup>ψ</sup>* \_\_\_

**2. Historical background of solitary waves and solitons**

<sup>∂</sup>*η*\_\_\_

propagate without change of form.

*i*

<sup>∂</sup><sup>2</sup> *<sup>ψ</sup>* \_\_\_

known (a bit flippantly) as the sine-Gordon equation.

equation, the nonlinear Schrödinger equation

where *t*,̂*x*, ̂*y*, ̂*z* are time and space coordinates in customary units (e.g., seconds and meters). ̂ Similarly, *m* is the mass of the electron, *c* is the speed of light, and *ℏ* is Planck's constant divided by 2*π*, also in customary units. Unit distance in our dimensionless coordinates corresponds to *ℏ*/(*mc*), which is about 3.86 × 10−13 m. That distance is known as the Compton wavelength of the electron (divided here by 2*π*), which is a fundamental unit of length for quantum-mechanical phenomena that involve electrons.

When one attempts to use the Klein-Gordon equation to describe the quantum-mechanical problem of an electron in the Coulomb field of a nucleus of atomic number *Z*, Eq. (1) becomes

$$\left(\frac{\partial}{\partial t} + i \frac{Z.a\_{\flat}}{r}\right)\left(\frac{\partial}{\partial t} + i \frac{Z.a\_{\flat}}{r}\right)\psi - \nabla^{2}\psi + \psi = 0,\tag{3}$$

where *αfs* is the fine structure constant (approximately 1/137), where *<sup>i</sup>* <sup>=</sup> <sup>√</sup> \_\_\_ −1, and where *r*2  = *x*<sup>2</sup>  + *y*<sup>2</sup>  + *z*<sup>2</sup> . Unit distance for the dimensionless quantity *r* is thus also *ℏ*/(*mc*). Although Eq. (3) reduces properly to Schrödinger's equation in the nonrelativistic approximation, it was realized almost immediately that Eq. (3) was in fact not appropriate for the quantummechanical description of the electron, and the Klein-Gordon equation fell even into a degree of disrepute, especially after the remarkable insight that led to Dirac's equation in 1928 [1, 2].

The Klein-Gordon equation is, nevertheless, a simple example of a relativistic partial differential equation and, as such, it is second in importance only to the wave equation. The relativistic invariance is preserved even when the *ψ* term is replaced with a term that depends nonlinearly on *ψ*, and the resulting equation can be used, for example, in the study of nonlinear waves [3, 4]. This chapter will deal primarily with the nonlinear Klein-Gordon (NKG) equation in the form

$$\frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \mathcal{W} \left( \psi \overline{\psi} \right) \psi = 0,\tag{4}$$

where *ψ* depends on time *t* and spatial coordinates *x*, *y*, *z*. Overbar denotes the complex conjugate since *ψ* will usually be taken as complex-valued. The real-valued function *W* is used to introduce an appropriate nonlinearity, and *W*′ (*ψψ*¯) denotes the derivative of *W(ψψ*¯*)* with respect to its argument *ψψ*¯. Let us take *W*(0) = 0, and with suitable scaling we can also take *W*′ (0) = 1, so that Eq. (4) will agree with Eq. (1) when *ψ* is small.

Here, we are interested in the soliton solutions that occur in the NKG equation, so Section 2 is a brief historical overview of solitary waves and solitons, followed by the historical context in Section 3 that motivates the use of solitons in classical field theories. Before coupling the NKG equation to Maxwell's equations in Section 8, we need to examine the NKG equation itself in more detail in Sections 5 and 6. An explicit example of a soliton solution is given in Section 7. In Sections 9–11, we summarize a recently proposed model in which a second NKG equation is introduced so as to model solitons that can be thought of as protons as well as electrons. In that model, the proton and electron-like solitons attract and repel each other in the desired way. Finally, in Section 12, we present new results from the same model where it is shown that small oscillations in the neighborhood of a proton-like soliton emit the wellknown frequencies of the hydrogen spectrum into the electromagnetic field. Since emission of the hydrogen spectrum is a phenomenon that would normally be expected to occur only in a quantum-mechanical setting and not in the present classical field model, we interpret this result in Section 13 and suggest the future direction of the present research.

#### **2. Historical background of solitary waves and solitons**

equation." Within the year of 1926, Klein, Gordon, and at least three others independently proposed the use of Eq. (1) as the relativistic generalization of Schrödinger's equation. Eq. (1) has since become known as the Klein-Gordon equation. Instead of the original time and space

<sup>ℏ</sup> *<sup>x</sup>*̂, *<sup>y</sup>* <sup>=</sup> \_\_\_

Similarly, *m* is the mass of the electron, *c* is the speed of light, and *ℏ* is Planck's constant divided by 2*π*, also in customary units. Unit distance in our dimensionless coordinates corresponds to *ℏ*/(*mc*), which is about 3.86 × 10−13 m. That distance is known as the Compton wavelength of the electron (divided here by 2*π*), which is a fundamental unit of length for

When one attempts to use the Klein-Gordon equation to describe the quantum-mechanical problem of an electron in the Coulomb field of a nucleus of atomic number *Z*, Eq. (1) becomes

Eq. (3) reduces properly to Schrödinger's equation in the nonrelativistic approximation, it was realized almost immediately that Eq. (3) was in fact not appropriate for the quantummechanical description of the electron, and the Klein-Gordon equation fell even into a degree of disrepute, especially after the remarkable insight that led to Dirac's equation in 1928 [1, 2]. The Klein-Gordon equation is, nevertheless, a simple example of a relativistic partial differential equation and, as such, it is second in importance only to the wave equation. The relativistic invariance is preserved even when the *ψ* term is replaced with a term that depends nonlinearly on *ψ*, and the resulting equation can be used, for example, in the study of nonlinear waves [3, 4]. This chapter will deal primarily with the nonlinear Klein-Gordon (NKG)

̂*z* that might be used in a quantum-mechanical context, we have chosen to write

*mc*

̂*z* are time and space coordinates in customary units (e.g., seconds and meters).

. Unit distance for the dimensionless quantity *r* is thus also *ℏ*/(*mc*). Although

<sup>ℏ</sup> *<sup>y</sup>*̂, *<sup>z</sup>* <sup>=</sup> \_\_\_

*mc*

*<sup>Z</sup> <sup>α</sup>fs* \_\_\_\_\_\_\_\_\_ *<sup>r</sup>* ) *<sup>ψ</sup>* <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>ψ</sup>* <sup>+</sup> *<sup>ψ</sup>* <sup>=</sup> <sup>0</sup>, (3)

(*ψψ*¯ )*ψ* = 0, (4)

(*ψψ*¯) denotes the derivative of *W(ψψ*¯*)* with

<sup>ℏ</sup> *<sup>z</sup>*̂, (2)

\_\_\_

−1, and where

variables *t*,̂*x*,

where *t*,̂*x*,

*r*2  = *x*<sup>2</sup>  + *y*<sup>2</sup>  + *z*<sup>2</sup>

*W*′

̂*y*,

̂*y*,

̂

(\_\_<sup>∂</sup>

equation in the form

<sup>∂</sup><sup>2</sup> *<sup>ψ</sup>* \_\_\_

to introduce an appropriate nonlinearity, and *W*′

(0) = 1, so that Eq. (4) will agree with Eq. (1) when *ψ* is small.

̂

*t* = *<sup>m</sup> <sup>c</sup>* <sup>2</sup> \_\_\_\_

Eq. (1) using dimensionless variables *t*, *x*, *y*, *z* defined as

114 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

quantum-mechanical phenomena that involve electrons.

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> *<sup>i</sup>*

*<sup>Z</sup> <sup>α</sup>fs* \_\_\_\_\_\_\_\_\_ *<sup>r</sup>* )(\_\_<sup>∂</sup>

where *αfs* is the fine structure constant (approximately 1/137), where *<sup>i</sup>* <sup>=</sup> <sup>√</sup>

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>ψ</sup>* <sup>+</sup> *<sup>W</sup>*′

where *ψ* depends on time *t* and spatial coordinates *x*, *y*, *z*. Overbar denotes the complex conjugate since *ψ* will usually be taken as complex-valued. The real-valued function *W* is used

respect to its argument *ψψ*¯. Let us take *W*(0) = 0, and with suitable scaling we can also take

Here, we are interested in the soliton solutions that occur in the NKG equation, so Section 2 is a brief historical overview of solitary waves and solitons, followed by the historical context in Section 3 that motivates the use of solitons in classical field theories. Before coupling the NKG equation to Maxwell's equations in Section 8, we need to examine the NKG equation

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> *<sup>i</sup>*

<sup>ℏ</sup> *<sup>t</sup>*̂, *<sup>x</sup>* <sup>=</sup> \_\_\_

*mc*

The study of solitary waves has a long history [4], which originated with observations by Scott Russell ([5], [6, p. 118]), of an isolated wave moving in a canal. Following along on horseback, he was able to observe that the wave traveled a great distance with little change in form and only gradually died out. The linear theories of water waves available at that time predicted, however, that such waveshapes would necessarily disperse, so for some time Scott Russell's observations were not taken seriously. The full equations for water waves, even in the irrotational, inviscid case, present quite a formidable boundary value problem, so various simple model equations have been examined in much greater detail. Strangely in contrast to the linear theory, a certain nonlinear model of water waves for shallow water predicted that waves of all shapes would steepen and ultimately break. Eventually, various simple, nonlinear model equations such as the equation of Korteweg and deVries (often known as the KdV equation)

$$\frac{\partial \eta}{\partial t} + \left(c\_0 + c\_1 \,\eta\right) \frac{\partial \eta}{\partial x} + \nu \frac{\partial^3 \eta}{\partial x^3} = \,\, 0,\tag{5}$$

were formulated, where both the dispersion and steepening effects are included. In these equations, the two effects balance each other and do, in fact, allow an isolated waveshape to propagate without change of form.

The term soliton was later introduced to emphasize that solitary waves, like particles, tend to maintain their identity. When a nonlinear term is introduced into the usual linear Schrödinger equation, the nonlinear Schrödinger equation

$$\mathrm{i}\,\frac{\partial\psi}{\partial t} - \nabla^2\psi + \mathcal{W}\left(\psi\overline{\psi}\right)\psi = 0,\tag{6}$$

is obtained. It has soliton solutions and is of particular interest in relation to the ideas of de Broglie and Bohm [7, 8]. Soliton solutions occur in the NKG Eq. (4) and also in the related equation

$$\frac{\partial^2 \psi}{\partial t^2} - \frac{\partial^2 \psi}{\partial x^2} + \sin \psi = 0 \tag{7}$$

known (a bit flippantly) as the sine-Gordon equation.

Remarkable methods, such as the inverse scattering method, have been devised for use with certain model equations. With these methods, it is possible to find exact solutions that show solitons colliding and passing through each other while still maintaining their identity. The subject of solitary waves and solitons has been extensively described in books and review articles [9–13]. For more recent work, see [14–24] and references therein.

**4. Topological and nontopological solitons**

be interested in possible solutions of the form

where *r*<sup>2</sup>

 = *x*<sup>2</sup>  + *y*<sup>2</sup>  + *z*<sup>2</sup>

*ψ* = *U*(*r*)*ei<sup>t</sup>*

It was found early on that if one tries to use the NKG Eq. (4) to form a localized, particle-like solution, where *ψ* is a static, real-valued function of spatial coordinates *x*, *y*, *z*, that such a solution turns out to be wildly unstable. Even the linear stability analysis shows that there is a mode that grows in time, so the solution either tends to collapse or to grow arbitrarily large. Hobart [27] and Derrick [28] concluded from energy considerations that a wide class of field equations would turn out to be unstable in this way. One possible way around this difficulty is for the solution to have a suitable topological property. Scott [29] and Rubinstein [30] have examined a simple example of such a topological soliton in the sine-Gordon Eq. (7). Scott's interpretation by means of a mechanical model is especially easy to visualize, where *ψ* repre-

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

Here, however, we will be concerned with another approach to stability that uses the NKG Eq. (4) but with a complex-valued variable *ψ* that has a time-dependent phase. Thus, we will

The solution is stationary in the sense that *U* does not depend on *t*, but it is not static since *ψ* is in effect rotating with angular frequency *ω* in the complex plane. Some of the early works along these lines were done by Glasko et al. [31], Zastavenko [32], and Rosen [33]. It turns out that orbital stability can be achieved in suitable circumstances. Roughly speaking, the rotation allows stability to occur just as a spinning gyroscope can be stable in a situation that would otherwise be unstable from energetic considerations. Solitons of this nature are often called nontopological solitons, in contrast to the topological solitons that occur in examples such as the sine-Gordon Eq. (7). The term soliton is sometimes reserved for particle-like solutions that have been shown to be orbitally stable; but here, we will allow the use of the term for particlelike solutions in general. One difficulty with the use of nontopological solitons as particle models is that a whole family of particle-like solutions typically occurs, as will be seen in the example of Section 7. This problem will be discussed further in Section 10. Existence and stability questions have been extensively studied for the NKG equation [15, 34]. More recently,

soliton solutions with nonzero angular momentum have also been of interest [18, 19].

equation can be derived from a variational principle. The Lagrangian density is

To understand soliton solutions for the NKG equation, it is important first to see that the NKG

<sup>2</sup> <sup>∇</sup>*<sup>ψ</sup>* <sup>∙</sup> <sup>∇</sup>*ψ*¯ <sup>−</sup> \_\_1

**5. Conservation equations for the NKG equation**

2 ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯ \_\_\_ ∂*t* − \_\_1

and the corresponding Euler operator (with respect to *ψ*) is

*L* = \_\_1

and where the real-valued function *U*(*r*) is exponentially small for large *r*.

, (8)

http://dx.doi.org/10.5772/intechopen.71215

117

<sup>2</sup> *W*(*ψψ*¯), (9)

sents an angle in the *y*, *z* plane, and the solution loops around the *x*-axis.

#### **3. Use of solitons as particle models**

The motivation for the use of solitons as particle models can best be understood by comparison with the historical development of physical theories, especially electromagnetism and quantum mechanics. Despite the successes of Maxwell's theory of electromagnetism in the late 1800s and the clarified understanding of its transformation properties according to Einstein's special theory of relativity, classical physics seemed to reach an impasse at the beginning of the twentieth century. The notion of the electron as a point particle was immediately inconsistent with electromagnetism since the electromagnetic energy around a charged point particle is easily calculated to be infinite. Moreover, if an electron were to orbit a proton like a planet around the sun, it would radiate into the electromagnetic field at a frequency based on the orbital speed and quickly spiral inward toward the nucleus.

Early in the twentieth century, however, Planck, Einstein, and Bohr advanced daring new hypotheses to describe blackbody radiation, the photoelectric effect, and the hydrogen atom, respectively. Dirac, Heisenberg, and others later developed various versions of a new quantum mechanics more sophisticated but certainly no more intuitively comprehensible. Quantum mechanics has been perpetually troubled by divergences, as when Bethe [25] somewhat jokingly referred to a quantity that "…comes out infinite in all existing theories, and has therefore always been ignored." Remarkable progress was later achieved in quantum electrodynamics by Tomonaga, Schwinger, Feynman, and others, but still only with the aid of a rather arbitrary (some would say procrustean) renormalization procedure. Dirac [2] concludes his book with the thought that "It would seem that we have followed as far as possible the path of logical development of the ideas of quantum mechanics as they are at present understood. The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage from Bohr's orbit theory to the present quantum mechanics."

There have been various attempts, notably by de Broglie [7, 26], to gain insight into a possible "drastic change in the foundations of the theory" as envisioned by Dirac, but so far such attempts have had little success. One possible approach, which apparently occurred to many people, is to ask whether solitons could be used to model actual particles as localized regions where the field is large. If point particles were replaced by solitons one could hope for a theory along more classical lines where problems with infinite, divergent integrals would be avoided in a natural way. The line of research described here is meant to be a step, if only a tiny step, in that direction. A recently proposed model is summarized in Sections 9–11. A new result is presented in Section 12, where a phenomenon suggestive of quantum-mechanics occurs, but in an unexpected context.

#### **4. Topological and nontopological solitons**

Remarkable methods, such as the inverse scattering method, have been devised for use with certain model equations. With these methods, it is possible to find exact solutions that show solitons colliding and passing through each other while still maintaining their identity. The subject of solitary waves and solitons has been extensively described in books and review

The motivation for the use of solitons as particle models can best be understood by comparison with the historical development of physical theories, especially electromagnetism and quantum mechanics. Despite the successes of Maxwell's theory of electromagnetism in the late 1800s and the clarified understanding of its transformation properties according to Einstein's special theory of relativity, classical physics seemed to reach an impasse at the beginning of the twentieth century. The notion of the electron as a point particle was immediately inconsistent with electromagnetism since the electromagnetic energy around a charged point particle is easily calculated to be infinite. Moreover, if an electron were to orbit a proton like a planet around the sun, it would radiate into the electromagnetic field at a frequency

Early in the twentieth century, however, Planck, Einstein, and Bohr advanced daring new hypotheses to describe blackbody radiation, the photoelectric effect, and the hydrogen atom, respectively. Dirac, Heisenberg, and others later developed various versions of a new quantum mechanics more sophisticated but certainly no more intuitively comprehensible. Quantum mechanics has been perpetually troubled by divergences, as when Bethe [25] somewhat jokingly referred to a quantity that "…comes out infinite in all existing theories, and has therefore always been ignored." Remarkable progress was later achieved in quantum electrodynamics by Tomonaga, Schwinger, Feynman, and others, but still only with the aid of a rather arbitrary (some would say procrustean) renormalization procedure. Dirac [2] concludes his book with the thought that "It would seem that we have followed as far as possible the path of logical development of the ideas of quantum mechanics as they are at present understood. The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage

There have been various attempts, notably by de Broglie [7, 26], to gain insight into a possible "drastic change in the foundations of the theory" as envisioned by Dirac, but so far such attempts have had little success. One possible approach, which apparently occurred to many people, is to ask whether solitons could be used to model actual particles as localized regions where the field is large. If point particles were replaced by solitons one could hope for a theory along more classical lines where problems with infinite, divergent integrals would be avoided in a natural way. The line of research described here is meant to be a step, if only a tiny step, in that direction. A recently proposed model is summarized in Sections 9–11. A new result is presented in Section 12, where a phenomenon suggestive of quantum-mechanics

articles [9–13]. For more recent work, see [14–24] and references therein.

116 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

based on the orbital speed and quickly spiral inward toward the nucleus.

from Bohr's orbit theory to the present quantum mechanics."

occurs, but in an unexpected context.

**3. Use of solitons as particle models**

It was found early on that if one tries to use the NKG Eq. (4) to form a localized, particle-like solution, where *ψ* is a static, real-valued function of spatial coordinates *x*, *y*, *z*, that such a solution turns out to be wildly unstable. Even the linear stability analysis shows that there is a mode that grows in time, so the solution either tends to collapse or to grow arbitrarily large. Hobart [27] and Derrick [28] concluded from energy considerations that a wide class of field equations would turn out to be unstable in this way. One possible way around this difficulty is for the solution to have a suitable topological property. Scott [29] and Rubinstein [30] have examined a simple example of such a topological soliton in the sine-Gordon Eq. (7). Scott's interpretation by means of a mechanical model is especially easy to visualize, where *ψ* represents an angle in the *y*, *z* plane, and the solution loops around the *x*-axis.

Here, however, we will be concerned with another approach to stability that uses the NKG Eq. (4) but with a complex-valued variable *ψ* that has a time-dependent phase. Thus, we will be interested in possible solutions of the form

$$
\psi = \mathcal{U}(r) \, e^{i\alpha t},\tag{8}
$$

where *r*<sup>2</sup>  = *x*<sup>2</sup>  + *y*<sup>2</sup>  + *z*<sup>2</sup> and where the real-valued function *U*(*r*) is exponentially small for large *r*. The solution is stationary in the sense that *U* does not depend on *t*, but it is not static since *ψ* is in effect rotating with angular frequency *ω* in the complex plane. Some of the early works along these lines were done by Glasko et al. [31], Zastavenko [32], and Rosen [33]. It turns out that orbital stability can be achieved in suitable circumstances. Roughly speaking, the rotation allows stability to occur just as a spinning gyroscope can be stable in a situation that would otherwise be unstable from energetic considerations. Solitons of this nature are often called nontopological solitons, in contrast to the topological solitons that occur in examples such as the sine-Gordon Eq. (7). The term soliton is sometimes reserved for particle-like solutions that have been shown to be orbitally stable; but here, we will allow the use of the term for particlelike solutions in general. One difficulty with the use of nontopological solitons as particle models is that a whole family of particle-like solutions typically occurs, as will be seen in the example of Section 7. This problem will be discussed further in Section 10. Existence and stability questions have been extensively studied for the NKG equation [15, 34]. More recently, soliton solutions with nonzero angular momentum have also been of interest [18, 19].

#### **5. Conservation equations for the NKG equation**

To understand soliton solutions for the NKG equation, it is important first to see that the NKG equation can be derived from a variational principle. The Lagrangian density is

$$L = \frac{1}{2} \frac{\partial \psi}{\partial t} \frac{\partial \overline{\psi}}{\partial t} - \frac{1}{2} \ \nabla \psi \cdot \nabla \overline{\psi} - \frac{1}{2} \ \mathcal{W}(\psi \overline{\psi}),\tag{9}$$

and the corresponding Euler operator (with respect to *ψ*) is

$$
\bullet \, p\_{\psi} = -\frac{\partial^2 \vec{\psi}}{\partial t^2} + \nabla^2 \vec{\psi} - \mathcal{W}^{'} (\psi \vec{\psi}) \, \overline{\psi}. \tag{10}
$$

is conserved, except for any momentum that enters or leaves through the outer surface.

Next, because the variational principle is invariant when *ψ* is rotated in the complex plane, the

will be defined as the charge (contained within the region in question), for reasons that will become clear when the NKG equation is coupled to Maxwell's equations. Again, by the divergence theorem, the charge within the region must remain constant in time except for any

Now let us look for solutions of the NKG equation that are rotating in the complex plane with

where *U* is a real function that depends on *x*, *y*, *z*, but is independent of *t*. Then, *U* needs to

<sup>∇</sup><sup>2</sup> *<sup>U</sup>* <sup>+</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>U</sup>* <sup>−</sup> *<sup>W</sup>*′(*U*<sup>2</sup>)*<sup>U</sup>* <sup>=</sup> <sup>0</sup>. (18)

Generally, we will be interested in localized solutions, and *U* will typically be exponentially small when *r* is large. Substitution of expression (17) in Eqs. (16) and (12) shows that the

(*ω*<sup>2</sup> *U*<sup>2</sup> + ∇*U* ∙ ∇*U* + *W*(*U*<sup>2</sup>)) *d*<sup>3</sup>

where the integrals are now taken over all of *x*, *y*, *z* space. From Eq. (19), it is clear that the charge will be positive or negative depending on the sign of *ω*. We are not interested here in the case where *Q* is zero since we want *U* to be nontrivial, and *ω* needs to be nonzero when

*iψ*\_\_ <sup>2</sup> <sup>∇</sup>*ψ*¯¯ <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

<sup>2</sup> ∇*ψ*). (15)

119

<sup>∂</sup>*<sup>t</sup>* ) *<sup>d</sup>*<sup>3</sup> *<sup>x</sup>* (16)

http://dx.doi.org/10.5772/intechopen.71215

, (17)

*x*, (19)

*x*, (20)

Conservation of momentum in the *y* and *z* directions is similar.

<sup>2</sup> (*<sup>ψ</sup> <sup>O</sup> <sup>p</sup><sup>ψ</sup>* <sup>−</sup>¯¯¯ *<sup>ψ</sup> <sup>O</sup> <sup>p</sup><sup>ψ</sup>* ) <sup>=</sup> \_\_<sup>∂</sup>

charge that enters or leaves through the outer surface.

*ψ* = *U ei<sup>t</sup>*

*<sup>Q</sup>* <sup>=</sup> <sup>−</sup>*<sup>ω</sup>*∫*U*<sup>2</sup> *<sup>d</sup>*<sup>3</sup>

2

**6. Rotating nonlinear solutions for the NKG equation**

*Q* = ∫(

angular frequency *ω*, that is

charge of such a solution is

<sup>ℰ</sup> <sup>=</sup> <sup>∫</sup> \_\_1

we try for an orbitally stable soliton solution.

and the energy is

satisfy

(*<sup>ψ</sup> <sup>O</sup> <sup>p</sup><sup>ψ</sup>* <sup>−</sup>¯¯¯ *<sup>ψ</sup> <sup>O</sup> <sup>p</sup><sup>ψ</sup>* )/ <sup>2</sup> can be put in the conservation form

<sup>∂</sup>*t*( *iψ*¯¯ \_\_ 2 ∂*ψ*\_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>ψ*\_\_ 2 ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* ) <sup>+</sup> <sup>∇</sup>∙(

> *iψ*¯¯ \_\_ 2 ∂*ψ*\_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>ψ*\_\_ 2 ∂*ψ*¯¯ \_\_\_

quantity *i*

\_\_*<sup>i</sup>*

The quantity

When we set the Euler operator (10) to zero (and take the negative of the complex conjugate), we obtain the NKG Eq. (4). It is well known from Noether's theorem [4, 35, 36] that each symmetry of the variational principle leads to an equation in what is referred to as conservation form. Once the equation is in conservation form, it is straightforward to integrate in order to show that a certain quantity is conserved. Symmetry with respect to time translation thus leads to conservation of energy, and symmetry with respect to translation in *x*, *y*, *z* space leads to conservation of momentum.

Since the variational principle that follows from Eq. (9) is unchanged under time translation, it follows that energy is conserved. Specifically, we can rewrite the expression <sup>∂</sup>*ψ*/∂*<sup>t</sup>* <sup>+</sup> *<sup>O</sup>*¯*<sup>p</sup><sup>ψ</sup>* <sup>∂</sup>*ψ*¯/∂*<sup>t</sup>*

<sup>−</sup>(*<sup>O</sup> <sup>p</sup><sup>ψ</sup>* )/ <sup>2</sup> in the following conservation form:

 −\_\_1 <sup>2</sup>(*O p<sup>ψ</sup>* ∂*ψ*\_\_\_ ∂*t* + *O*¯ *pψ* ∂*ψ*¯ \_\_\_ ∂*t* ) <sup>=</sup> \_\_<sup>∂</sup> ∂*t* \_\_1 2( ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯ \_\_\_ ∂*t* + ∇*ψ* ∙ ∇*ψ*¯ + *W*(*ψψ*¯)) −\_\_1 <sup>2</sup>∇∙( ∂*ψ*\_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>∇</sup>*ψ*¯¯ <sup>+</sup> ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>∇</sup>*ψ*). (11)

Let

$$\mathcal{S}' = \int\_{\mathcal{Z}} \frac{1}{2} \left( \frac{\partial \psi}{\partial t} \frac{\partial \bar{\psi}}{\partial t} + \nabla \psi \cdot \nabla \bar{\psi} + \mathcal{W}(\psi \bar{\psi}) \right) d^3 \mathbf{x},\tag{12}$$

where the integral is to be taken over a certain region in *x*, *y*, *z* space and where *d*<sup>3</sup> *x* denotes a volume element. Then, ℰ is defined as the energy in that region. Let us suppose that Eq. (4) is satisfied, so that the right-hand side of Eq. (11) must be equal to zero. Then, if Eq. (11) is integrated over the region in question, it is clear from the divergence theorem that the energy within that region will remain constant in time, except for any energy that enters or leaves the region through the outer surface.

Similarly, since the variational principle is invariant under translation in the *x* direction, a conservation equation can be found for momentum in the *x* direction. We can thus rewrite the expression (*Op<sup>ψ</sup>* <sup>∂</sup>*ψ*/∂*<sup>x</sup>* <sup>+</sup> *<sup>O</sup>*¯*<sup>p</sup><sup>ψ</sup>* <sup>∂</sup>*ψ*¯/∂*x*)/ <sup>2</sup> in the following conservation form:

$$
\begin{split} & \left\langle \nabla p\_{\psi} \bar{\psi} \right\rangle / 2 \text{ in the following conservation form:} \\ & \frac{1}{2} \left( O \int p\_{\psi} \frac{\partial \psi}{\partial x} + \overline{O} \frac{\partial \bar{\psi}}{\partial x} \right) = \frac{\partial}{\partial t} \Big( -\frac{1}{2} \frac{\partial \psi}{\partial t} \frac{\partial \bar{\psi}}{\partial x} - \frac{1}{2} \frac{\partial \bar{\psi}}{\partial t} \frac{\partial \psi}{\partial x} \Big) \\ & \qquad + \frac{\partial}{\partial x} \Big( \frac{1}{2} \frac{\partial \psi}{\partial t} \frac{\partial \bar{\psi}}{\partial t} - \frac{1}{2} \ \nabla \psi \cdot \nabla \bar{\psi} - \frac{1}{2} \mathcal{W} (\psi \bar{\psi}) \Big) \\ & \qquad + \nabla \cdot \Big( \frac{1}{2} \frac{\partial \bar{\psi}}{\partial x} \, \nabla \psi + \frac{1}{2} \frac{\partial \psi}{\partial x} \, \nabla \bar{\psi} \Big). \end{split} \tag{13}
$$

Then

$$P\_x = \int \left( -\frac{1}{2} \frac{\partial \psi}{\partial t} \frac{\partial \bar{\psi}}{\partial x} - \frac{1}{2} \frac{\partial \bar{\psi}}{\partial t} \frac{\partial \psi}{\partial x} \right) d^3 x \tag{14}$$

is defined as the momentum in the *x* direction (as contained within the region in question). Again, the use of the divergence theorem shows that momentum within the region is conserved, except for any momentum that enters or leaves through the outer surface. Conservation of momentum in the *y* and *z* directions is similar.

Next, because the variational principle is invariant when *ψ* is rotated in the complex plane, the quantity *i* (*<sup>ψ</sup> <sup>O</sup> <sup>p</sup><sup>ψ</sup>* <sup>−</sup>¯¯¯ *<sup>ψ</sup> <sup>O</sup> <sup>p</sup><sup>ψ</sup>* )/ <sup>2</sup> can be put in the conservation form

$$\frac{i}{2}\left(\psi\bullet\mathcal{O}p\_{\psi}-\overline{\psi\bullet\mathcal{O}p\_{\psi}}\right) = \frac{\partial}{\partial t}\left(\frac{i\overline{\psi}}{2}\frac{\partial\psi}{\partial t}-\frac{i\psi}{2}\frac{\partial\overline{\psi}}{\partial t}\right)+\nabla\cdot\left(\frac{i\psi}{2}\ \nabla\overline{\psi}-\frac{i\overline{\psi}}{2}\ \nabla\psi\right).\tag{15}$$

The quantity

*O p<sup>ψ</sup>* = −

118 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

to conservation of momentum.

<sup>2</sup>(*O p<sup>ψ</sup>*

<sup>ℰ</sup> <sup>=</sup> <sup>∫</sup> \_\_1

region through the outer surface.

\_\_1 <sup>2</sup>(*O p<sup>ψ</sup>*

*Px* <sup>=</sup> <sup>∫</sup>(−\_\_1

∂*ψ*\_\_\_ ∂*t* + *O*¯ *pψ* ∂*ψ*¯ \_\_\_ ∂*t*

<sup>∂</sup>*ψ*/∂*<sup>t</sup>* <sup>+</sup> *<sup>O</sup>*¯*<sup>p</sup><sup>ψ</sup>* <sup>∂</sup>*ψ*¯/∂*<sup>t</sup>*

−\_\_1

<sup>−</sup>(*<sup>O</sup> <sup>p</sup><sup>ψ</sup>*

Let

Then

∂2 *ψ*¯ \_\_\_ <sup>∂</sup>*t*2 + <sup>∇</sup><sup>2</sup> *ψ*̶ <sup>−</sup> *<sup>W</sup>*′

When we set the Euler operator (10) to zero (and take the negative of the complex conjugate), we obtain the NKG Eq. (4). It is well known from Noether's theorem [4, 35, 36] that each symmetry of the variational principle leads to an equation in what is referred to as conservation form. Once the equation is in conservation form, it is straightforward to integrate in order to show that a certain quantity is conserved. Symmetry with respect to time translation thus leads to conservation of energy, and symmetry with respect to translation in *x*, *y*, *z* space leads

Since the variational principle that follows from Eq. (9) is unchanged under time translation,

<sup>2</sup>∇∙( ∂*ψ*\_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>∇</sup>*ψ*¯¯ <sup>+</sup>

volume element. Then, ℰ is defined as the energy in that region. Let us suppose that Eq. (4) is satisfied, so that the right-hand side of Eq. (11) must be equal to zero. Then, if Eq. (11) is integrated over the region in question, it is clear from the divergence theorem that the energy within that region will remain constant in time, except for any energy that enters or leaves the

Similarly, since the variational principle is invariant under translation in the *x* direction, a conservation equation can be found for momentum in the *x* direction. We can thus rewrite the

> ∂*t*(−\_\_1 2 ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>x</sup>* <sup>−</sup> \_\_1 2 ∂*ψ*¯¯ \_\_\_ ∂*t* ∂*ψ*\_\_\_ <sup>∂</sup>*x*)

<sup>2</sup> <sup>∇</sup>*<sup>ψ</sup>* <sup>∙</sup> <sup>∇</sup>*ψ*¯¯ <sup>−</sup> \_\_1

<sup>+</sup> \_\_\_<sup>∂</sup> <sup>∂</sup>*x*(\_\_1

2*W*(*ψψ*¯¯))

<sup>∂</sup>*x*) *<sup>d</sup>*<sup>3</sup> *<sup>x</sup>* (14)

it follows that energy is conserved. Specifically, we can rewrite the expression

)/ <sup>2</sup> in the following conservation form:

−\_\_1

2( ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯¯ \_\_\_

expression (*Op<sup>ψ</sup>* <sup>∂</sup>*ψ*/∂*<sup>x</sup>* <sup>+</sup> *<sup>O</sup>*¯*<sup>p</sup><sup>ψ</sup>* <sup>∂</sup>*ψ*¯/∂*x*)/ <sup>2</sup> in the following conservation form:

∂*ψ*\_\_\_ ∂*x* + *O*¯ *pψ* ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*x*) <sup>=</sup> \_\_<sup>∂</sup>

2 ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> \_\_1

<sup>+</sup> <sup>∇</sup>∙(\_\_1 2 ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>x</sup>* <sup>∇</sup>*<sup>ψ</sup>* <sup>+</sup> \_\_1 2 ∂*ψ*\_\_\_ <sup>∂</sup>*<sup>x</sup>* <sup>∇</sup>*ψ*¯¯).

> 2 ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>x</sup>* <sup>−</sup> \_\_1 2 ∂*ψ*¯¯ \_\_\_ ∂*t* ∂*ψ*\_\_\_

is defined as the momentum in the *x* direction (as contained within the region in question). Again, the use of the divergence theorem shows that momentum within the region

) <sup>=</sup> \_\_<sup>∂</sup> ∂*t* \_\_1 2( ∂*ψ*\_\_\_ ∂*t* ∂*ψ*¯ \_\_\_ ∂*t*

where the integral is to be taken over a certain region in *x*, *y*, *z* space and where *d*<sup>3</sup>

(*ψψ*¯)*ψ*¯. (10)

<sup>∂</sup>*<sup>t</sup>* <sup>∇</sup>*ψ*). (11)

*x* denotes a

(13)

+ ∇*ψ* ∙ ∇*ψ*¯ + *W*(*ψψ*¯))

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>∇</sup>*<sup>ψ</sup>* <sup>∙</sup> <sup>∇</sup>*ψ*¯¯ <sup>+</sup> *<sup>W</sup>*(*ψψ*¯¯)) *<sup>d</sup>*<sup>3</sup> *<sup>x</sup>*, (12)

∂*ψ*¯¯ \_\_\_

$$\mathbf{Q} = \int \left( \frac{i\overline{\psi}}{2} \frac{\partial \psi}{\partial t} - \frac{i\psi}{2} \frac{\partial \overline{\psi}}{\partial t} \right) d^3 \mathbf{x} \tag{16}$$

will be defined as the charge (contained within the region in question), for reasons that will become clear when the NKG equation is coupled to Maxwell's equations. Again, by the divergence theorem, the charge within the region must remain constant in time except for any charge that enters or leaves through the outer surface.

#### **6. Rotating nonlinear solutions for the NKG equation**

Now let us look for solutions of the NKG equation that are rotating in the complex plane with angular frequency *ω*, that is

$$
\psi = \mathcal{U}e^{i\alpha t},\tag{17}
$$

where *U* is a real function that depends on *x*, *y*, *z*, but is independent of *t*. Then, *U* needs to satisfy

$$
\nabla^2 \mathcal{U} + \omega^2 \mathcal{U} - \mathcal{W}^\dagger (\mathcal{U}^2) \mathcal{U} = \begin{array}{c} \mathbf{0} \\ \end{array} \tag{18}
$$

Generally, we will be interested in localized solutions, and *U* will typically be exponentially small when *r* is large. Substitution of expression (17) in Eqs. (16) and (12) shows that the charge of such a solution is

$$\mathcal{Q} = -\omega \oint \mathcal{U}^2 d^3 \mathbf{x}\_{\prime} \tag{19}$$

and the energy is

$$\mathcal{S}' = \int \frac{1}{2} (\omega^2 \, \mathrm{L}^2 + \nabla \mathrm{L} \cdot \nabla \mathrm{L} + \mathcal{W}(\mathrm{L}^2)) \, d^3 \mathbf{x}\_{\prime} \tag{20}$$

where the integrals are now taken over all of *x*, *y*, *z* space. From Eq. (19), it is clear that the charge will be positive or negative depending on the sign of *ω*. We are not interested here in the case where *Q* is zero since we want *U* to be nontrivial, and *ω* needs to be nonzero when we try for an orbitally stable soliton solution.

If we look for a radially symmetric solution *U*(*r*), where *r*<sup>2</sup>  = *x*<sup>2</sup>  + *y*<sup>2</sup>  + *z*<sup>2</sup> , we note that

$$\nabla^2 \mathcal{U} = \frac{1}{r} \frac{d^2 \mathcal{U} D}{dr^2} \tag{21}$$

*U* = \_\_

*U* = \_\_

*r*̂ = √

(*U*<sup>2</sup>

*bW*, as well as a value of the parameter *θ*<sup>0</sup>

*x*, *y*, *z* space, will be larger when *bW* is smaller.

 = −√

we need to work instead with the scalar potential *A*<sup>0</sup>

defined in terms of the relations

<sup>∇</sup>*A*<sup>0</sup> <sup>−</sup> \_\_\_ <sup>∂</sup>**<sup>A</sup>**

*Q* = −*ω* ∫<sup>0</sup>

cates as ∣*Q*∣ is increased.

for *r*̂<sup>≤</sup> *<sup>r</sup>*̂ 0 and

for *r*̂<sup>&</sup>gt; *<sup>r</sup>*̂ 0

on *θ*<sup>0</sup>

Then, we need *W*′

2 and *W*′

. The result has been written in terms of scaled quantities

It turns out that the integral for the charge can be worked out explicitly as

\_\_\_\_\_\_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> *<sup>b</sup> <sup>W</sup>* cos<sup>2</sup> *<sup>θ</sup>*<sup>0</sup>

<sup>∞</sup> *U*<sup>2</sup> 4*π r* <sup>2</sup> *dr*

If a graph of charge *Q* is plotted (say for *bW* = 0.1) with respect to the parameter *θ*<sup>0</sup>

**8. Coupling of the NKG equation to the electromagnetic field**

\_\_\_ *bW r*, *r*̂

) = *aW* for *U*<sup>2</sup> <sup>&</sup>lt; *UW*

*ω*<sup>2</sup> = *aW* − *bW* cos<sup>2</sup> *θ*<sup>0</sup>

*<sup>r</sup>* <sup>=</sup> *<sup>θ</sup>*<sup>0</sup> *UW* sin(*r*̂sin*θ*0) \_\_\_\_\_\_\_\_\_\_\_\_\_ *r*̂sin2 *θ*<sup>0</sup>

*<sup>r</sup>* <sup>=</sup> *<sup>θ</sup>*<sup>0</sup> *UW <sup>e</sup>*(*r*−̂*r*̂

<sup>0</sup> = √ \_\_\_

(*U*<sup>2</sup>

The shape of the solution, as given by the dependence of *U* on *r* in Eqs. ̂ (27) and (28), depends

The charge *Q* and energy ℰ of such a soliton solution follow from the integrals (19) and (20).

that a minimum value of ∣*Q*∣ occurs. Expressed in another way, the family of solutions bifur-

Next, we want to couple the NKG equation to electromagnetism to obtain the nonlinear Klein-Gordon-Maxwell (NKGM) system of equations, as has been done by Rosen [37], Morris [41], and others. Again it is important to express the coupling of the NKG equation to the electromagnetic field in terms of a variational principle so that the conservation laws of interest can be obtained from Noether's theorem. Although Maxwell's equations are usually expressed in terms of the quantities **E**, **D**, **H**, **B**, it turns out that, to get a suitable variational principle,

 but not on *aW* or *bW*. For now, let us take *aW* = 1. It is easy to obtain *U* in terms of the original variable *r* using Eq. (29). Eq. (29) shows that the physical size of the soliton, i.e., its size in

2*π*(*θ*<sup>0</sup> cos*θ*<sup>0</sup> − sin *θ*0) *bW*

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (sin *<sup>θ</sup>*0)<sup>5</sup> cos*θ*<sup>0</sup>

0 )cos*θ*<sup>0</sup> \_\_\_\_\_\_\_\_\_\_\_ *r*̂sin *θ*<sup>0</sup>

*bW r*<sup>0</sup> = *θ*<sup>0</sup> /sin*θ*<sup>0</sup>

) = *aW* − *bW* for *U*<sup>2</sup> <sup>&</sup>gt; *UW*

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

, we find that *ω* is to be determined from the equation

−3/2 *UW* <sup>2</sup> *θ*<sup>0</sup> 2 (27)

121

(28)

http://dx.doi.org/10.5772/intechopen.71215

. (30)

and the vector potential **A**, which are

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> **<sup>E</sup>**, <sup>∇</sup> <sup>×</sup> **<sup>A</sup>** <sup>=</sup> **<sup>B</sup>**. (32)

. (29)

<sup>2</sup> . Given values of *aW* and

. (31)

, it is seen

so that Eq. (18) can be rewritten as

$$
\frac{d^2\mathcal{U}}{dr^2} + \omega^2 \mathcal{U} - \mathcal{W}'(\mathcal{U}^2)\mathcal{U} = 0,\tag{22}
$$

with = *rU*.

#### **7. An explicit soliton solution for the NKG equation**

To fix ideas, let us derive the following standard result in somewhat more than the usual detail. An expedient that has been frequently used [37–40] is to let *W*′ be a step function, in which case a simple example of a soliton solution can be worked out explicitly. Let us suppose that *W*′ has a fixed value *aW* when *r* is greater than a certain *r*<sup>0</sup> . Then, we want *aW* − *ω*<sup>2</sup>  > 0, so we can find a solution of Eq. (22) of the form = *B e* <sup>−</sup>*b*(*r*−*<sup>r</sup>* 0) for *r* > *r*<sup>0</sup> , where *B* is a constant and *b*2  = *aW* − *ω*<sup>2</sup> . To satisfy an appropriate boundary condition at infinity, we have chosen the solution where approaches 0 for large *r*. Next, let us suppose that *W*′ has a somewhat smaller fixed value *aW* − *bW* when *r* < *r*<sup>0</sup> . We want the constant *bW* to be a suitable value so that *aW* − *bW* − *ω*<sup>2</sup> will be negative. Then, we can find a solution of Eq. (22) in the form = *A* sin(*ar*) for *r* < *r*<sup>0</sup> ,where *A* is a constant and *a*<sup>2</sup>  = *ω*<sup>2</sup>  − *aW* + *bW* . We have chosen the sine rather than the cosine solution so as to satisfy an appropriate boundary condition at *r* = 0. Then, we note that

$$a^2 + b^2 = b\_{\mathcal{W}}.\tag{23}$$

We need *A* sin(*ar*<sup>0</sup> ) = *B* to make continuous at *r*<sup>0</sup> , and we need *Aa* cos(*ar*<sup>0</sup> ) =  − *Bb* to make *d*/*dr* continuous at *r*<sup>0</sup> , so, altogether, we want

$$a\cos\langle a\,r\_0\rangle = -b\sin\langle a\,r\_0\rangle.\tag{24}$$

There will be a whole family of rotating solutions for Eq. (18), as was mentioned in Section 4. It is convenient to use *θ*<sup>0</sup>  = *ar*<sup>0</sup> as a parameter for this family. Using Eqs. (23) and (24), we find that *a*<sup>2</sup>  = *bW* sin2 *θ*<sup>0</sup> and *b*<sup>2</sup>  = *bW* cos<sup>2</sup>*θ*<sup>0</sup> . We will be looking for solutions with *θ*<sup>0</sup> between *π*/2 and *π* (where sin *θ*<sup>0</sup> is positive but cos *θ*<sup>0</sup> is negative) so we find

$$a = \sqrt{b\_{\mathcal{W}}} \sin \theta\_{o'} \quad b = -\sqrt{b\_{\mathcal{W}}} \cos \theta\_{o'} \tag{25}$$

If we wish to have *U* take the value *UW* at *r*<sup>0</sup> , then

$$B = \frac{\mathcal{U}\_{\rm W} \theta\_o}{a}, \quad A = \frac{\mathcal{U}\_{\rm W} \theta\_o}{a \sin \theta\_o}. \tag{26}$$

Collecting the above results, we have

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models http://dx.doi.org/10.5772/intechopen.71215 121

$$
\Delta U = \frac{\eta\_{\ell}}{r} = \frac{\theta\_{o} \,\mathrm{l}\mathcal{U}\_{\mathrm{W}} \sin(\mathfrak{r} \sin \theta\_{o})}{\hat{r} \sin^{2} \theta\_{o}} \tag{27}
$$

for *r*̂<sup>≤</sup> *<sup>r</sup>*̂ 0 and

If we look for a radially symmetric solution *U*(*r*), where *r*<sup>2</sup>

120 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

**7. An explicit soliton solution for the NKG equation**

pose that *W*′ has a fixed value *aW* when *r* is greater than a certain *r*<sup>0</sup>

tion where approaches 0 for large *r*. Next, let us suppose that *W*′

) = *B* to make continuous at *r*<sup>0</sup>

, so, altogether, we want

 = *ar*<sup>0</sup>

is positive but cos *θ*<sup>0</sup>

 = *bW* cos<sup>2</sup>*θ*<sup>0</sup>

and *b*<sup>2</sup>

If we wish to have *U* take the value *UW* at *r*<sup>0</sup>

*<sup>B</sup>* <sup>=</sup> *UW <sup>θ</sup>* \_\_\_\_\_0

*a* = √

Collecting the above results, we have

so we can find a solution of Eq. (22) of the form = *B e* <sup>−</sup>*b*(*r*−*<sup>r</sup>*

 = *ω*<sup>2</sup>

<sup>∇</sup><sup>2</sup> *<sup>U</sup>* <sup>=</sup> \_\_1

so that Eq. (18) can be rewritten as

with = *rU*.

*b*2

 = *aW* − *ω*<sup>2</sup>

value *aW* − *bW* when *r* < *r*<sup>0</sup>

*A* is a constant and *a*<sup>2</sup>

We need *A* sin(*ar*<sup>0</sup>

It is convenient to use *θ*<sup>0</sup>

 = *bW* sin2 *θ*<sup>0</sup>

continuous at *r*<sup>0</sup>

*π* (where sin *θ*<sup>0</sup>

that *a*<sup>2</sup>

*<sup>d</sup>*\_\_\_\_ <sup>2</sup>

 = *x*<sup>2</sup>  + *y*<sup>2</sup>  + *z*<sup>2</sup>

*dr* <sup>2</sup> <sup>+</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>W</sup>*′(*U*<sup>2</sup>) <sup>=</sup> <sup>0</sup>, (22)

0) for *r* > *r*<sup>0</sup>

. We want the constant *bW* to be a suitable value so that *aW* − *bW* − *ω*<sup>2</sup>

 − *aW* + *bW* . We have chosen the sine rather than the cosine solution so

, and we need *Aa* cos(*ar*<sup>0</sup>

as a parameter for this family. Using Eqs. (23) and (24), we find

. We will be looking for solutions with *θ*<sup>0</sup>

\_\_\_ *bW* cos*θ*<sup>0</sup>

*a* sin*θ*<sup>0</sup>

. To satisfy an appropriate boundary condition at infinity, we have chosen the solu-

*r d*2(*rU*) \_\_\_\_\_

To fix ideas, let us derive the following standard result in somewhat more than the usual detail. An expedient that has been frequently used [37–40] is to let *W*′ be a step function, in which case a simple example of a soliton solution can be worked out explicitly. Let us sup-

be negative. Then, we can find a solution of Eq. (22) in the form = *A* sin(*ar*) for *r* < *r*<sup>0</sup>

*a* <sup>2</sup> + *b*<sup>2</sup> = *bW*. (23)

*a* cos(*a r*0) = −*b* sin(*a r*0). (24)

There will be a whole family of rotating solutions for Eq. (18), as was mentioned in Section 4.

is negative) so we find

, *b* = −√

*<sup>a</sup>* , *<sup>A</sup>* <sup>=</sup> *UW <sup>θ</sup>* \_\_\_\_\_\_\_\_\_\_\_ <sup>0</sup>

, then

as to satisfy an appropriate boundary condition at *r* = 0. Then, we note that

\_\_\_ *bW* sin*θ*<sup>0</sup> , we note that

. Then, we want *aW* − *ω*<sup>2</sup>

, where *B* is a constant and

has a somewhat smaller fixed

) =  − *Bb* to make *d*/*dr*

between *π*/2 and

. (25)

. (26)

 > 0,

will

,where

*<sup>d</sup> <sup>r</sup>* <sup>2</sup> (21)

$$
\mathcal{U} = \frac{\mathcal{U}}{r} = \frac{\theta\_0 \mathcal{U}\_W e^{0 + \theta\_0 \hbar \alpha \theta\_i}}{\hat{r} \sin \theta\_0} \tag{28}
$$

for *r*̂<sup>&</sup>gt; *<sup>r</sup>*̂ 0 . The result has been written in terms of scaled quantities

$$
\hat{r} = \sqrt{b\_W} \, r, \quad \hat{r}\_o = \sqrt{b\_W} \, r\_o = \, \theta\_o/\sin\theta\_o. \tag{29}
$$

Then, we need *W*′ (*U*<sup>2</sup> ) = *aW* for *U*<sup>2</sup> <sup>&</sup>lt; *UW* 2 and *W*′ (*U*<sup>2</sup> ) = *aW* − *bW* for *U*<sup>2</sup> <sup>&</sup>gt; *UW* <sup>2</sup> . Given values of *aW* and *bW*, as well as a value of the parameter *θ*<sup>0</sup> , we find that *ω* is to be determined from the equation

$$
\omega^2 = a\_W - b\_W \cos^2 \theta\_o. \tag{30}
$$

The shape of the solution, as given by the dependence of *U* on *r* in Eqs. ̂ (27) and (28), depends on *θ*<sup>0</sup> but not on *aW* or *bW*. For now, let us take *aW* = 1. It is easy to obtain *U* in terms of the original variable *r* using Eq. (29). Eq. (29) shows that the physical size of the soliton, i.e., its size in *x*, *y*, *z* space, will be larger when *bW* is smaller.

The charge *Q* and energy ℰ of such a soliton solution follow from the integrals (19) and (20). It turns out that the integral for the charge can be worked out explicitly as

$$\begin{split} Q &= -\omega \int\_0^\omega \mathcal{U}^2 \, 4\pi \, r^2 \, dr \\ &= -\sqrt{1 - b\_W \cos^2 \theta\_0} \frac{2\pi (\theta\_o \cos \theta\_o - \sin \theta\_0) b\_W \,^{\text{\textquotedbl{}}\Omega} \, \mathcal{U}^2\_W \theta\_0^2}{\{\sin \theta\_0\}^3 \cos \theta\_0} . \end{split} \tag{31}$$

If a graph of charge *Q* is plotted (say for *bW* = 0.1) with respect to the parameter *θ*<sup>0</sup> , it is seen that a minimum value of ∣*Q*∣ occurs. Expressed in another way, the family of solutions bifurcates as ∣*Q*∣ is increased.

#### **8. Coupling of the NKG equation to the electromagnetic field**

Next, we want to couple the NKG equation to electromagnetism to obtain the nonlinear Klein-Gordon-Maxwell (NKGM) system of equations, as has been done by Rosen [37], Morris [41], and others. Again it is important to express the coupling of the NKG equation to the electromagnetic field in terms of a variational principle so that the conservation laws of interest can be obtained from Noether's theorem. Although Maxwell's equations are usually expressed in terms of the quantities **E**, **D**, **H**, **B**, it turns out that, to get a suitable variational principle, we need to work instead with the scalar potential *A*<sup>0</sup> and the vector potential **A**, which are defined in terms of the relations

$$
\nabla A\_0 - \frac{\partial \mathbf{A}}{\partial t} = \mathbf{E}, \quad \nabla \times \mathbf{A} = \mathbf{B}. \tag{32}
$$

Here, the notation *A*<sup>0</sup> is used for the scalar potential to emphasize that it can be thought of as the time component of a Lorentz 4-vector, while **A** = (*A*<sup>1</sup> , *A*<sup>2</sup> , *A*<sup>3</sup> ) gives the three spatial components. The gauge condition can then be taken in the form

$$\frac{\partial A\_o}{\partial t} - \nabla \cdot \mathbf{A} = \mathbf{0}.\tag{33}$$

**9. A model with interacting solitons**

In addition to Eq. (34) for *ψ*, we now have an equation

\_\_∂

*<sup>A</sup>*<sup>0</sup> <sup>=</sup> *<sup>i</sup>ψ*\_\_ 2 ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_ 2 ∂*ψ*\_\_\_

**<sup>A</sup>** <sup>=</sup> *<sup>i</sup>ψ*\_\_

plane and *φ* = 0. Also, we will take the scalar field *A*<sup>0</sup>

∇<sup>2</sup> *V* = (−*ω* + *V*) *U*<sup>2</sup>

<sup>2</sup> <sup>∇</sup>*ψ*¯¯ <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_

where *U* and *V* are real-valued. Then, Eqs. (34) and (40)–(42) reduce to

for *φ*, and Eqs. (37) and (38) are to be replaced by

<sup>∂</sup><sup>2</sup> *<sup>φ</sup>* \_\_\_

\_\_∂ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup> <sup>A</sup>*0)(

> \_\_\_\_ 2 *A*0 <sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup>

\_\_\_\_ <sup>2</sup> **A** <sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup>

**10. Electron-like solitons**

*ψ* = *U*(*x*, *y*, *z*)*ei<sup>t</sup>*

(

<sup>∂</sup>

<sup>∂</sup>

respectively.

that is

and

Recently, a model (call it Model One) has been proposed [24] in which a second NKG equation

is also coupled to the electromagnetic field, but scaled with the constant *M* so that solitons in the *φ* field can be thought of as protons, whereas solitons in the *ψ* field are to be regarded as electrons. It must be stated at the outset that Model One is a very primitive model that clearly does not represent the universe in which we live; nevertheless, as an example in applied mathematics it has interesting features that seem worth examining. Although Model One is purely classical in nature, we will show in Section 12 that it exhibits behavior suggestive of a phenomenon that is generally presumed to occur only in a quantum-mechanical context.

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup> <sup>A</sup>*0)*<sup>φ</sup>* <sup>−</sup> (∇−*i***A**) <sup>∙</sup> (∇−*i***A**)*<sup>φ</sup>* <sup>+</sup> *<sup>M</sup>*<sup>2</sup> *<sup>W</sup>*′

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>A</sup>*<sup>0</sup> *ψψ*¯¯ <sup>+</sup>

<sup>2</sup> ∇*ψ* − **A***ψψ*¯¯ +

Let us first investigate solutions of the system (34), (40)–(42) where *ψ* rotates in the complex

<sup>∇</sup><sup>2</sup> *<sup>U</sup>* <sup>+</sup> (*<sup>ω</sup>* <sup>−</sup> *<sup>V</sup>*)<sup>2</sup> *<sup>U</sup>* <sup>−</sup> *<sup>W</sup>*′(*U*<sup>2</sup>)*<sup>U</sup>* <sup>=</sup> <sup>0</sup>, (44)

*iφ*\_\_ 2 ∂*φ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>φ*¯¯ \_\_ 2 ∂*φ*\_\_\_

*iφ*\_\_ <sup>2</sup> <sup>∇</sup>*ϕ*¯¯ <sup>−</sup> *<sup>i</sup>ϕ*¯¯ \_\_

( *φφ*¯¯ \_\_\_

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

*<sup>M</sup>*<sup>2</sup>)*<sup>φ</sup>* <sup>=</sup> <sup>0</sup>, (39)

http://dx.doi.org/10.5772/intechopen.71215

123

( *φφ*¯¯ \_\_\_

*<sup>M</sup>*<sup>2</sup>)*<sup>φ</sup>* <sup>=</sup> <sup>0</sup> (40)

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>A</sup>*<sup>0</sup> *φφ*¯¯ (41)

<sup>2</sup> ∇*φ* − **A***φϕ*¯¯, (42)

to be constant in time and set **A** to zero,

. (45)

, *φ* = 0, *A*<sup>0</sup> = *V*(*x*, *y*, *z*), **A** = 0, (43)

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>φ</sup>* <sup>+</sup> *<sup>M</sup>*<sup>2</sup> *<sup>W</sup>*′

It should be noted that *A*<sup>0</sup> = −Φ has the opposite sign from the symbol Φ that is often used to designate the scalar potential [42]. Eq. (4) can be coupled to the electromagnetic field in the usual gauge invariant way by replacing ∂/∂*t* by ∂/∂*t* − *iA*<sup>0</sup> and by replacing ∇ by ∇ − *i***A**, where *i* = √ \_\_\_\_\_\_\_\_\_ −1 . The resulting equation for *ψ* is

$$\left(\frac{\partial}{\partial t} - i\, A\_{\text{v}}\right)\left(\frac{\partial}{\partial t} - i\, A\_{\text{v}}\right)\psi - \left(\nabla - i\mathbf{A}\right) \cdot \left(\nabla - i\mathbf{A}\right)\psi + W\left(\psi\overline{\psi}\right)\psi = 0. \tag{34}$$

When oscillations of *ψ* are small so that *W*′ (*ψψ*¯¯) is effectively 1, we want Eq. (34) to reduce to the linear Klein-Gordon equation. In particular, for the example of the Coulomb field of a nucleus of atomic number *Z*, where **A** will be 0, we find that Eq. (34) reduces to

$$(\frac{\partial}{\partial t} - iA\_0) \left(\frac{\partial}{\partial t} - iA\_0\right) \psi - \nabla^2 \psi + \psi = 0. \tag{35}$$

Then by comparison with Eq. (3), it is clear that a nucleus of atomic number *Z* should generate a potential well

$$A\_{\rm o} = -\frac{Z\_{\rm \parallel}a\_{\rm \parallel}}{r}.\tag{36}$$

The Lagrangian density for the electromagnetic field can be expressed in terms of *A*<sup>0</sup> and **A**, as mentioned, but it involves a tensor quantity that is not convenient to express in terms of the usual vector notation. Details are given in [24]. When the equations for *ψ*, *A*<sup>0</sup> , and **A** are derived as Euler equations of a variational principle, as in [24], it turns out that *A*<sup>0</sup> and **A** are to satisfy

$$\frac{\partial^2 A\_0}{\partial t^2} - \nabla^2 A\_0 = \frac{i\mu}{2} \frac{\partial \bar{\psi}}{\partial t} - \frac{i\bar{\psi}}{2} \frac{\partial \psi}{\partial t} - A\_0 \psi \bar{\psi} \tag{37}$$

and

$$\frac{\partial^2 \mathbf{A}}{\partial t^2} - \nabla^2 \mathbf{A} = \frac{i\nu}{2} \,\, \nabla \overline{\psi} - \frac{i\overline{\psi}}{2} \,\, \nabla \psi - \mathbf{A} \psi \overline{\psi} \,\, \tag{38}$$

respectively. Substantial work has been done regarding existence of solitons in the NKGM system [43–45], in some cases even with nonzero angular momentum [16]. Stability of solitons in the NKGM system is a difficult subject, but some progress has been made [17, 21, 44], especially in the case of small coupling of the NKG equation to the electromagnetic field. The effect of an external field on a soliton has also been examined in the NKGM context by several authors [20, 46, 47].

#### **9. A model with interacting solitons**

Recently, a model (call it Model One) has been proposed [24] in which a second NKG equation

$$\frac{\partial^2 \phi}{\partial t^2} - \nabla^2 \phi + M^2 \, \mathcal{W} \left( \frac{\phi \bar{\phi}}{M^2} \right) \phi = 0,\tag{39}$$

is also coupled to the electromagnetic field, but scaled with the constant *M* so that solitons in the *φ* field can be thought of as protons, whereas solitons in the *ψ* field are to be regarded as electrons. It must be stated at the outset that Model One is a very primitive model that clearly does not represent the universe in which we live; nevertheless, as an example in applied mathematics it has interesting features that seem worth examining. Although Model One is purely classical in nature, we will show in Section 12 that it exhibits behavior suggestive of a phenomenon that is generally presumed to occur only in a quantum-mechanical context.

In addition to Eq. (34) for *ψ*, we now have an equation

$$\left(\frac{\partial}{\partial t} - iA\_0\right)\left(\frac{\partial}{\partial t} - iA\_0\right)\phi - (\nabla - i\mathbf{A}) \cdot (\nabla - i\mathbf{A})\phi + M^2 \,\mathcal{W}\left(\frac{\phi\phi}{M^2}\right)\phi = 0\tag{40}$$

for *φ*, and Eqs. (37) and (38) are to be replaced by

$$\frac{\partial^2 A\_v}{\partial t^2} - \nabla^2 A\_v = \frac{i\psi}{2} \frac{\partial \bar{\psi}}{\partial t} - \frac{i\bar{\psi}}{2} \frac{\partial \psi}{\partial t} - A\_v \psi \bar{\psi} + \frac{i\phi}{2} \frac{\partial \bar{\phi}}{\partial t} - \frac{i\bar{\phi}}{2} \frac{\partial \phi}{\partial t} - A\_v \phi \bar{\phi} \tag{41}$$

and

Here, the notation *A*<sup>0</sup>

It should be noted that *A*<sup>0</sup>

(

a potential well

to satisfy

and

*i* = √ \_\_\_\_\_\_\_\_\_

the time component of a Lorentz 4-vector, while **A** = (*A*<sup>1</sup>

\_\_\_\_

−1 . The resulting equation for *ψ* is

When oscillations of *ψ* are small so that *W*′

*A*<sup>0</sup> = −

\_\_∂ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup> <sup>A</sup>*0)(

(

<sup>∂</sup>

<sup>∂</sup>

authors [20, 46, 47].

nents. The gauge condition can then be taken in the form

122 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

usual gauge invariant way by replacing ∂/∂*t* by ∂/∂*t* − *iA*<sup>0</sup>

\_\_∂

\_\_∂ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup> <sup>A</sup>*0)(

\_\_\_\_ <sup>2</sup> *A*<sup>0</sup>

\_\_\_\_ <sup>2</sup> **A**

∂*A*0

designate the scalar potential [42]. Eq. (4) can be coupled to the electromagnetic field in the

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup> <sup>A</sup>*0)*<sup>ψ</sup>* <sup>−</sup> (∇<sup>−</sup> *<sup>i</sup>***A**) <sup>∙</sup> (∇<sup>−</sup> *<sup>i</sup>***A**) *<sup>ψ</sup>* <sup>+</sup> *<sup>W</sup>*′

the linear Klein-Gordon equation. In particular, for the example of the Coulomb field of a

Then by comparison with Eq. (3), it is clear that a nucleus of atomic number *Z* should generate

as mentioned, but it involves a tensor quantity that is not convenient to express in terms of

nucleus of atomic number *Z*, where **A** will be 0, we find that Eq. (34) reduces to

\_\_∂

The Lagrangian density for the electromagnetic field can be expressed in terms of *A*<sup>0</sup>

the usual vector notation. Details are given in [24]. When the equations for *ψ*, *A*<sup>0</sup>

derived as Euler equations of a variational principle, as in [24], it turns out that *A*<sup>0</sup>

*i* 2 ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_ 2 ∂*ψ*\_\_\_

*i* <sup>2</sup> <sup>∇</sup>*ψ*¯¯ <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_

respectively. Substantial work has been done regarding existence of solitons in the NKGM system [43–45], in some cases even with nonzero angular momentum [16]. Stability of solitons in the NKGM system is a difficult subject, but some progress has been made [17, 21, 44], especially in the case of small coupling of the NKG equation to the electromagnetic field. The effect of an external field on a soliton has also been examined in the NKGM context by several

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>A</sup>*<sup>0</sup> <sup>=</sup> \_\_

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup> **<sup>A</sup>** <sup>=</sup> \_\_

is used for the scalar potential to emphasize that it can be thought of as

, *A*<sup>2</sup> , *A*<sup>3</sup>

= −Φ has the opposite sign from the symbol Φ that is often used to

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> <sup>∇</sup> <sup>∙</sup> **<sup>A</sup>** <sup>=</sup> <sup>0</sup>. (33)

(*ψψ*¯¯) is effectively 1, we want Eq. (34) to reduce to

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup> <sup>A</sup>*0)*<sup>ψ</sup>* <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>ψ</sup>* <sup>+</sup> *<sup>ψ</sup>* <sup>=</sup> <sup>0</sup>. (35)

*<sup>Z</sup> <sup>α</sup>fs* \_\_\_\_\_\_\_\_\_ *<sup>r</sup>* . (36)

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>A</sup>*<sup>0</sup> *ψψ*¯¯ (37)

<sup>2</sup> ∇*ψ* − **A***ψψ*¯¯, (38)

) gives the three spatial compo-

and by replacing ∇ by ∇ − *i***A**, where

(*ψψ*¯¯)*ψ* = 0. (34)

and **A**,

, and **A** are

and **A** are

$$\frac{\partial^2 \mathbf{A}}{\partial t^2} - \nabla^2 \mathbf{A} = \frac{i\psi}{2} \left( \nabla \overline{\psi} - \frac{i\overline{\psi}}{2} \left( \nabla \psi - \mathbf{A} \psi \overline{\psi} + \frac{i\phi}{2} \right. \right. \left. \nabla \overline{\phi} - \frac{i\overline{\phi}}{2} \left. \nabla \phi - \mathbf{A} \varphi \overline{\phi} \right. \tag{42}$$

respectively.

#### **10. Electron-like solitons**

Let us first investigate solutions of the system (34), (40)–(42) where *ψ* rotates in the complex plane and *φ* = 0. Also, we will take the scalar field *A*<sup>0</sup> to be constant in time and set **A** to zero, that is

$$\psi = \mathcal{U}(\mathbf{x}, \mathbf{y}, \mathbf{z}) \, e^{i\mathbf{x}t}, \quad \phi = 0, \quad A\_0 = V(\mathbf{x}, \mathbf{y}, \mathbf{z}), \quad \mathbf{A} = \mathbf{0}, \tag{43}$$

where *U* and *V* are real-valued. Then, Eqs. (34) and (40)–(42) reduce to

$$
\nabla^2 \mathcal{U} + (\omega - V)^2 \mathcal{U} - \mathcal{W}'(\mathcal{U}^2)\mathcal{U} = 0,\tag{44}
$$

$$
\nabla^2 V = (-\omega + V) \,\mathrm{d}I^2.\tag{45}
$$

We want to find a soliton solution that satisfies both Eqs. (44) and (45) and can be thought of as an electron. Suppose that we start with a soliton solution for Eq. (18) of the kind that was previously discussed. We want the coupling to the electromagnetic field in Eqs. (44) and (45) to result in only a small change, so that *U* will still be spherically symmetric and localized, with *U* exponentially small for large *r*. The right-hand side of Eq. (45) is then nearly zero for large *r*, so the solution for *V* will be nearly proportional to 1/*r* for large *r*. If we can arrange for *V* to approach *αfs* /*r* for large *r* we see by comparison with Eq. (36) that the soliton will have one negative unit of elementary charge, so we can think of it as an electron-like soliton.

In the present NKGM context, it turns out Eq. (16) is to be replaced by

$$Q = \int \left(\frac{i\overline{\psi}}{2}\frac{\partial \psi}{\partial t} - \frac{i\psi}{2}\frac{\partial \overline{\psi}}{\partial t} + \psi \overline{\psi} \, A\_0\right) d^3 \,\mathbf{x} \tag{46}$$

so for a solution of the form (43), the charge is now

$$Q = f(-\omega + V) \text{ } \mathcal{U}^2 \, d^3 \mathbf{x}. \tag{47}$$

for real-valued functions *U*

*U*

gives a solution for Eq. (49) with *ω*

such as attraction of like charges.

arc spectrum of hydrogen.

except for very small values of *r*.

) = 1 and factor Eq. (44) to get

set *W*′ (*U*<sup>2</sup>

**12. An unexpected result of the model**

∇<sup>2</sup>

*ω*, then

~ and *V* ~

clockwise rotation in the complex plane but still allow *ω*

*U* <sup>~</sup> <sup>+</sup> (*<sup>ω</sup>* ~ + *V* ~) 2 *U*

. Then *U* ~ and *V* ~

<sup>~</sup> <sup>−</sup> *<sup>M</sup>*<sup>2</sup> *<sup>W</sup>*′ (*<sup>U</sup>*

<sup>~</sup> <sup>=</sup> *<sup>M</sup> <sup>f</sup>*(*Mr*), *<sup>V</sup>*

~2 \_\_\_ *<sup>M</sup>*<sup>2</sup>)*<sup>U</sup>*

If a solution *U* = *f*(*r*) and *V* = *g*(*r*) has been found for Eqs. (44) and (45) with a certain value of

so we have purposely introduced the minus sign in the solution form (48) so as to obtain

solution has one positive unit of elementary charge, but its energy is larger by a factor of *M* than that of the electron-like soliton (43). We want to interpret a solution of the form (48) as a proton, so we will refer to *φ* as the proton field, and take *M* to be the appropriate value,

Solitons in the electron field *ψ* and the proton field *φ* interact at a distance through the electromagnetic field. It turns out that, as desired, like charges repel and opposite charges attract, and the magnitude of the interaction agrees with Coulomb's law to a good approximation when *W* satisfies an appropriate condition. Details are available in [24]. By contrast, some early attempts to investigate interaction of solitons [48, 49] led to obviously undesired results

Although Model One was set up from purely classical considerations to describe the dynamical interaction of electron and proton-like solitons, a startling and perhaps unexpected result occurs. When small oscillations in the electron field occur in the vicinity of a proton-like soliton, it turns out that, because of the nature of the coupling to the electromagnetic field, only the difference frequencies are radiated, and it will be seen that the radiation is just the familiar

Because of the scaling (50) needed for Eq. (39), the spatial size of a proton-like soliton is much smaller (by a factor of 1836) than that of an electron-like soliton even though the magnitude of the charge is the same. Consequently, in the vicinity of a proton-like soliton, Eq. (36) with *Z* = 1 will be a good approximation, and the electric potential *V* will be nearly equal to −*αfs*/*r*

Now let us suppose that small oscillations are excited in the *ψ* field in the vicinity of a protonlike soliton, so that various modes of oscillation occur. Since the oscillations are small, we can

∇<sup>2</sup> *U* + (*ω* − *V* − 1)(*ω* − *V* + 1)*U* = 0, (51)

approximately 1836, to give the desired mass ratio between the proton and electron.

need to satisfy

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

*V* <sup>~</sup> <sup>=</sup> (*<sup>ω</sup>* ~ + *V* <sup>~</sup>) *<sup>U</sup>* ~2

~ <sup>=</sup> *<sup>M</sup>*. We are now looking for a positively charged soliton,

<sup>~</sup> <sup>=</sup> <sup>−</sup>*<sup>M</sup> <sup>g</sup>*(*Mr*) (50)

~ to be taken as positive. The new

http://dx.doi.org/10.5772/intechopen.71215

. (49)

125

<sup>~</sup> <sup>=</sup> <sup>0</sup>, <sup>∇</sup><sup>2</sup>

The divergence theorem together with Eq. (45) shows that for large *r*, where *U* is small, *V* is approximately −*Q*/(4*πr*), and so, by comparison with Eq. (36), *Q* = 4*παfs* corresponds to one unit of elementary charge. Positive *ω* in a solution of the form (43), which gives counterclockwise rotation in the complex plane, thus gives negative *Q* in agreement with the usual convention for electron charge. There will be a corresponding solution with negative *ω* and clockwise rotation which can then be thought of as a positron.

As mentioned in Section 4, there is a fundamental difficulty with the use of nontopological solitons as particle models. A whole family of soliton solutions typically exists, and the different solutions will have various values of charge, as was illustrated in the example of Section 7. Thus, it is not yet clear whether a preferred solution occurs in practice that could be regarded as defining one unit of elementary charge. Morris [41] has made a suggestion in this regard, but it seems that the resolution of the question is beyond the scope of present research on models of this nature since it may well involve complex, possibly chaotic, interactions of many solitons. A second problem is that the stability proofs currently available apply only in the limit of small coupling, so it is not clear whether orbital stability is achieved when the coupling is sufficient to correspond to a meaningful physical case. Future research should resolve this question, although perhaps in a somewhat nonrigorous manner based at least in part on numerical calculation. Despite these two difficulties let us assume for present purposes that a suitable orbitally stable solution of the form (43) is available which can then be thought of as an electron.

#### **11. Proton-like solitons**

Now let us look for solutions of the form

$$
\psi = 0, \quad \phi = \tilde{\mathcal{U}}(\mathbf{x}, y, \mathbf{z}) \, e^{-i\tilde{\omega}t}, \quad A\_0 = \tilde{\mathcal{V}}(\mathbf{x}, y, \mathbf{z}), \quad \mathbf{A} = \mathbf{0}, \tag{48}
$$

for real-valued functions *U* ~ and *V* ~ . Then *U* ~ and *V* ~ need to satisfy

We want to find a soliton solution that satisfies both Eqs. (44) and (45) and can be thought of as an electron. Suppose that we start with a soliton solution for Eq. (18) of the kind that was previously discussed. We want the coupling to the electromagnetic field in Eqs. (44) and (45) to result in only a small change, so that *U* will still be spherically symmetric and localized, with *U* exponentially small for large *r*. The right-hand side of Eq. (45) is then nearly zero for large *r*, so the solution for *V* will be nearly proportional to 1/*r* for large *r*. If we can arrange for *V* to approach *αfs* /*r* for large *r* we see by comparison with Eq. (36) that the soliton will have one

*Q* = ∫(−*ω* + *V*) *U*<sup>2</sup> *d*<sup>3</sup> *x*. (47)

The divergence theorem together with Eq. (45) shows that for large *r*, where *U* is small, *V* is approximately −*Q*/(4*πr*), and so, by comparison with Eq. (36), *Q* = 4*παfs* corresponds to one unit of elementary charge. Positive *ω* in a solution of the form (43), which gives counterclockwise rotation in the complex plane, thus gives negative *Q* in agreement with the usual convention for electron charge. There will be a corresponding solution with negative *ω* and clockwise

As mentioned in Section 4, there is a fundamental difficulty with the use of nontopological solitons as particle models. A whole family of soliton solutions typically exists, and the different solutions will have various values of charge, as was illustrated in the example of Section 7. Thus, it is not yet clear whether a preferred solution occurs in practice that could be regarded as defining one unit of elementary charge. Morris [41] has made a suggestion in this regard, but it seems that the resolution of the question is beyond the scope of present research on models of this nature since it may well involve complex, possibly chaotic, interactions of many solitons. A second problem is that the stability proofs currently available apply only in the limit of small coupling, so it is not clear whether orbital stability is achieved when the coupling is sufficient to correspond to a meaningful physical case. Future research should resolve this question, although perhaps in a somewhat nonrigorous manner based at least in part on numerical calculation. Despite these two difficulties let us assume for present purposes that a suitable orbitally stable solution of the form (43) is available which can then be thought of as an electron.

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> *ψψ*¯¯ *<sup>A</sup>*0) *<sup>d</sup>*<sup>3</sup> *<sup>x</sup>* (46)

negative unit of elementary charge, so we can think of it as an electron-like soliton.

In the present NKGM context, it turns out Eq. (16) is to be replaced by

124 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

*iψ*¯¯ \_\_ 2 ∂*ψ*\_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>ψ*\_\_ 2 ∂*ψ*¯¯ \_\_\_

*Q* = ∫(

**11. Proton-like solitons**

Now let us look for solutions of the form

~

(*x*, *y*, *z*)*e* <sup>−</sup>*i<sup>ω</sup>*

~*t*

,     *A*<sup>0</sup> = *V* ~

(*x*, *y*, *z*),     **A** = 0, (48)

*ψ* = 0,     *φ* = *U*

so for a solution of the form (43), the charge is now

rotation which can then be thought of as a positron.

$$
\nabla^2 \tilde{\boldsymbol{L}} + \left(\tilde{\boldsymbol{\omega}} + \tilde{\boldsymbol{V}}\right)^2 \tilde{\boldsymbol{L}} - M^2 \boldsymbol{W} \left(\frac{\tilde{\boldsymbol{L}}^2}{M^2}\right) \tilde{\boldsymbol{L}} = \boldsymbol{0}, \quad \nabla^2 \tilde{\boldsymbol{V}} = \left(\tilde{\boldsymbol{\omega}} + \tilde{\boldsymbol{V}}\right) \tilde{\boldsymbol{L}}^2. \tag{49}
$$

If a solution *U* = *f*(*r*) and *V* = *g*(*r*) has been found for Eqs. (44) and (45) with a certain value of *ω*, then

$$
\tilde{M} = M \, f(Mr), \quad \tilde{V} = -M \, g(Mr) \tag{50}
$$

gives a solution for Eq. (49) with *ω* ~ <sup>=</sup> *<sup>M</sup>*. We are now looking for a positively charged soliton, so we have purposely introduced the minus sign in the solution form (48) so as to obtain clockwise rotation in the complex plane but still allow *ω* ~ to be taken as positive. The new solution has one positive unit of elementary charge, but its energy is larger by a factor of *M* than that of the electron-like soliton (43). We want to interpret a solution of the form (48) as a proton, so we will refer to *φ* as the proton field, and take *M* to be the appropriate value, approximately 1836, to give the desired mass ratio between the proton and electron.

Solitons in the electron field *ψ* and the proton field *φ* interact at a distance through the electromagnetic field. It turns out that, as desired, like charges repel and opposite charges attract, and the magnitude of the interaction agrees with Coulomb's law to a good approximation when *W* satisfies an appropriate condition. Details are available in [24]. By contrast, some early attempts to investigate interaction of solitons [48, 49] led to obviously undesired results such as attraction of like charges.

#### **12. An unexpected result of the model**

Although Model One was set up from purely classical considerations to describe the dynamical interaction of electron and proton-like solitons, a startling and perhaps unexpected result occurs. When small oscillations in the electron field occur in the vicinity of a proton-like soliton, it turns out that, because of the nature of the coupling to the electromagnetic field, only the difference frequencies are radiated, and it will be seen that the radiation is just the familiar arc spectrum of hydrogen.

Because of the scaling (50) needed for Eq. (39), the spatial size of a proton-like soliton is much smaller (by a factor of 1836) than that of an electron-like soliton even though the magnitude of the charge is the same. Consequently, in the vicinity of a proton-like soliton, Eq. (36) with *Z* = 1 will be a good approximation, and the electric potential *V* will be nearly equal to −*αfs*/*r* except for very small values of *r*.

Now let us suppose that small oscillations are excited in the *ψ* field in the vicinity of a protonlike soliton, so that various modes of oscillation occur. Since the oscillations are small, we can set *W*′ (*U*<sup>2</sup> ) = 1 and factor Eq. (44) to get

$$
\nabla^2 \mathcal{L} + (\omega - V - 1)(\omega - V + 1)\mathcal{L} = 0,\tag{51}
$$

where *V* now represents the potential well caused by the *φ* field of the proton-like soliton. Any contribution to the *V* field caused by the *ψ* field can be neglected if the oscillations in the *ψ* field are sufficiently small. Let us suppose that ∣*V*∣ is typically much smaller than 1 and that *ω* is near to 1. Then, we can approximate *ω* − *V* + 1 as 2 in the Klein-Gordon Eq. (51) to obtain the usual Schrödinger equation approximation:

$$-\frac{1}{2}\nabla^2\mathcal{U} + V\mathcal{U} = (\omega - 1)\mathcal{U}.\tag{52}$$

when the *ψ* field is restricted to a single mode of oscillation. The nonzero value of *S*<sup>0</sup>

have used the original values of *A*<sup>0</sup>

field, so that

Since *S*<sup>0</sup>

and **A** caused by the *ψ* oscillation will be of higher order).

*ψ* = *U*(*x*, *y*, *z*) *ei<sup>t</sup>* + *U*̂

Substitution in Eqs. (55) and (56) shows

*S*<sup>0</sup> = (*ω* − *V*) *U*<sup>2</sup> + (*ω*̂− V) *U*̂

only occur at the difference frequency *<sup>ω</sup>* <sup>−</sup> *<sup>ω</sup>*̂.

**13. Interpretation and summary**

a change in the static *V* potential field, but the change in the potential well is small since the *ψ* oscillation is taken as small (in the evaluation of Eqs. (55) and (56) for present purposes, we

Next, let us consider the case where small oscillations occur in two different modes in the *ψ*

2

**S** = (*U*̂ ∇*U* − *U* ∇*U*̂) sin(*ω* − *ω*̂)*t*. (59)

a possibility, but from the form of the sine and cosine terms it turns out that the radiation can

It should be noted that charge is not necessarily quantized in Model One, and that the small oscillations in the *ψ* field indeed represent a small amount of charge. In this example, the *V* field is not to be noticeably changed from that of the proton-like soliton, however, so the small oscillations in the *ψ* field must correspond to only a minuscule amount of charge, much less than one elementary charge. Then, Eqs. (41) and (42) are to be solved using the source terms (55), (56) in the form (58), (59). Since charge cannot be radiated into the electromagnetic field, small amounts of charge must be transferred between the trapped modes in the *ψ* field, and the charge must go to those modes in which the ratio of energy to charge is smaller. It is expected that there will be a selection rule since the two modes in question will not effectively radiate into the electromagnetic field in some cases because of their relative symmetry.

Thus, we find in Model One that radiation is emitted into the electromagnetic field when small oscillations in the *ψ* field occur in the potential well created by a proton-like soliton. That radiation turns out to be the well-known spectrum of hydrogen, namely, difference frequencies of the form *<sup>ω</sup>* <sup>−</sup> *<sup>ω</sup>*̂, where *ω* and *ω*̂ are given by Eq. (54) with different values of *n*.

Although Model One, Eqs. (34) and (40)–(42), was set up to let solitons mimic the dynamical behavior of electrons and protons, it turns out that small oscillations of the electron field *ψ* in the vicinity of a proton-like soliton have modes of oscillation that correspond to the various terms in the Grotrian diagram for hydrogen. Thus, a quantum-like phenomenon automatically occurs even though the model is of a purely classical nature. It has seemed

and **S** are now time-varying it is clear that radiation into the electromagnetic field is

and **A** from the proton solution (48) since any change in *A*<sup>0</sup>

. (57)

http://dx.doi.org/10.5772/intechopen.71215

+ (*ω* + *ω*̂− 2*V*) *UU*̂cos(*ω* − *ω*̂)*t*, (58)

(*x*, *y*, *z*) *ei<sup>ω</sup>*̂*<sup>t</sup>*

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

causes

127

Here, *V* describes a potential well, but the quantity *ω* − 1 is related to an actual oscillation frequency *ω*, which is unlike the traditional use of Schrödinger's equation, where the right-hand side of the equation is regarded as related to an energy.

The main point of interest here is that solutions of the soliton equations lead in a natural way to Eq. (52), which of course results in frequencies of the hydrogen spectrum. For the *V* =  − *αfs*/*r* potential well of a proton-like soliton, it is then straightforward to solve Eq. (52) according to the well-known solution of Schrödinger's equation for a hydrogen atom [50]. Eigenfunctions occur in the form

$$\mathcal{U} = \mathcal{R}(r) \, P\_{\parallel}^{u}(\cos \theta) (\cos \omicron \mu \varphi) + \sin \langle m \varphi \rangle\_{\flat} \tag{53}$$

where the spherical harmonics are written here in terms of the associated Legendre polynomials *Pl <sup>m</sup>* and the usual spherical coordinates *θ* and . In Eq. (53), the usual factor *eim* has been replaced by sinusoids since a real-valued formulation for *U* is desired. A real-valued solution for the radial function *R*(*r*) can be obtained in the standard way. The corresponding frequencies are

$$
\omega = 1 - \frac{a\_{\flat}^2}{2n^{2\prime}} \quad n = 1, 2, \dots,\tag{54}
$$

with *l* = 0, 1, …, *n* − 1 and *m* =  − *l*, …, *l*.

When small oscillations in the electron field *ψ* are allowed in the vicinity of a proton-like soliton, the overall solution (48) will be slightly changed, so we need to investigate how the electron field source terms

$$S\_0 = \frac{i\psi}{2}\frac{\partial\bar{\psi}}{\partial t} - \frac{i\bar{\psi}}{2}\frac{\partial\psi}{\partial t} - A\_v\psi\bar{\psi} \tag{55}$$

and

$$\mathbf{S} = \frac{i\psi}{2} \,\, \nabla \overline{\psi} - \frac{i\overline{\psi}}{2} \,\, \nabla \psi - \mathbf{A} \psi \overline{\psi} \tag{56}$$

in Eqs. (41) and (42) affect the electromagnetic field *A*<sup>0</sup> , **A**. First, let us try *ψ* = *U*(*x*, *y*, *z*)*eiωt*, where *U* is a solution of the form (53) and *ω* is the corresponding frequency given by Eq. (54). Since the oscillation is to be small, we will assume that an appropriate small constant is absorbed in *R*(*r*). Substitution in Eqs. (55) and (56) shows that *S*<sup>0</sup>  = (*ω* − *V*)*U*<sup>2</sup> and **S** = 0. Since the time dependence has canceled out, we conclude that no radiation into the electromagnetic field occurs when the *ψ* field is restricted to a single mode of oscillation. The nonzero value of *S*<sup>0</sup> causes a change in the static *V* potential field, but the change in the potential well is small since the *ψ* oscillation is taken as small (in the evaluation of Eqs. (55) and (56) for present purposes, we have used the original values of *A*<sup>0</sup> and **A** from the proton solution (48) since any change in *A*<sup>0</sup> and **A** caused by the *ψ* oscillation will be of higher order).

Next, let us consider the case where small oscillations occur in two different modes in the *ψ* field, so that

$$
\psi = \mathcal{L}\{\mathbf{x}, \mathbf{y}, \mathbf{z}\} \, e^{i\alpha t} + \hat{\mathcal{L}}\{\mathbf{x}, \mathbf{y}, \mathbf{z}\} \, e^{i\alpha t}.\tag{57}
$$

Substitution in Eqs. (55) and (56) shows

where *V* now represents the potential well caused by the *φ* field of the proton-like soliton. Any contribution to the *V* field caused by the *ψ* field can be neglected if the oscillations in the *ψ* field are sufficiently small. Let us suppose that ∣*V*∣ is typically much smaller than 1 and that *ω* is near to 1. Then, we can approximate *ω* − *V* + 1 as 2 in the Klein-Gordon Eq. (51) to obtain

Here, *V* describes a potential well, but the quantity *ω* − 1 is related to an actual oscillation frequency *ω*, which is unlike the traditional use of Schrödinger's equation, where the right-hand

The main point of interest here is that solutions of the soliton equations lead in a natural way to Eq. (52), which of course results in frequencies of the hydrogen spectrum. For the *V* =  − *αfs*/*r* potential well of a proton-like soliton, it is then straightforward to solve Eq. (52) according to the well-known solution of Schrödinger's equation for a hydrogen atom [50]. Eigenfunctions occur

where the spherical harmonics are written here in terms of the associated Legendre polynomi-

2 \_\_\_

When small oscillations in the electron field *ψ* are allowed in the vicinity of a proton-like soliton, the overall solution (48) will be slightly changed, so we need to investigate how the

> 2 ∂*ψ*¯¯ \_\_\_ <sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_ 2 ∂*ψ*\_\_\_

<sup>2</sup> <sup>∇</sup>*ψ*¯¯ <sup>−</sup> *<sup>i</sup>ψ*¯¯ \_\_

*U* is a solution of the form (53) and *ω* is the corresponding frequency given by Eq. (54). Since the oscillation is to be small, we will assume that an appropriate small constant is absorbed in

dence has canceled out, we conclude that no radiation into the electromagnetic field occurs

 = (*ω* − *V*)*U*<sup>2</sup>

*<sup>m</sup>* and the usual spherical coordinates *θ* and . In Eq. (53), the usual factor *eim* has been replaced by sinusoids since a real-valued formulation for *U* is desired. A real-valued solution for the radial function *R*(*r*) can be obtained in the standard way. The corresponding frequen-

<sup>2</sup> <sup>∇</sup><sup>2</sup> *<sup>U</sup>* <sup>+</sup> *VU* <sup>=</sup> (*<sup>ω</sup>* <sup>−</sup> <sup>1</sup>)*U*. (52)

*<sup>m</sup>*(cos*θ*)(cos(*m*) + sin(*m*)), (53)

<sup>2</sup> *<sup>n</sup>*<sup>2</sup>, *<sup>n</sup>* <sup>=</sup> <sup>1</sup>, <sup>2</sup>, …, (54)

<sup>∂</sup>*<sup>t</sup>* <sup>−</sup> *<sup>A</sup>*<sup>0</sup> *ψψ*¯¯ (55)

<sup>2</sup> ∇*ψ* − **A***ψψ*¯¯ (56)

, **A**. First, let us try *ψ* = *U*(*x*, *y*, *z*)*eiωt*, where

and **S** = 0. Since the time depen-

the usual Schrödinger equation approximation:

side of the equation is regarded as related to an energy.

126 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

−\_\_1

*U* = *R*(*r*) *Pl*

*<sup>ω</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>α</sup>fs*

with *l* = 0, 1, …, *n* − 1 and *m* =  − *l*, …, *l*.

*<sup>S</sup>*<sup>0</sup> <sup>=</sup> *<sup>i</sup>ψ*\_\_

**<sup>S</sup>** <sup>=</sup> *<sup>i</sup>ψ*\_\_

in Eqs. (41) and (42) affect the electromagnetic field *A*<sup>0</sup>

*R*(*r*). Substitution in Eqs. (55) and (56) shows that *S*<sup>0</sup>

electron field source terms

in the form

als *Pl*

cies are

and

$$S\_0 = \left(\omega - \mathcal{V}\right)\mathcal{U}^2 + \left(\hat{\omega} - \mathcal{V}\right)\hat{\mathcal{U}}^2 + \left(\omega + \hat{\omega} - 2\mathcal{V}\right)\mathcal{U}\hat{\mathcal{U}}\cos\left(\omega - \hat{\omega}\right)\mathbf{t},\tag{58}$$

$$\mathbf{S} = \begin{pmatrix} \hat{\mathbf{U}} \ \nabla \mathbf{U} - \mathbf{U} \ \nabla \hat{\mathbf{U}} \end{pmatrix} \sin(\omega - \hat{\omega}) \mathbf{t}. \tag{59}$$

Since *S*<sup>0</sup> and **S** are now time-varying it is clear that radiation into the electromagnetic field is a possibility, but from the form of the sine and cosine terms it turns out that the radiation can only occur at the difference frequency *<sup>ω</sup>* <sup>−</sup> *<sup>ω</sup>*̂.

It should be noted that charge is not necessarily quantized in Model One, and that the small oscillations in the *ψ* field indeed represent a small amount of charge. In this example, the *V* field is not to be noticeably changed from that of the proton-like soliton, however, so the small oscillations in the *ψ* field must correspond to only a minuscule amount of charge, much less than one elementary charge. Then, Eqs. (41) and (42) are to be solved using the source terms (55), (56) in the form (58), (59). Since charge cannot be radiated into the electromagnetic field, small amounts of charge must be transferred between the trapped modes in the *ψ* field, and the charge must go to those modes in which the ratio of energy to charge is smaller. It is expected that there will be a selection rule since the two modes in question will not effectively radiate into the electromagnetic field in some cases because of their relative symmetry.

Thus, we find in Model One that radiation is emitted into the electromagnetic field when small oscillations in the *ψ* field occur in the potential well created by a proton-like soliton. That radiation turns out to be the well-known spectrum of hydrogen, namely, difference frequencies of the form *<sup>ω</sup>* <sup>−</sup> *<sup>ω</sup>*̂, where *ω* and *ω*̂ are given by Eq. (54) with different values of *n*.

#### **13. Interpretation and summary**

Although Model One, Eqs. (34) and (40)–(42), was set up to let solitons mimic the dynamical behavior of electrons and protons, it turns out that small oscillations of the electron field *ψ* in the vicinity of a proton-like soliton have modes of oscillation that correspond to the various terms in the Grotrian diagram for hydrogen. Thus, a quantum-like phenomenon automatically occurs even though the model is of a purely classical nature. It has seemed in the past that such a result was beyond the realm of a possible classical model since physical oscillation of a charge would presumably radiate away energy at the actual oscillation frequency, whereas only difference frequencies are in fact observed in the hydrogen spectrum.

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Here, it should be noted that three circumstances intrinsic to Model One contribute to allow this quantum-like result to occur. First, the standard, i.e., gauge invariant, coupling of the *ψ* and *φ* fields to the electromagnetic field dictates that the electric potential created by the proton-like soliton is experienced by the *ψ* field as a potential well that allows trapped oscillations. Secondly, the great difference in scale (by a factor of 1836) needed in Eq. (50) assures that the *ψ* field automatically experiences almost a pure *V* =  − *αfs* /*r* potential well, which leads to frequencies that correspond to the well-known terms in the Grotrian diagram for hydrogen. Finally, standard coupling of the *ψ* field to the electromagnetic field again automatically implies that only the difference frequencies are radiated.

The emission of the hydrogen spectrum in Model One has some interesting aspects but it occurs in an unexpected context. When oscillations of the *ψ* field occur in the vicinity of a proton-like soliton, only small oscillations are allowed or the *V* =  − *αfs*/*r* potential well created by the proton-like soliton would be substantially changed and the spectrum typical of hydrogen would not be observed. Thus, in the situation in question, an electron-like soliton need not, indeed must not, be in the vicinity at all. When we investigate Model One, then, whether we like it or not, within that model the hydrogen spectrum is emitted from the vicinity of an isolated proton-like soliton. In other words, in Model One, the hydrogen arc spectrum is emitted from the equivalent of a hydrogen *ion* rather than a neutral hydrogen atom, which is in startling contrast to the accepted understanding of the actual physical phenomenon. It remains to be seen whether such a result might have any possible physical significance, but it seems of interest, nevertheless, to see how a purely classical model can behave in a way suggestive of a quantum mechanical phenomenon.

Relativistic effects have not been considered in the above treatment of Model One since it is well-known that the Klein-Gordon equation gives wrong answers for fine-structure corrections, especially those that involve the anomalous Zeeman effect. It is contemplated that the NKG equations might be replaced with nonlinear Dirac equations in a more advanced Model Two. Such a model will be comparatively difficult to examine, however, so it seems that further study of Model One will still be of interest.

#### **Author details**

Jon C. Luke

Address all correspondence to: jcluke@iupui.edu

Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana, USA

#### **References**

in the past that such a result was beyond the realm of a possible classical model since physical oscillation of a charge would presumably radiate away energy at the actual oscillation frequency, whereas only difference frequencies are in fact observed in the hydrogen

Here, it should be noted that three circumstances intrinsic to Model One contribute to allow this quantum-like result to occur. First, the standard, i.e., gauge invariant, coupling of the *ψ* and *φ* fields to the electromagnetic field dictates that the electric potential created by the proton-like soliton is experienced by the *ψ* field as a potential well that allows trapped oscillations. Secondly, the great difference in scale (by a factor of 1836) needed in Eq. (50) assures that the *ψ* field automatically experiences almost a pure *V* =  − *αfs* /*r* potential well, which leads to frequencies that correspond to the well-known terms in the Grotrian diagram for hydrogen. Finally, standard coupling of the *ψ* field to the electromagnetic field again automatically

The emission of the hydrogen spectrum in Model One has some interesting aspects but it occurs in an unexpected context. When oscillations of the *ψ* field occur in the vicinity of a proton-like soliton, only small oscillations are allowed or the *V* =  − *αfs*/*r* potential well created by the proton-like soliton would be substantially changed and the spectrum typical of hydrogen would not be observed. Thus, in the situation in question, an electron-like soliton need not, indeed must not, be in the vicinity at all. When we investigate Model One, then, whether we like it or not, within that model the hydrogen spectrum is emitted from the vicinity of an isolated proton-like soliton. In other words, in Model One, the hydrogen arc spectrum is emitted from the equivalent of a hydrogen *ion* rather than a neutral hydrogen atom, which is in startling contrast to the accepted understanding of the actual physical phenomenon. It remains to be seen whether such a result might have any possible physical significance, but it seems of interest, nevertheless, to see how a purely classical model can behave in a way sug-

Relativistic effects have not been considered in the above treatment of Model One since it is well-known that the Klein-Gordon equation gives wrong answers for fine-structure corrections, especially those that involve the anomalous Zeeman effect. It is contemplated that the NKG equations might be replaced with nonlinear Dirac equations in a more advanced Model Two. Such a model will be comparatively difficult to examine, however, so it seems that fur-

Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis,

implies that only the difference frequencies are radiated.

128 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

gestive of a quantum mechanical phenomenon.

ther study of Model One will still be of interest.

Address all correspondence to: jcluke@iupui.edu

**Author details**

Indianapolis, Indiana, USA

Jon C. Luke

spectrum.


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2013;**10**:1-23


**Chapter 8**

**Provisional chapter**

**A Perturbation Theory for Nonintegrable Equations**

We describe an approach called the "weak asymptotics method" to construct multisoliton asymptotic solutions for essentially nonintegrable equations with small dispersion. This paper contains a detailed review of the method and a perturbation theory to describe the interaction of distorted solitons for equations with small perturbations. All constructions

**Keywords:** generalized Korteweg-de Vries equation, soliton, interaction, perturbation,

We consider the problem of propagation and interaction of soliton-type solutions of nonlinear equations. Our basic example is the nonhomogeneous version of the generalized KdV

<sup>∂</sup>*x*<sup>3</sup> <sup>=</sup> *<sup>f</sup>*(*u*, *<sup>ε</sup>* \_\_\_ <sup>∂</sup>*<sup>u</sup>*

where *μ* ∈ (1, 5), *ε* ≪ 1, *f*(*u*, *z*) is a known smooth function such that *f*(0, 0) = 0. Note that the restriction on *μ* implies both the soliton-type solution and the stability of the equation with

In the special case *f* ≡ 0 and *μ* = 2 (*μ* = 3), Eq. (1) is the famous KdV (modified KdV) equation. It is well known that KdV (mKdV) solitons are stable and interact in the elastic manner: after

have been realized for the gKdV equation with the nonlinearity *uμ*, *μ* ∈ (1, 5).

**2010 Mathematics Subject Classification:** 35D30, 35Q53, 46F10

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>∂</sup>*uμ* \_\_\_

respect to initial data (see, for example [1, 2]).

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>ε</sup>* <sup>2</sup> <sup>∂</sup><sup>3</sup> \_\_\_*<sup>u</sup>*

**A Perturbation Theory for Nonintegrable Equations** 

DOI: 10.5772/intechopen.71030

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>∂</sup>*x*), *<sup>x</sup>* <sup>∈</sup> *<sup>R</sup>*<sup>1</sup>

, *t* > 0, (1)

and reproduction in any medium, provided the original work is properly cited.

**with Small Dispersion**

**with Small Dispersion**

http://dx.doi.org/10.5772/intechopen.71030

weak asymptotics method

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Georgy Omel'yanov

**Abstract**

**1. Introduction**

\_\_\_ <sup>∂</sup>*<sup>u</sup>*

equation

Georgy Omel'yanov

**Provisional chapter**

#### **A Perturbation Theory for Nonintegrable Equations with Small Dispersion with Small Dispersion**

**A Perturbation Theory for Nonintegrable Equations** 

DOI: 10.5772/intechopen.71030

Georgy Omel'yanov Georgy Omel'yanov Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71030

**Abstract**

We describe an approach called the "weak asymptotics method" to construct multisoliton asymptotic solutions for essentially nonintegrable equations with small dispersion. This paper contains a detailed review of the method and a perturbation theory to describe the interaction of distorted solitons for equations with small perturbations. All constructions have been realized for the gKdV equation with the nonlinearity *uμ*, *μ* ∈ (1, 5).

**Keywords:** generalized Korteweg-de Vries equation, soliton, interaction, perturbation, weak asymptotics method

**2010 Mathematics Subject Classification:** 35D30, 35Q53, 46F10

#### **1. Introduction**

We consider the problem of propagation and interaction of soliton-type solutions of nonlinear equations. Our basic example is the nonhomogeneous version of the generalized KdV equation

$$\frac{\partial u}{\partial t} + \frac{\partial u^{\mu}}{\partial x} + \varepsilon^{2} \frac{\partial^{3} u}{\partial x^{3}} = f \left( u, \varepsilon \frac{\partial u}{\partial x} \right), \quad \mathbf{x} \in \mathbb{R}^{1}, \quad t \ge 0,\tag{1}$$

where *μ* ∈ (1, 5), *ε* ≪ 1, *f*(*u*, *z*) is a known smooth function such that *f*(0, 0) = 0. Note that the restriction on *μ* implies both the soliton-type solution and the stability of the equation with respect to initial data (see, for example [1, 2]).

In the special case *f* ≡ 0 and *μ* = 2 (*μ* = 3), Eq. (1) is the famous KdV (modified KdV) equation. It is well known that KdV (mKdV) solitons are stable and interact in the elastic manner: after

> Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

the collision, they preserve the original amplitudes and velocities shifting the trajectories only (see [3] and other bibliographies devoted to the inverse scattering transform (IST) method). In the case of *μ* = 2 (*μ* = 3) but with *f* ≠ 0, Eq. (1) is a nonintegrable one. However, using the smallness of *ε* (or of *f* for other scaling), it is possible to create a perturbation theory that describes the evolution of distorted solitons (see the approaches by Karpman and E. Maslov [4] and Kaup and Newell [5] on the basis of the IST method, and the "direct" method by V. Maslov and Omel'yanov [6]). Moreover, the approach by V. Maslov and Omel'yanov [6] can be easily extended to essentially nonintegrable equations (*μ* ≠ 2, 3), but for a single soliton only. In fact, it is impossible to use any direct method in the classical sense for the general problem of the wave interaction. To explain this proposition, let us consider the homogeneous gKdV equation

$$
\frac{
\partial u
}{
\partial t
} + \frac{
\partial u^{\mu}
}{
\partial x
} + \varepsilon^{2} \frac{
\partial^{\lambda} u
}{
\partial x^{\lambda}
} = 0, \quad x \in \mathbb{R}^{1}, \quad t \ge 0. \tag{2}
$$

**2.** Let *A*<sup>2</sup>

term *W*<sup>0</sup>

Since *ϕ* ̇ <sup>1</sup> ≠ *ϕ* ̇ 2

**3.** The amplitudes *A*<sup>2</sup>

(*τ*1 , *τ*<sup>2</sup> , *t*):

*A*̂*W*<sup>0</sup> + *B*̂

 > *A*<sup>1</sup>

*W*0 *<sup>μ</sup>* + *B*̂ 3

, we can pass to new variables, *<sup>η</sup>* <sup>=</sup> (*τ*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*2)/(*<sup>ϕ</sup>*

∂*W*<sup>0</sup> ∂*η* + ∂*W*<sup>0</sup> *μ* \_\_\_\_ <sup>∂</sup>*<sup>ζ</sup>* <sup>+</sup> ∂ \_\_\_\_\_ <sup>3</sup> *W*<sup>0</sup>

*u*(*x*, *t*, *ε*) = *W*((*x* − *ϕ*<sup>1</sup>

form equation (9) to the gKdV form (2) again

sible for any essentially nonintegrable case.

\_\_\_\_

distorted solitons for equation (1).

**2. Weak asymptotics method**

\_\_\_

 = *u*<sup>2</sup>

*Q*<sup>1</sup> = *u*, *P*<sup>1</sup> = *uμ*, *Q*<sup>2</sup> = *u*<sup>2</sup>

**2.1. Main definitions**

 = *u*, *R*<sup>2</sup>

and *R*<sup>1</sup>

 − *A*<sup>1</sup> ≪ 1. We write again the ansatz in the form (7), where *ν* = *A*<sup>2</sup>

we assume *ν*/*ε* ≪ 1. In fact, this case coincides with the problem considered in [7].

(*t*))/*ε*, (*x* − *ϕ*<sup>2</sup>

*i*=1 2 *ϕ*̇ *i* \_\_\_∂ ∂*τ<sup>i</sup>*

> ̇ <sup>2</sup> − *ϕ* ̇ 1), *<sup>ζ</sup>* <sup>=</sup> (*<sup>ϕ</sup>* ̇ <sup>1</sup> *τ*<sup>2</sup> − *ϕ* ̇ <sup>2</sup> *τ*1)/(*ϕ* ̇ <sup>1</sup> − *ϕ* ̇

without any additional parameter. Substituting (8) into equation (2), we obtain for the leading

*W*<sup>0</sup> = 0, *A*̂ = −∑

Therefore, to construct two-phase asymptotics, we should solve (10) explicitly what is impos-

This difficulty can be overcome by using the weak asymptotics method. The main point here is that solitons tend to distributions as *ε* → 0. Thus, it is possible to pass to the weak description of the problem, ignore the actual shape of the multiwave solutions, and find only the main solution characteristics, that is, the time dynamics of wave amplitudes and velocities. The weak asymptotics method has been proposed at first for shock wave type solutions [8] and for soliton-type solutions [9] many years ago. Further generalizations, modifications, and adaptations to other problems can be found in publications by M. Colombeau, Danilov, Mitrovic,

Omel'yanov, Shelkovich, and others, see, for example, [10–20] and references therein.

The contents of the paper are the following: in Section 2, we present a detailed survey of the weak asymptotics method application to the problem of multisoliton asymptotics and Section 3 contains new results, namely a perturbation theory to describe the evolution and collision of

Let us associate equation (2) with first two conservation laws written in the differential form:

<sup>∂</sup><sup>3</sup> *<sup>R</sup>* \_\_\_\_*<sup>j</sup>*

∂*Qj* <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> ∂*P*\_\_\_*j* <sup>∂</sup>*<sup>x</sup>* <sup>=</sup> *<sup>ε</sup>* <sup>2</sup>

. Next, we define smallness in the weak sense:

are arbitrary numbers. Then, we should write a two-phase ansatz

, *B*̂ = ∑ *i*=1 <sup>2</sup> \_\_\_∂ ∂*τ<sup>i</sup>*

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

<sup>∂</sup>*ζ*<sup>3</sup> <sup>=</sup> <sup>0</sup>. (10)

<sup>∂</sup>*x*<sup>3</sup> , *<sup>j</sup>* <sup>=</sup> <sup>1</sup>, 2, (11)

2

, (12)

, *P*<sup>2</sup> = 2*μ uμ*+<sup>1</sup> /(*μ* + 1) − 3 (*ε ux*)

 − *A*<sup>1</sup> ≪ 1 now, and

135

. (9)

2), and trans-

(*t*))/*ε*, *t*, *x*, *ε*) (8)

http://dx.doi.org/10.5772/intechopen.71030

It is easy to find the explicit soliton solution of (2),

$$u(\mathbf{x}, t, \varepsilon) = A a \langle \beta(\mathbf{x} - Vt)/\varepsilon \rangle, \quad \omega(\eta) = \cosh^{-\gamma}(\eta/\gamma). \tag{3}$$

$$\gamma = \Im/(\mu - 1), \quad V = \beta^2, \quad A^{\mu - 1} = V(\mu + 1)/2. \tag{4}$$

Next let us consider two-soliton initial data

$$\boldsymbol{\omega} \parallel\_{\boldsymbol{\omega} = 0} = \sum\_{i=1}^{2} A\_i \,\omega \{ \beta\_{\boldsymbol{\beta}} (\mathbf{x} - \mathbf{x}\_{\boldsymbol{(\beta, 0)}}) / \boldsymbol{\varepsilon} \},\tag{5}$$

where *x*(1, 0) > *x*(2, 0) and *A*<sup>2</sup>  > *A*<sup>1</sup> . Obviously, since (*x*(2, 0) − *x*(1, 0))/*ε* → ∞ as *ε* → 0, the sum of the waves (3)

$$\boldsymbol{\mu} = \sum\_{i=1}^{2} A\_i \,\omega \{ \beta\_i (\mathbf{x} - \mathbf{V}\_i \mathbf{t} - \mathbf{x}\_{(i,0)})/\varepsilon \}\tag{6}$$

approximates the problem (2), (5) solution with the precision *O*(*ε*∞) but for *t* ≪ 1 only. Conversely, the sum (6) does not satisfy the gKdV equation for *t* ∼ *O*(1) in view of the trajectories *x* = *Vi t* + *x*(*i*, 0) intersection at a point (*x*\* , *t*\* ).

Let us consider shortly how it is possible to analyze the problem (2), (5). There are some different cases:

**1.** Let *A*<sup>1</sup> ≪ *A*<sup>2</sup> . Then, one can construct an asymptotic solution

$$\mathbf{u} = \mathbf{W}[(\mathbf{x} - \varphi\_2(t)) / \varepsilon, t, \mathbf{x}, \varepsilon, \nu], \tag{7}$$

where *ν* = *A*<sup>1</sup> /*A*<sup>2</sup> ≪ 1 and *W*((*x* − *ϕ*<sup>2</sup> (*t*))/*ε*, *t*, *x*, *ε*, *ν*) = *A*<sup>2</sup> *ω*(*β*<sup>2</sup> (*x* − *V*<sup>2</sup> *t* − *x*(2, 0))/*ε*) + *O*(*ν* + *ε*). Thus, to find the leading term of the asymptotics, we obtain an equation with nonlinear ordinary differential operator; whereas to construct the corrections, it is enough to analyze the linearization of this operator. This construction (with a little bit of other viewpoints) has been realized by Ostrovsky et al. [7].


$$u(\mathbf{x},t,\varepsilon) = \mathcal{W}[(\mathbf{x} - \varphi\_1(t)) / \varepsilon, \langle \mathbf{x} - \varphi\_2(t) \rangle / \varepsilon, t, \mathbf{x}, \varepsilon] \tag{8}$$

without any additional parameter. Substituting (8) into equation (2), we obtain for the leading term *W*<sup>0</sup> (*τ*1 , *τ*<sup>2</sup> , *t*):

$$
\hat{A}W\_0 + \hat{B}W\_0^\mu + \hat{B}^3W\_0 = 0, \quad \hat{A} = -\sum\_{i=1}^2 \dot{\varphi}\_i \frac{\partial}{\partial \tau\_i}, \quad \hat{B} = \sum\_{i=1}^2 \frac{\partial}{\partial \tau\_i}.\tag{9}
$$

Since *ϕ* ̇ <sup>1</sup> ≠ *ϕ* ̇ 2 , we can pass to new variables, *<sup>η</sup>* <sup>=</sup> (*τ*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*2)/(*<sup>ϕ</sup>* ̇ <sup>2</sup> − *ϕ* ̇ 1), *<sup>ζ</sup>* <sup>=</sup> (*<sup>ϕ</sup>* ̇ <sup>1</sup> *τ*<sup>2</sup> − *ϕ* ̇ <sup>2</sup> *τ*1)/(*ϕ* ̇ <sup>1</sup> − *ϕ* ̇ 2), and transform equation (9) to the gKdV form (2) again

$$\frac{\partial \mathcal{W}\_0}{\partial \eta} + \frac{\partial \mathcal{W}\_0^\*}{\partial \zeta} + \frac{\partial^3 \mathcal{W}\_0}{\partial \zeta^3} = \mathbf{0}. \tag{10}$$

Therefore, to construct two-phase asymptotics, we should solve (10) explicitly what is impossible for any essentially nonintegrable case.

This difficulty can be overcome by using the weak asymptotics method. The main point here is that solitons tend to distributions as *ε* → 0. Thus, it is possible to pass to the weak description of the problem, ignore the actual shape of the multiwave solutions, and find only the main solution characteristics, that is, the time dynamics of wave amplitudes and velocities. The weak asymptotics method has been proposed at first for shock wave type solutions [8] and for soliton-type solutions [9] many years ago. Further generalizations, modifications, and adaptations to other problems can be found in publications by M. Colombeau, Danilov, Mitrovic, Omel'yanov, Shelkovich, and others, see, for example, [10–20] and references therein.

The contents of the paper are the following: in Section 2, we present a detailed survey of the weak asymptotics method application to the problem of multisoliton asymptotics and Section 3 contains new results, namely a perturbation theory to describe the evolution and collision of distorted solitons for equation (1).

#### **2. Weak asymptotics method**

#### **2.1. Main definitions**

the collision, they preserve the original amplitudes and velocities shifting the trajectories only (see [3] and other bibliographies devoted to the inverse scattering transform (IST) method). In the case of *μ* = 2 (*μ* = 3) but with *f* ≠ 0, Eq. (1) is a nonintegrable one. However, using the smallness of *ε* (or of *f* for other scaling), it is possible to create a perturbation theory that describes the evolution of distorted solitons (see the approaches by Karpman and E. Maslov [4] and Kaup and Newell [5] on the basis of the IST method, and the "direct" method by V. Maslov and Omel'yanov [6]). Moreover, the approach by V. Maslov and Omel'yanov [6] can be easily extended to essentially nonintegrable equations (*μ* ≠ 2, 3), but for a single soliton only. In fact, it is impossible to use any direct method in the classical sense for the general problem of the wave interaction. To explain this proposition, let us consider the homogeneous gKdV equation

<sup>∂</sup>*x*<sup>3</sup> <sup>=</sup> <sup>0</sup>, *<sup>x</sup>* <sup>∈</sup> *<sup>R</sup>*<sup>1</sup>

, *t* > 0. (2)

, *Aμ*−1 = *V*(*μ* + 1)/2. (4)

*Ai ω*(*βi*(*x* − *x*(*i*,0))/*ε*), (5)

*Ai ω*(*βi*(*x* − *Vi t* − *x*(*i*,0))/*ε*) (6)

(*t*))/*ε*, *t*, *x*, *ε*, *ν*), (7)

*t* − *x*(2, 0))/*ε*) + *O*(*ν* + *ε*). Thus, to

. Obviously, since (*x*(2, 0) − *x*(1, 0))/*ε* → ∞ as *ε* → 0, the sum of the

(*η*/*γ*), (3)

\_\_\_ <sup>∂</sup>*<sup>u</sup>*

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>∂</sup>*uμ* \_\_\_

134 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

*u*(*x*, *t*, *ε*) = *A*(*β*(*x* − *Vt*)/*ε*), *ω*(*η*) = cosh<sup>−</sup>*<sup>γ</sup>*

It is easy to find the explicit soliton solution of (2),

*γ* = 2/(*μ* − 1), *V* = *β*<sup>2</sup>

 > *A*<sup>1</sup>

*t* + *x*(*i*, 0) intersection at a point (*x*\*

*u* = *W*((*x* − *ϕ*<sup>2</sup>

/*A*<sup>2</sup> ≪ 1 and *W*((*x* − *ϕ*<sup>2</sup>

Next let us consider two-soliton initial data

*u*|*<sup>t</sup>*=<sup>0</sup> = ∑

*u* = ∑

where *x*(1, 0) > *x*(2, 0) and *A*<sup>2</sup>

waves (3)

tories *x* = *Vi*

different cases:

**1.** Let *A*<sup>1</sup> ≪ *A*<sup>2</sup>

where *ν* = *A*<sup>1</sup>

by Ostrovsky et al. [7].

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>ε</sup>* <sup>2</sup> <sup>∂</sup><sup>3</sup> \_\_\_*<sup>u</sup>*

*i*=1 2

approximates the problem (2), (5) solution with the precision *O*(*ε*∞) but for *t* ≪ 1 only. Conversely, the sum (6) does not satisfy the gKdV equation for *t* ∼ *O*(1) in view of the trajec-

, *t*\* ). Let us consider shortly how it is possible to analyze the problem (2), (5). There are some

(*t*))/*ε*, *t*, *x*, *ε*, *ν*) = *A*<sup>2</sup>

find the leading term of the asymptotics, we obtain an equation with nonlinear ordinary differential operator; whereas to construct the corrections, it is enough to analyze the linearization of this operator. This construction (with a little bit of other viewpoints) has been realized

*ω*(*β*<sup>2</sup>

(*x* − *V*<sup>2</sup>

*i*=1 2

. Then, one can construct an asymptotic solution

Let us associate equation (2) with first two conservation laws written in the differential form:

$$\frac{\partial Q\_{\dot{\gamma}}}{\partial t} + \frac{\partial P\_{\dot{\gamma}}}{\partial x} = \varepsilon^2 \frac{\partial^3 R\_{\dot{\gamma}}}{\partial x^3}, \quad \dot{j} = 1, 2,\tag{11}$$

$$Q\_1 = u, \quad P\_1 = u^{\mu}, \quad Q\_2 = u^2, \quad P\_2 = 2\mu \, u^{\mu+1} / (\mu+1) - 3 \, (\varepsilon \, u\_z)^2,\tag{12}$$

and *R*<sup>1</sup>  = *u*, *R*<sup>2</sup>  = *u*<sup>2</sup> . Next, we define smallness in the weak sense: **Definition 1.** *A function v*(*t*, *x*, *ε*) *is said to be of the value <sup>O</sup>*' (*ε* ϰ) *if the relation* ∫ −∞ <sup>∞</sup> *v*(*t*, *x*, *ε*)*Ψ*(*x*)*dx* = *O*(*ε* ϰ) *holds uniformly in t for any test function <sup>Ψ</sup>* <sup>∈</sup> (ℝ*<sup>x</sup>* 1 )*. The right-hand side here is a* <sup>∞</sup> -*function for ε* = *const* > 0 *and a piecewise continuous function uniformly in ε* ≥ 0.

Following [9, 17, 18], we define two-soliton weak asymptotics:

**Definition 2.** *A sequence u*(*t*, *x*, *ε*)*, belonging to* <sup>∞</sup> (0, *T*; <sup>∞</sup> (ℝ*<sup>x</sup>* 1 )) *for ε* = const > 0 *and belonging to* (0, *T*; ' (ℝ*<sup>x</sup>* 1 )) *uniformly in ε, is called a weak asymptotic mod <sup>O</sup>*' (*<sup>ε</sup>* <sup>2</sup>) *solution of (2) if the relations (11) hold uniformly in t with the accuracy <sup>O</sup>*' (*<sup>ε</sup>* <sup>2</sup>).

Let us consider the interaction of two solitary waves for the model (2) with the initial data (5).

Following [9, 17, 18] again, we write the asymptotic ansatz in the form:

$$\boldsymbol{\mu} = \sum\_{i=1}^{2} \mathbf{G}\_{\rangle}(\mathbf{r}) \boldsymbol{\omega} \big( \boldsymbol{\beta} \big( \mathbf{x} - \boldsymbol{\varphi} \big( \mathbf{t}, \tau, \boldsymbol{\varepsilon} \big) \big) / \boldsymbol{\varepsilon} \big), \quad \mathbf{G}\_{\rangle}(\mathbf{r}) = \boldsymbol{A}\_{\rangle} + \mathbf{S}\_{\rfloor}(\boldsymbol{\tau}).\tag{13}$$

Here *ϕ<sup>i</sup>*  = *ϕ<sup>i</sup>*<sup>0</sup> (*t*) + *εϕ<sup>i</sup>*<sup>1</sup> (*τ*), where *ϕ<sup>i</sup>*<sup>0</sup>  = *Vi t* + *x*(*i*, 0) are the trajectories of noninteracting solitary waves, *τ* = *ψ*<sup>0</sup> (*t*)/*ε* denotes the "fast time", *ψ*<sup>0</sup> (*t*) = *β*<sup>1</sup> (*ϕ*20(*t*) − *ϕ*10(*t*)), and the phase and amplitude corrections *ϕ<sup>i</sup>*<sup>1</sup> , *Si* are smooth functions such that with exponential rates

 *ϕ<sup>i</sup>*<sup>1</sup> (*τ*) → 0 *as τ* → −∞, *ϕ<sup>i</sup>*<sup>1</sup> (*τ*) → *ϕ<sup>i</sup>*<sup>1</sup> <sup>∞</sup> = *consti as τ* → +∞, (14)

$$S\_!(\tau) \to 0 \quad \text{as} \quad \tau \to \pm \infty. \tag{15}$$

*η*<sup>12</sup> = − *σ*, *η*<sup>22</sup> = *η*, *σ* = *β*1(*ϕ*<sup>1</sup> − *ϕ*2)/*ε*, *θ* = *β*<sup>1</sup> /*β*<sup>2</sup>

nentially fast as ∣*ϕ*<sup>1</sup>

where *ψ*̇

final form:

where

ℜ*<sup>F</sup>*

precision *O*'

where

obtain modulo *O*'

*u* = *ε* ∑

*u*<sup>2</sup> = *ε* ∑

*Ki*

*a*<sup>1</sup> *Vi Ki*<sup>0</sup>

∑

pass to the system:

 − *ϕ*<sup>2</sup>

*τ*/*ψ*̇ <sup>0</sup> + *ϕ<sup>i</sup>*<sup>1</sup>

*δ*(*x* − *ϕ<sup>i</sup>*

(*<sup>ε</sup>* <sup>3</sup>). Thus, using (22) with *F*(*u*) = *u*<sup>2</sup>

(*ε* <sup>3</sup>), ℜ¯¯ *<sup>F</sup>* = *χ*<sup>2</sup> ℜ*<sup>F</sup>*

) − *ε aF*,*i* (1) \_\_\_ *βi δ*′ (*x* − *x*\*)

(*n*) = ∫ −∞

*Gi ω*(*η<sup>i</sup>*2), ∑ *i*=1 2

*δ*(*x* − *ϕ<sup>i</sup>*

*δ*(*x* − *ϕ<sup>i</sup>*

*<sup>n</sup>* /*β<sup>i</sup>* , *Ki*<sup>0</sup>

*ϕ<sup>i</sup>*<sup>0</sup> = *x*\* + *Vi*

*f*(*τ*)*δ*(*x* − *ϕ<sup>i</sup>*

which holds for each *ϕ<sup>i</sup>*

*F*(*u*, *ε ux*) = *ε* ∑

*i*=1 2 { *aF*,*i* (0) \_\_\_ *βi*

+ *O*'

*aF*,*<sup>i</sup>*

(*n*) = ∫ −∞ <sup>∞</sup> *η<sup>n</sup>* {*F*(<sup>∑</sup> *i*=1 2

Here, we take into account that to define ∂ *u*<sup>2</sup>

*i*=1 2 *a*<sup>1</sup> *Ki*<sup>0</sup> (1)

> *i*=1 2 *a*<sup>2</sup> *Ki*<sup>0</sup> (2)

(*n*) = *Gi*

(1) − *aP*<sup>1</sup> ,*i* (0)

> *i*=1 2 *Ki*1

(*<sup>ε</sup>* <sup>3</sup>):

(11), we obtain linear combinations of *εδ*′

<sup>0</sup> <sup>=</sup> *<sup>β</sup>*1(*V*<sup>2</sup> <sup>−</sup> *<sup>V</sup>*1), *χ<sup>i</sup>* <sup>=</sup> *Vi*

We take into account that the second integrand in the right-hand side of (18) vanishes expo-

(*t* − *t*\*) = *x*\* + *ε Vi τ*/*ψ*̇

) = *f*(*τ*)*δ*(*x* − *x*\*) − *<sup>i</sup> f*(*τ*) *δ*′

Schwartz space. Moreover, the second term in the right-hand side of (21) is *O*'

(0) + ℜ*<sup>F</sup>* (1) /*β*2

∣ grows; thus, its main contribution is at the point *x*\*

. It remains to apply the formula

of the form (20) with slowly increasing *χ<sup>i</sup>*

the assumptions (14) and (15), we obtain the weak asymptotic expansion of *F*(*u*, *ε*∂*u*/∂*x*) in the

<sup>∞</sup> *η<sup>n</sup> F*(*Ai ω*(*η*), *β<sup>i</sup> Ai ω*′

/∂*<sup>t</sup>* mod*O*'

*β<sup>i</sup> Gi ω*′

) + *ε* ∑ *i*=1 2 *a*<sup>1</sup> *Ki*<sup>1</sup> (1)

) <sup>+</sup> \_\_*<sup>ε</sup> β*2 {ℜ*u*<sup>2</sup> (0)

(*n*) = *Ai*

(*x* − *ϕ<sup>i</sup>*

/*β<sup>i</sup>* = 0, *a*<sup>2</sup> *Vi Ki*<sup>0</sup>

(1) = 0, ℜ*u*<sup>2</sup>

*<sup>n</sup>* /*β<sup>i</sup>* , *Ki*<sup>1</sup>

> (2) − *aP*<sup>2</sup> ,*i* (0)

Calculating weak expansions for other terms from Definition 2 and substituting them into

} <sup>+</sup> \_\_*<sup>ε</sup> β*2 {ℜ*<sup>F</sup>* (0)

(*η<sup>i</sup>*2)) <sup>−</sup> <sup>∑</sup> *i*=1 2

<sup>0</sup> and *ϕ<sup>i</sup>* = *x*\* + *<sup>i</sup>*

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*\*) <sup>+</sup> *<sup>O</sup>*'

*δ*(*x* − *x*\*) − *ε* ℜ¯¯ *<sup>F</sup> δ*′

*F*(*Ai ω*(*η<sup>i</sup>*2), *β<sup>i</sup> Ai ω*′

{*δ*(*x* − *x*\*) − *<sup>i</sup> δ*′

*δ*(*x* − *x*\*) − *ε*ℜ¯¯ *<sup>u</sup>*<sup>2</sup> *δ*′

(*n*) = *Ki* (*n*) − *Ki*<sup>0</sup> (*n*)

), *i* = 1, 2, *δ*(*x* − *x*\*

(*<sup>ε</sup>* <sup>2</sup>), it is necessary to calculate *u*<sup>2</sup>

and transforming (16) with the help of (21), we

(*x* − *x*\*)

(*x* − *x*\*)

), and *εδ*′

(0) = 0, *i* = 1, 2, (29)

(*x* − *x*\*

/*β<sup>i</sup>* = 0, *i* = 1, 2, (28)

, (22)

(*η*))*d* , (23)

. (19)

137

, (20)

. We write

http://dx.doi.org/10.5772/intechopen.71030

(*ε* <sup>2</sup>), (21)

and for *f*(*τ*) from the

(*x* − *x*\*) }

(*η<sup>i</sup>*2))}*<sup>d</sup>* . (24)

}, (25)

}, (26)

). Therefore, we

. (27)

with the

(*ε*). Thus, under

#### **2.2. Two-wave asymptotic construction**

To construct the asymptotics, we should calculate the weak expansions of the terms from the left-hand sides of the relations (11). It is easy to check that

$$u = \ \varepsilon \sum\_{i=1}^{2} a\_i \frac{G\_i}{\beta\_i} \delta(\mathbf{x} - \varphi\_i) + O\_\varpi(\varepsilon^3),\tag{16}$$

where *δ*(*x*) is the Dirac delta-function. Here and in what follows, we use the notation

$$a\_k \stackrel{\text{def}}{=} \int\_{\omega}^{\omega} \left(\omega(\eta)\right)^k d\eta, \quad k \ge 0, \quad a\_2' \stackrel{\text{def}}{=} \int\_{\omega}^{\omega} (\omega'(\eta))^2 d\eta. \tag{17}$$

At the same time for any *F*(*u*, *ε*∂*u*/∂*x*) ∈ *C*<sup>1</sup> , we have

$$\begin{split} & \int\_{-\omega}^{\omega} F\left(\sum\_{i=1}^{2} \mathbf{G}\_{i} \omega \left(\boldsymbol{\beta}\_{i} \frac{\boldsymbol{x} - \boldsymbol{\varphi}\_{i}}{\varepsilon}\right), \sum\_{i=1}^{2} \boldsymbol{\beta}\_{i} \mathbf{G}\_{i} \omega^{\prime} \left(\boldsymbol{\beta}\_{i} \frac{\boldsymbol{x} - \boldsymbol{\varphi}\_{i}}{\varepsilon}\right)\right) \boldsymbol{\psi}(\boldsymbol{\alpha}) d\mathbf{x} \\ & = \varepsilon \sum\_{i=1}^{2} \frac{1}{\beta\_{i}} \int\_{\boldsymbol{\beta}\_{i}}^{\omega} F\left(\boldsymbol{A}\_{i} \boldsymbol{\omega}(\boldsymbol{\eta}) \boldsymbol{\eta}, \boldsymbol{\beta}\_{i} \boldsymbol{A}\_{i} \boldsymbol{\omega}^{\prime}(\boldsymbol{\eta})\right) \boldsymbol{\psi}\left(\boldsymbol{\varrho}\_{i} + \varepsilon \frac{\boldsymbol{\eta}}{\beta\_{i}}\right) d\boldsymbol{\eta} \\ & + \frac{\varepsilon}{\beta\_{\omega}} \int\_{-\omega}^{\omega} \left\{ F\left(\sum\_{i=1}^{2} \mathbf{G}\_{i} \boldsymbol{\omega}(\boldsymbol{\eta}\_{i2}) \sum\_{i=1}^{2} \boldsymbol{\beta}\_{i} \mathbf{G}\_{i} \boldsymbol{\omega}^{\prime}(\boldsymbol{\eta}\_{i2})\right) \right. \\ & \left. - \sum\_{i=1}^{2} F\left(\boldsymbol{A}\_{i} \boldsymbol{\omega}(\boldsymbol{\eta}\_{i2}) \boldsymbol{\beta}\_{i} \boldsymbol{A}\_{i} \boldsymbol{\omega}^{\prime}(\boldsymbol{\eta}\_{i2})\right)\right\} \boldsymbol{\psi}\left(\boldsymbol{\varrho}\_{2} + \varepsilon \frac{\boldsymbol{\eta}}{\beta\_{\omega}}\right) d\boldsymbol{\eta}\_{i} \end{split} \tag{18}$$

where

$$
\eta\_{12} = \theta \eta - \sigma, \quad \eta\_{22} = \eta\_{\prime} \quad \sigma = \beta\_1 (\varphi\_1 - \varphi\_2)/\varepsilon, \quad \Theta = \beta\_1/\beta\_2. \tag{19}
$$

We take into account that the second integrand in the right-hand side of (18) vanishes exponentially fast as ∣*ϕ*<sup>1</sup>  − *ϕ*<sup>2</sup> ∣ grows; thus, its main contribution is at the point *x*\* . We write

$$
\varphi\_{i0} = \mathbf{x}^\* + V\_i(t - t^\*) = \mathbf{x}^\* + \varepsilon \, V\_i \, \pi/\psi\_0 \quad \text{and} \quad \varphi\_i = \mathbf{x}^\* + \varepsilon \, \chi\_i \tag{20}
$$

where *ψ*̇ <sup>0</sup> <sup>=</sup> *<sup>β</sup>*1(*V*<sup>2</sup> <sup>−</sup> *<sup>V</sup>*1), *χ<sup>i</sup>* <sup>=</sup> *Vi τ*/*ψ*̇ <sup>0</sup> + *ϕ<sup>i</sup>*<sup>1</sup> . It remains to apply the formula

$$f(\mathbf{r})\delta(\mathbf{x}-\boldsymbol{\varphi}) = f(\mathbf{r})\delta(\mathbf{x}-\mathbf{x}') - \varepsilon\,\chi\_{\parallel}f(\mathbf{r})\,\delta(\mathbf{x}-\mathbf{x}') + O\_{\boldsymbol{\varphi}}(\varepsilon^2),\tag{21}$$

which holds for each *ϕ<sup>i</sup>* of the form (20) with slowly increasing *χ<sup>i</sup>* and for *f*(*τ*) from the Schwartz space. Moreover, the second term in the right-hand side of (21) is *O*' (*ε*). Thus, under the assumptions (14) and (15), we obtain the weak asymptotic expansion of *F*(*u*, *ε*∂*u*/∂*x*) in the final form:

$$F(\boldsymbol{\mu}, \boldsymbol{\varepsilon} | \boldsymbol{u}\_{\boldsymbol{x}}) = \boldsymbol{\varepsilon} \sum\_{i=1}^{2} \left\{ \frac{a\_{\boldsymbol{r},i}^{(0)}}{\beta\_{i}} \delta(\mathbf{x} - \boldsymbol{\varphi}\_{i}) - \boldsymbol{\varepsilon} \frac{a\_{\boldsymbol{r},i}^{(0)}}{\beta\_{i}} \delta(\mathbf{x} - \mathbf{x}^{\prime}) \right\} + \frac{\boldsymbol{\varepsilon}}{\beta\_{\boldsymbol{r}}} \{ \mathfrak{R}\_{\boldsymbol{r}}^{(0)} \delta(\mathbf{x} - \mathbf{x}^{\prime}) - \boldsymbol{\varepsilon} \cdot \overline{\mathfrak{R}}\_{\boldsymbol{r}} \delta^{\circ}(\mathbf{x} - \mathbf{x}^{\prime}) \} $$
 
$$+ \mathcal{O}\_{\boldsymbol{\omega}}(\boldsymbol{\varepsilon}^{3}), \quad \overline{\mathfrak{R}}\_{\boldsymbol{r}} = \chi\_{2} \mathfrak{R}\_{\boldsymbol{r}}^{(0)} + \mathfrak{R}\_{\boldsymbol{r}}^{(0)} / \beta\_{2\boldsymbol{r}} \tag{22}$$

where

**Definition 1.** *A function v*(*t*, *x*, *ε*) *is said to be of the value <sup>O</sup>*'

136 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

*ε* = *const* > 0 *and a piecewise continuous function uniformly in ε* ≥ 0. Following [9, 17, 18], we define two-soliton weak asymptotics:

)) *uniformly in ε, is called a weak asymptotic mod <sup>O</sup>*'

Following [9, 17, 18] again, we write the asymptotic ansatz in the form:

(*<sup>ε</sup>* <sup>2</sup>).

(*x* − *ϕ<sup>i</sup>*

 = *Vi*

*holds uniformly in t for any test function <sup>Ψ</sup>* <sup>∈</sup> (ℝ*<sup>x</sup>*

**Definition 2.** *A sequence u*(*t*, *x*, *ε*)*, belonging to* <sup>∞</sup>

*i*=1 2 *Gi* (*τ*)*ω*(*β<sup>i</sup>*

(*τ*), where *ϕ<sup>i</sup>*<sup>0</sup>

(*t*)/*ε* denotes the "fast time", *ψ*<sup>0</sup>

(*τ*) → 0 *as τ* → −∞, *ϕ<sup>i</sup>*<sup>1</sup>

left-hand sides of the relations (11). It is easy to check that

def ∫ −∞

∞

(*ω*(*η*))

*Gi ω*(*β<sup>i</sup>*

*hold uniformly in t with the accuracy <sup>O</sup>*'

*u* = ∑

(*t*) + *εϕ<sup>i</sup>*<sup>1</sup>

*Si*

**2.2. Two-wave asymptotic construction**

*u* = *ε* ∑

At the same time for any *F*(*u*, *ε*∂*u*/∂*x*) ∈ *C*<sup>1</sup>

∫ −∞ ∞ *F*(∑ *i*=1 2

= *ε* ∑ *i*=1 <sup>2</sup> \_\_1 *βi* ∫ −∞

 + \_*ε β*2 ∫ −∞ ∞ {*F*(∑ *i*=1 2

 −∑ *i*=1 2

*a<sup>k</sup>* =

where

, *Si*

 = *ϕ<sup>i</sup>*<sup>0</sup>

tude corrections *ϕ<sup>i</sup>*<sup>1</sup>

*ϕ<sup>i</sup>*<sup>1</sup>

(0, *T*; ' (ℝ*<sup>x</sup>* 1

Here *ϕ<sup>i</sup>*

waves, *τ* = *ψ*<sup>0</sup>

(*ε* ϰ) *if the relation* ∫

(*τ*) = *Ai* + *Si*

*t* + *x*(*i*, 0) are the trajectories of noninteracting solitary

(*τ*) → 0 *as τ* → ±∞. (15)

(*ϕ*20(*t*) − *ϕ*10(*t*)), and the phase and ampli-

<sup>∞</sup> = *consti as τ* → +∞, (14)

(*ε* <sup>3</sup>), (16)

(*η*))<sup>2</sup> *d*. (17)

(18)

)*. The right-hand side here is a* <sup>∞</sup>

1

Let us consider the interaction of two solitary waves for the model (2) with the initial data (5).

(*t*, *τ*, *ε*))/*ε*), *Gi*

(*t*) = *β*<sup>1</sup>

are smooth functions such that with exponential rates

(*τ*) → *ϕ<sup>i</sup>*<sup>1</sup>

To construct the asymptotics, we should calculate the weak expansions of the terms from the

*δ*(*x* − *ϕ<sup>i</sup>*

*<sup>k</sup> d*, *k* > 0, *a*<sup>2</sup>

, we have

*β<sup>i</sup> Gi ω*′ (*βi* \_\_\_\_ *x* − *ϕ<sup>i</sup>*

) + *O*'

′ = def ∫ −∞

(*η*))*ψ*(*ϕ<sup>i</sup>* <sup>+</sup> *<sup>ε</sup>*

*β<sup>i</sup> Gi ω*′

(*η<sup>i</sup>*2))}*ψ*(*ϕ*<sup>2</sup> <sup>+</sup> *<sup>ε</sup>*

∞(*ω*′

*<sup>ε</sup>* ))*ψ*(*x*)*dx*

*η*\_\_ *βi* )*d* 

*η*\_ *<sup>β</sup>*2)*d*,

(*η<sup>i</sup>*2))

*i*=1 2 *a*1 \_\_ *Gi βi*

\_\_\_\_ *x* − *ϕ<sup>i</sup> <sup>ε</sup>* ), ∑ *i*=1 2

<sup>∞</sup> *F*(*Ai ω*(*η*), *β<sup>i</sup> Ai ω*′

*F*(*Ai ω*(*η<sup>i</sup>*2), *β<sup>i</sup> Ai ω*′

*Gi ω*(*η<sup>i</sup>*2), ∑ *i*=1 2

where *δ*(*x*) is the Dirac delta-function. Here and in what follows, we use the notation

(0, *T*; <sup>∞</sup> (ℝ*<sup>x</sup>* 1 −∞

)) *for ε* = const > 0 *and belonging to*

(*<sup>ε</sup>* <sup>2</sup>) *solution of (2) if the relations (11)* 

(*τ*). (13)

<sup>∞</sup> *v*(*t*, *x*, *ε*)*Ψ*(*x*)*dx* = *O*(*ε* ϰ)


$$a\_{\mathbb{F},i}^{(\alpha)} = \int\_{\sim} \prescript{f\_{\alpha}}{}{\eta}^\* \, F\{A\_i \,\omega(\eta)\_\nu \beta\_i A\_i \,\omega^i(\eta)\} d\eta,\tag{23}$$

$$\mathfrak{R}\_{\rm F}^{(n)} = \int\_{\rightharpoonup}^{\omega} \eta^{n} \Big[ \mathsf{F} \Big( \sum\_{l=1}^{2} \mathsf{G}\_{l} \,\omega(\eta\_{l\mathfrak{L}}) \cdot \sum\_{l=1}^{2} \beta\_{l} \,\mathsf{G}\_{l} \,\omega^{\prime}(\eta\_{l\mathfrak{L}}) \Big) - \sum\_{l=1}^{2} \mathsf{F} \Big( A\_{l} \,\omega(\eta\_{l\mathfrak{L}}) \cdot \beta\_{l} A\_{l} \,\omega^{\prime}(\eta\_{l\mathfrak{L}}) \Big) d\eta. \tag{24}$$

Here, we take into account that to define ∂ *u*<sup>2</sup> /∂*<sup>t</sup>* mod*O*' (*<sup>ε</sup>* <sup>2</sup>), it is necessary to calculate *u*<sup>2</sup> with the precision *O*' (*<sup>ε</sup>* <sup>3</sup>). Thus, using (22) with *F*(*u*) = *u*<sup>2</sup> and transforming (16) with the help of (21), we obtain modulo *O*' (*<sup>ε</sup>* <sup>3</sup>):

$$\boldsymbol{\mu} = \boldsymbol{\varepsilon} \sum\_{\boldsymbol{\mu} \boldsymbol{1}}^{2} \boldsymbol{a}\_{\boldsymbol{1}} \boldsymbol{K}\_{\boldsymbol{0}}^{(1)} \boldsymbol{\delta}(\mathbf{x} - \boldsymbol{\varphi}) + \boldsymbol{\varepsilon} \sum\_{\boldsymbol{\mu} \boldsymbol{1}}^{2} \boldsymbol{a}\_{\boldsymbol{1}} \boldsymbol{K}\_{\boldsymbol{1}}^{(1)} [\boldsymbol{\delta}(\mathbf{x} - \mathbf{x}^{\prime}) - \boldsymbol{\varepsilon} \boldsymbol{\chi}] \boldsymbol{\delta}(\mathbf{x} - \mathbf{x}^{\prime}) \tag{25}$$

$$
\mu^2 = \varepsilon \sum\_{i=1}^2 a\_{\varepsilon} K\_{i0}^{(2)} \delta(\mathbf{x} - \boldsymbol{\varphi}\_i) + \frac{\varepsilon}{\beta\_2} \{ \Re\_{\boldsymbol{\mu}^\*} \delta(\mathbf{x} - \mathbf{x}^\*) - \varepsilon \Re\_{\boldsymbol{\mu}^\*} \delta'(\mathbf{x} - \mathbf{x}^\*) \},
\tag{26}
$$

where

$$\mathbf{K}\_{\rangle}^{(n)} = \mathbf{G}\_{\rangle}^{\*} / \boldsymbol{\beta}\_{\prime} \quad \mathbf{K}\_{\, 0}^{(n)} = \mathbf{A}\_{\, \prime}^{\*} / \boldsymbol{\beta}\_{\prime} \quad \mathbf{K}\_{\, \, 0}^{(n)} = \mathbf{K}\_{\, \, \prime}^{(n)} - \mathbf{K}\_{\, \, 0}^{(n)}. \tag{27}$$

Calculating weak expansions for other terms from Definition 2 and substituting them into (11), we obtain linear combinations of *εδ*′ (*x* − *ϕ<sup>i</sup>* ), *i* = 1, 2, *δ*(*x* − *x*\* ), and *εδ*′ (*x* − *x*\* ). Therefore, we pass to the system:

$$a\_1 \, V\_i \, K\_{i0}^{(1)} - a\_{p\_\rho}^{(0)} / \beta\_i = 0, \quad a\_2 \, V\_i \, K\_{i0}^{(2)} - a\_{p\_\rho}^{(0)} / \beta\_i = 0, \quad i = 1, 2,\tag{28}$$

$$\sum\_{i=1}^{2} K\_{\boldsymbol{\alpha}}^{(0)} = 0, \quad \Re\_{\boldsymbol{u}^{\boldsymbol{z}}} = 0, \quad i = 1, 2,\tag{29}$$

$$\dot{\boldsymbol{\Psi}}\_{0} \frac{d}{d\tau} \sum\_{i=1}^{2} \left[ \mathbf{K}\_{i0}^{(1)} \boldsymbol{\varphi}\_{\boldsymbol{\Omega}} + \boldsymbol{\chi}\_{i} \boldsymbol{K}\_{i1}^{(1)} \right] = \boldsymbol{f}\_{\prime} \cdot \dot{\boldsymbol{\Psi}}\_{0} \frac{d}{d\tau} \Big| \sum\_{i=1}^{2} \boldsymbol{a}\_{2} \boldsymbol{K}\_{i0}^{(2)} \boldsymbol{\varphi}\_{\boldsymbol{\Omega}} + \overline{\boldsymbol{\mathfrak{R}}}\_{\boldsymbol{u}^{\circ}} \Big| = \boldsymbol{F} \tag{30}$$

where

$$f = \frac{1}{a\_1 \beta\_2} \Re\_{p\_i}^{(0)}, \quad F = \frac{1}{\beta\_2} \Re\_{p\_i}^{(0)} - a\_i \psi\_0 \sum\_{l=1}^2 \varphi\_{l1} \frac{dK\_{il}^{\ominus}}{d\tau}. \tag{31}$$

where *Qj*

, *Pj*

*Q*<sup>3</sup> = (*εux*)

*<sup>Q</sup>*<sup>4</sup> <sup>=</sup> \_\_<sup>1</sup>

*P*<sup>4</sup> = 12*u*<sup>3</sup> (*ε* <sup>2</sup>

**Theorem 2.** *Let us denote Ai*

*Then, under the assumption*

*β*<sup>2</sup> /*β*<sup>3</sup> = *ν* <sup>3</sup>

*elastic scenario of the solitary waves interaction*.

**Figure 2.** Evolution of the soliton triplet with *μ* = 4, *ε* = 0.1.

**Figure 2** depicts the evolution of a three-wave solution [14].

*that Ai* + 1 > *Ai*

, *j* = 1, 2, coincide with (12) for *μ* = 4, *Ki*

<sup>2</sup> (*ε* <sup>2</sup>

Note that the nondivergent "production" *ε*−1*K*<sup>4</sup>

for rapidly varying functions) as the first ones in (32).

<sup>2</sup> − \_\_2 <sup>5</sup> *u*<sup>5</sup>

*uxx*)<sup>2</sup> <sup>+</sup> \_\_<sup>5</sup>

*uxx*)<sup>2</sup> − 19*u* (*ε ux*)

[18, 19] for the problem of three soliton collisions for the gKdV-4 equation.

*, x*(*i*, 0) > *x*(*i* + 1, 0), and *i* = 1, 2*. Let all trajectories x* = *ϕ<sup>i</sup>*<sup>0</sup>

, *P*<sup>3</sup> = 16 *u*<sup>3</sup>

<sup>21</sup> *<sup>u</sup>*<sup>8</sup> <sup>−</sup> \_\_<sup>10</sup>

<sup>3</sup> *u*<sup>3</sup>

<sup>4</sup> − \_\_3 <sup>2</sup> (*ε* <sup>3</sup>

The formal scheme of the asymptotic construction is similar to the one described above: we write the ansatz of the form (13) but with *N* summands, found weak representations for all terms in (32), and pass to a system similar to (28)–(30). The main obstacle here is the proof that this system admits a solution with the properties of (14), (15). This idea has been realized in

, *β*<sup>1</sup> /*β*<sup>3</sup> = *ν* <sup>3</sup>(3+*α*)/2

*with sufficiently small ν < 1, the three-phase asymptotic solution exists and describes mod <sup>O</sup>*'

 = 0, *i* = 1, 2, 3,

<sup>2</sup> − *u*<sup>8</sup> − 3 (*ε* <sup>2</sup> *uxx*)<sup>2</sup>

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

, *K*<sup>4</sup> = −(*εux*)

<sup>231</sup> *<sup>u</sup>*<sup>11</sup> <sup>−</sup> \_\_\_ <sup>100</sup>

 *the amplitudes of the original solitons and x*(*i*, 0) *their initial positions such* 

5

<sup>3</sup> *u*<sup>6</sup>

has the same value *O*(*ε*−1) (in the *C*-sense and

(*εux*) 2

http://dx.doi.org/10.5772/intechopen.71030

(*t*) *have an intersection point* (*x*\*

, *α* ∈ [0, 1) (36)

, (33)

139

, (34)

. (35)

, *t*\* )*.* 

(*<sup>ε</sup>* <sup>2</sup>) *the* 

(*εux*)

(*εux*) 2

*uxxx*)<sup>2</sup> <sup>+</sup> \_\_\_ <sup>160</sup>

The first four algebraic equations (28) imply again the relation (4) among *Ai* , *β<sup>i</sup>* , and *Vi* . Furthermore, there exists a number *θ*\* ∈ (0, 1) such that equations (29), (30) have the required solution *Si* , *ϕ<sup>i</sup>*<sup>1</sup> with the properties (14) and (15) under the sufficient condition *θ* ≤ *θ*\* (see [9, 17]). It is obvious that the existence of the weak asymptotics (13) with the properties (14) and (15) implies that the solitary waves interact like the KdV solitons at least in the leading term.

**Theorem 1.** *Let θ* ≤ *θ*\* *. Then (13) describes mod <sup>O</sup>*' (*<sup>ε</sup>* <sup>2</sup>) *the elastic scenario of the solitary waves interaction for the μ-gKdV equation (2).*

Numerical simulations ([14, 15, 17]) confirm the traced analysis, see **Figure 1**. Note that a small oscillating tail appears after the soliton collision, see [15] for detail. Obviously, this effect is similar to the "radiation" appearance for the perturbed KdV [21].

#### **2.3. Multisoliton interaction**

*N*-wave solutions of the form similar to waves (13) contain 2*N* free functions *Si* , *ϕ<sup>i</sup>*<sup>1</sup> . Thus, to describe an *N*-soliton collision, we should consider *N* conservation laws. However, nonintegrability implies the existence of a finite number of conservation laws only. For this reason, we need to involve into the consideration balance laws. For the gKdV-4 equation, the first conservation and balance laws have the form

$$\frac{\partial Q\_{\uparrow}}{\partial t} + \frac{\partial P\_{\downarrow}}{\partial x} + \varepsilon^{-1} K\_{\uparrow} = \mathcal{O}\_{\varpi}(\varepsilon^2), \tag{32}$$

**Figure 1.** Evolution of two solitary waves for *μ* = 4 and *ε* = 0.1.

where *Qj* , *Pj* , *j* = 1, 2, coincide with (12) for *μ* = 4, *Ki*  = 0, *i* = 1, 2, 3,

*ψ*̇

where

solution *Si*

0 \_\_\_*d <sup>d</sup>* ∑ *i*=1 2 {*Ki*<sup>0</sup> (1)

*f* = \_\_\_\_ <sup>1</sup>

, *ϕ<sup>i</sup>*<sup>1</sup>

**Theorem 1.** *Let θ* ≤ *θ*\*

*tion for the μ-gKdV equation (2).*

**2.3. Multisoliton interaction**

conservation and balance laws have the form

**Figure 1.** Evolution of two solitary waves for *μ* = 4 and *ε* = 0.1.

\_\_\_

*ϕ<sup>i</sup>*<sup>1</sup> + *χ<sup>i</sup> Ki*<sup>1</sup> (1)

*a*<sup>1</sup> *β*<sup>2</sup> ℜ*<sup>P</sup>*<sup>1</sup> (0)

138 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

*. Then (13) describes mod <sup>O</sup>*'

effect is similar to the "radiation" appearance for the perturbed KdV [21].

∂*Qj* <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> ∂*P*\_\_\_*j*

*N*-wave solutions of the form similar to waves (13) contain 2*N* free functions *Si*

} = *f*, *ψ*̇ 0 \_\_\_*d <sup>d</sup>*{∑ *i*=1 2 *a*<sup>2</sup> *Ki*<sup>0</sup> (2)

Furthermore, there exists a number *θ*\* ∈ (0, 1) such that equations (29), (30) have the required

17]). It is obvious that the existence of the weak asymptotics (13) with the properties (14) and (15) implies that the solitary waves interact like the KdV solitons at least in the leading term.

Numerical simulations ([14, 15, 17]) confirm the traced analysis, see **Figure 1**. Note that a small oscillating tail appears after the soliton collision, see [15] for detail. Obviously, this

describe an *N*-soliton collision, we should consider *N* conservation laws. However, nonintegrability implies the existence of a finite number of conservation laws only. For this reason, we need to involve into the consideration balance laws. For the gKdV-4 equation, the first

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>ε</sup>* −1 *Kj* <sup>=</sup> *<sup>O</sup>*'

with the properties (14) and (15) under the sufficient condition *θ* ≤ *θ*\*

, *F* = \_\_<sup>1</sup> *β*2 ℜ*<sup>P</sup>*<sup>2</sup> (0) − *a*<sup>1</sup> *ψ*̇ <sup>0</sup> ∑ *i*=1 2 *ϕi*1 *dKi*<sup>1</sup> (2) \_\_\_\_

The first four algebraic equations (28) imply again the relation (4) among *Ai*

*ϕ<sup>i</sup>*<sup>1</sup> + ℜ¯¯ *<sup>u</sup>*<sup>2</sup>

(*<sup>ε</sup>* <sup>2</sup>) *the elastic scenario of the solitary waves interac-*

} <sup>=</sup> *<sup>F</sup>*, (30)

, *β<sup>i</sup>*

, *ϕ<sup>i</sup>*<sup>1</sup>

(*ε* <sup>2</sup>), (32)

. Thus, to

, and *Vi* .

(see [9,

*<sup>d</sup>* . (31)

$$Q\_3 = \left\langle \varepsilon u\_{\rm x} \right\rangle^2 - \frac{2}{5} u^5, \quad P\_3 = 16 \, u^3 \left\langle \varepsilon u\_{\rm x} \right\rangle^2 - u^8 - 3 \left\langle \varepsilon^2 u\_{\rm x} \right\rangle^2 \tag{33}$$

$$Q\_4 = \frac{1}{2} \left( \varepsilon^2 \mu\_{\text{xx}} \right)^2 + \frac{5}{21} \mu^8 - \frac{10}{3} \mu^3 \left< \varepsilon \mu\_{\text{x}} \right>^2, \quad K\_4 = - \left< \varepsilon \mu\_{\text{x}} \right>^5,\tag{34}$$

$$P\_4 = 12u^3 \left\{ \varepsilon^2 u\_{\text{xx}} \right\}^2 - 19u \left\{ \varepsilon \left. u\_{\text{x}} \right\}^4 - \frac{3}{2} \left\{ \varepsilon^3 u\_{\text{xxx}} \right\}^2 + \frac{160}{231} u^{11} - \frac{100}{3} u^6 \left\{ \varepsilon u\_{\text{x}} \right\}^2. \tag{35}$$

Note that the nondivergent "production" *ε*−1*K*<sup>4</sup> has the same value *O*(*ε*−1) (in the *C*-sense and for rapidly varying functions) as the first ones in (32).

The formal scheme of the asymptotic construction is similar to the one described above: we write the ansatz of the form (13) but with *N* summands, found weak representations for all terms in (32), and pass to a system similar to (28)–(30). The main obstacle here is the proof that this system admits a solution with the properties of (14), (15). This idea has been realized in [18, 19] for the problem of three soliton collisions for the gKdV-4 equation.

**Theorem 2.** *Let us denote Ai the amplitudes of the original solitons and x*(*i*, 0) *their initial positions such that Ai* + 1 > *Ai , x*(*i*, 0) > *x*(*i* + 1, 0), and *i* = 1, 2*. Let all trajectories x* = *ϕ<sup>i</sup>*<sup>0</sup> (*t*) *have an intersection point* (*x*\* , *t*\* )*. Then, under the assumption*

$$
\beta\_2/\beta\_3 = \nu^3, \quad \beta\_1/\beta\_3 = \nu^{3(3+\alpha)/2}, \ a \in [0,1) \tag{36}
$$

*with sufficiently small ν < 1, the three-phase asymptotic solution exists and describes mod <sup>O</sup>*' (*<sup>ε</sup>* <sup>2</sup>) *the elastic scenario of the solitary waves interaction*.

**Figure 2** depicts the evolution of a three-wave solution [14].

**Figure 2.** Evolution of the soliton triplet with *μ* = 4, *ε* = 0.1.

#### **2.4. Asymptotic equivalence**

Let us come back to the case of two-phase asymptotics and transform the ansatz (13) to the following form:

$$\tilde{\boldsymbol{\mu}} = \sum\_{i=1}^{2} \left| \mathbf{G}\_{i}(\boldsymbol{\tau}) \boldsymbol{\omega} \left( \boldsymbol{\beta}\_{i} \frac{\mathbf{x} - \boldsymbol{\varphi}\_{i}(t, \boldsymbol{\tau}, \boldsymbol{\varepsilon})}{\boldsymbol{\varepsilon}} \right) + \mathfrak{S}\_{i}(\boldsymbol{\tau}) \mathbf{W} \left( \boldsymbol{\beta}\_{i} \frac{\mathbf{x} - \boldsymbol{\varphi}\_{i}(t, \boldsymbol{\tau}, \boldsymbol{\varepsilon})}{\boldsymbol{\varepsilon}} \right) \right| \tag{37}$$

where *<sup>i</sup>* (*τ*), *i* = 1, 2 are arbitrary functions from the Schwartz space,

$$\mathcal{W}(\eta) = \, d^{2l+1} \, \omega(\eta) / d \, \eta^{2l+1},\tag{38}$$

that there is not any rule and it is possible to use arbitrary combination of the laws. Thus, there appears the next question: what is the difference between such solutions? This problem has been discussed in [20] for two-phase asymptotic solutions of the gKdV-4 equation. Let us define two-

We consider now the nonhomogeneous version of the gKdV equation (1). It is easy to verify that, in the case of rapidly varying solutions, the right-hand side *f* can be treated as a "small

An approach to construct one-phase self-similar asymptotic solutions for (1) had been created in [6] (see also [17]). Let us generalize this approach to the multiphase case. From the beginning, we state that equation (1) is associated with balance laws, the first two of which are

*K*<sup>1</sup> = −*f*(*u*, *ε ux*), *K*<sup>2</sup> = −*uf*(*u*, *ε ux*). (45)

Let us first construct a two-phase version of self-similar asymptotics, which assumes a special initial data for (1) and discuss afterward how to treat it for more realistic initial data. By anal-

**Definition 5.** *Let a sequence u* = *u*(*t*, *x*, *ε*) *belong to the same functional space as in Definition 2.* 

(*ε*¯¯*q*

) + *ε*(*zi*

(*x*, *t*)ℌ(*η<sup>i</sup>*

(*t*)(*x* − *ϕ<sup>i</sup>*

) + *<sup>i</sup>*

)/*ε*, *ϕ<sup>i</sup>*

(*τ*)*ω*(*η<sup>i</sup>*

(*τ, t*)*ω*(*η<sup>i</sup>*

(*τ*), *η<sup>i</sup>* = *β<sup>i</sup>*

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *Kj* <sup>=</sup> *<sup>O</sup>*'

 = *u<sup>k</sup>*0, *k*<sup>1</sup>

1 , *k* 1

′ ∈ {2, 3, 4}.

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

in (32), productions here are regularly degenerating functions with

(*t*, *x*, *ε*) *belong to the same functional* 

http://dx.doi.org/10.5772/intechopen.71030

(*ε* <sup>2</sup>), *j* = 1, 2, (44)

) *solution of (1) if the relation (44) hold uniformly in* 

(*t*, *τ*, *ε*) = *ϕ<sup>i</sup>*<sup>0</sup> + *<sup>i</sup>*<sup>1</sup>

))}, (46)

. (47)

(*<sup>ε</sup>* <sup>2</sup>) *solution of (2) if* 

′ *of the problem* 

141

1

 ≤ 4 *and let a sequence u<sup>k</sup>*0, *k*<sup>1</sup>

**Theorem 3.** *Let θ be sufficiently small. Then, the weak asymptotic solutions u*1, *k*<sup>1</sup> *and <sup>u</sup>*1,*<sup>k</sup>*

*space as u*(*t*, *x*, *ε*) *in Definition 2. Then, uk*0, *k*<sup>1</sup> *is called a weak asymptotic mod <sup>O</sup>*'

phase asymptotics in the following manner:

 < *k*<sup>1</sup>

A detailed analysis implies the assertion [20].

**3. Collision of distorted solitons**

\_\_\_

and *Pj*

Note that, in contrast to *Kj*

*t* ∈ (0, *T*), ¯¯*<sup>q</sup>* <sup>=</sup> min{*μ*, 2}.

*Gi*

the value *O*(1) in the *C*-sense.

ogy with Definition 2, we write:

*u* = ∑

*Then u is called a weak asymptotic mod <sup>O</sup>*'

(*τ*, *t*) = *Ai*

Generalizing one-phase asymptotics, we write the ansatz as

*i*=1 2 {*Gi*

(*t*) + *Si*

*the relations (32) hold for j* = *k*<sup>0</sup> *and j* = *k*<sup>1</sup> *uniformly in t.*

*(2), (5) exist and they are asymptotically equivalent for all k*

∂*Qj* <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> ∂*P*\_\_\_*j*

coincide with ones described in (12),

**Definition 4.** *Let* 1 ≤ *k*<sup>0</sup>

perturbation."

where *Qj*

and *l* ≥ 1 is an arbitrary integer. Calculating the weak representations for *u*˜ and *u*˜ <sup>2</sup> , we obtain

by integer.\\
Calculating the weak representations for  $\mathfrak{k}$  and  $\mathfrak{k}$ ; we obtain

 $\widetilde{\mathfrak{u}} = \mathfrak{u} + O\_{\mathfrak{D}}(\varepsilon^{2l+2})$ ,

 $\widetilde{\mathfrak{u}}^{2} = \varepsilon \sum\_{i=1}^{2} a\_{2} K\_{i0}^{(2)} \delta(\mathfrak{x} - \varrho\_{i}) + \frac{\varepsilon}{\overline{\mathcal{B}}\_{2}} \mathfrak{R}\_{\widetilde{\mathfrak{u}}^{2}}^{(0)} \delta(\mathfrak{x} - \mathfrak{x}^{\*}) + O\_{\mathfrak{D}}(\varepsilon^{2})$ .

where

$$\mathfrak{R}\_{\mathbb{L}^z}^{(0)} = \int\_{\sim}^{\mathbb{L}} \left[ \left( \sum\_{i=1}^2 (\mathbb{G}\_i \,\omega \langle \eta\_{i2} \rangle + \mathfrak{S}\_i \,\mathcal{W} \langle \eta\_{i2} \rangle) \right)^2 - \sum\_{i=1}^2 \left( A\_i \,\omega \langle \eta\_{i2} \rangle \right)^2 \right] d\eta\_i \tag{40}$$

and *u* in the right-hand side in (39) is the representation (25). Thus, the difference between *u* of the forms (13) and (37) is arbitrarily small in the sense ' (ℝ*x*). At the same time, instead of (29), (30), we obtain

$$\sum\_{l=1}^{2} K\_{\text{il}}^{(1)} = 0, \quad \Re\_{\text{il}^z}^{(0)} = 0, \quad i = 1, 2,\tag{41}$$

$$\left\|\boldsymbol{\psi}\_{\boldsymbol{0}}\frac{d}{d\tau}\sum\_{\mu=1}^{2}\left[\boldsymbol{K}\_{\boldsymbol{0}0}^{(1)}\,\boldsymbol{\varphi}\_{\boldsymbol{\alpha}}+\boldsymbol{\chi}\_{\boldsymbol{\epsilon}}\boldsymbol{K}\_{\boldsymbol{\alpha}}^{(1)}\right]\right\| = \left.\widetilde{f}\_{\boldsymbol{\prime}}\,\boldsymbol{\psi}\_{\boldsymbol{0}}\frac{d}{d\tau}\Big|\sum\_{\mu=1}^{2}a\_{2}\,\boldsymbol{K}\_{\boldsymbol{\alpha}}^{(2)}\,\boldsymbol{\varphi}\_{\boldsymbol{\alpha}}+\widetilde{\boldsymbol{\mathsf{R}}}\_{\boldsymbol{\alpha}}\right\| = \left.\widetilde{\boldsymbol{F}}\right.\tag{42}$$

where *f* ˜, *F* ˜ differ from *f*, *F* in the same manner as ℜ*<sup>u</sup>*˜ <sup>2</sup> (0) differs from ℜ*u*<sup>2</sup> (0) . The system (41) and (42) have again a solution with the properties (14) and (15) [9, 12]; however, it differs from the solution of Eqs. (29) and (30) with the value *O*(1) in the *C*-sense. Moreover, the asymptotic solutions (13) and (37) differ with the precision *O*' (*ε*) in the sense of Definition 1. This implies the principal impossibility to describe explicitly neither the real shape of the waves at the time instant of the collision nor the real *ε*-size displacements of the trajectories after the interaction. However, the nonuniqueness of the value *O*(*ε*) is concentrated within *O*(*ε*1 − *<sup>ν</sup>* )-neighborhood of the time instant *t*\* of the interaction, *ν* > 0. Thus, it is small in the ' (*Rx*,*<sup>t</sup>* 2 ) sense. We set

**Definition 3.** *Functions u*<sup>1</sup> (*x*, *t*, *ε*) *and u*<sup>2</sup> (*x*, *t*, *ε*) *are said to be asymptotically equivalent if for any test function <sup>ψ</sup>* <sup>∈</sup> (*R*<sup>2</sup>)

$$\int\_{-\alpha}^{\varepsilon} \int\_{-\alpha}^{\varepsilon} \langle u\_1(\mathbf{x}, t, \varepsilon) - u\_2(\mathbf{x}, t, \varepsilon) \rangle \psi(\mathbf{x}, t) d\mathbf{x} \, dt \, = \, \mathcal{O}(\varepsilon^2). \tag{43}$$

In this sense, the solutions (13) and (37) are asymptotically equivalent.

We now focus attention on another question: how to choose, from the set of all possible conservation and balance laws, those that allow to construct a multiphase asymptotic solution? It seems that there is not any rule and it is possible to use arbitrary combination of the laws. Thus, there appears the next question: what is the difference between such solutions? This problem has been discussed in [20] for two-phase asymptotic solutions of the gKdV-4 equation. Let us define twophase asymptotics in the following manner:

**Definition 4.** *Let* 1 ≤ *k*<sup>0</sup>  < *k*<sup>1</sup>  ≤ 4 *and let a sequence u<sup>k</sup>*0, *k*<sup>1</sup>  = *u<sup>k</sup>*0, *k*<sup>1</sup> (*t*, *x*, *ε*) *belong to the same functional space as u*(*t*, *x*, *ε*) *in Definition 2. Then, uk*0, *k*<sup>1</sup> *is called a weak asymptotic mod <sup>O</sup>*' (*<sup>ε</sup>* <sup>2</sup>) *solution of (2) if the relations (32) hold for j* = *k*<sup>0</sup> *and j* = *k*<sup>1</sup> *uniformly in t.*

A detailed analysis implies the assertion [20].

**2.4. Asymptotic equivalence**

*u*˜ = ∑

ℜ*<sup>u</sup>*˜ <sup>2</sup>

(29), (30), we obtain

*ψ*̇

˜, *F*

of the time instant *t*\*

*function <sup>ψ</sup>* <sup>∈</sup> (*R*<sup>2</sup>)

**Definition 3.** *Functions u*<sup>1</sup>

∫

where *f*

∑

0 \_\_\_*d <sup>d</sup>* ∑ *i*=1 2 {*Ki*<sup>0</sup> (1)

solutions (13) and (37) differ with the precision *O*'

−∞ <sup>∞</sup> ∫ −∞ <sup>∞</sup> {*u*<sup>1</sup>

*i*=1 2 {*Gi* (*τ*)*ω*(*β<sup>i</sup>*

*W*(*η*) = *d*<sup>2</sup>*l*+<sup>1</sup> *ω*(*η*)/*d η*<sup>2</sup>*l*+<sup>1</sup>

140 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

*u*˜ = *u* + *O*'

(0) = ∫ −∞ ∞ {(∑ *i*=1 2

*i*=1 2 *a*<sup>2</sup> *Ki*<sup>0</sup> (2)

of the forms (13) and (37) is arbitrarily small in the sense '

*i*=1 2 *Ki*1

˜ differ from *f*, *F* in the same manner as ℜ*<sup>u</sup>*˜ <sup>2</sup>

(*x*, *t*, *ε*) *and u*<sup>2</sup>

*ϕ<sup>i</sup>*<sup>1</sup> + *χ<sup>i</sup> Ki*<sup>1</sup> (1) } = *f* ˜, *ψ*̇ 0 \_\_\_*d <sup>d</sup>*{∑ *i*=1 2 *a*<sup>2</sup> *Ki*<sup>0</sup> (2)

following form:

where *<sup>i</sup>*

where

Let us come back to the case of two-phase asymptotics and transform the ansatz (13) to the

(*τ*)*W*(*β<sup>i</sup>*

*x* − *ϕ<sup>i</sup>*

*<sup>δ</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*\*) <sup>+</sup> *<sup>O</sup>*'

(*Ai ω*(*η<sup>i</sup>*2))<sup>2</sup>

*ϕ<sup>i</sup>*<sup>1</sup> + ℜ¯¯ *<sup>u</sup>*˜ <sup>2</sup>

differs from ℜ*u*<sup>2</sup>

(0) = 0, *i* = 1, 2, (41)

} <sup>=</sup> *<sup>F</sup>*

(*ε*) in the sense of Definition 1. This implies

(*Rx*,*<sup>t</sup>* 2

(*x*, *t*, *ε*) *are said to be asymptotically equivalent if for any test* 

(*x*, *t*, *ε*)}*ψ*(*x*, *t*)*dx dt* = *O*(*ε* <sup>2</sup>). (43)

(0)

2 − ∑ *i*=1 2

(0)

(*t*, *<sup>τ</sup>*, *<sup>ε</sup>*) \_\_\_\_\_\_\_\_\_ *<sup>ε</sup>* )}, (37)

, (38)

(*<sup>ε</sup>* <sup>2</sup>), (39)

}*d*, (40)

˜, (42)

. The system (41) and

)-neighborhood

) sense. We set

(ℝ*x*). At the same time, instead of

, we obtain

(*t*, *<sup>τ</sup>*, *<sup>ε</sup>*) \_\_\_\_\_\_\_\_\_ *<sup>ε</sup>* ) <sup>+</sup> *<sup>i</sup>*

*x* − *ϕ<sup>i</sup>*

(*τ*), *i* = 1, 2 are arbitrary functions from the Schwartz space,

and *l* ≥ 1 is an arbitrary integer. Calculating the weak representations for *u*˜ and *u*˜ <sup>2</sup>

*δ*(*x* − *ϕ<sup>i</sup>*

(1) = 0, ℜ*<sup>u</sup>*˜ <sup>2</sup>

However, the nonuniqueness of the value *O*(*ε*) is concentrated within *O*(*ε*1 − *<sup>ν</sup>*

(*x*, *t*, *ε*) − *u*<sup>2</sup>

In this sense, the solutions (13) and (37) are asymptotically equivalent.

(*Gi <sup>ω</sup>*(*η<sup>i</sup>*2) <sup>+</sup> *<sup>i</sup> <sup>W</sup>*(*η<sup>i</sup>*2)))

) <sup>+</sup> \_\_*<sup>ε</sup> β*2 ℜ*<sup>u</sup>*˜ <sup>2</sup> (0)

and *u* in the right-hand side in (39) is the representation (25). Thus, the difference between *u*

(42) have again a solution with the properties (14) and (15) [9, 12]; however, it differs from the solution of Eqs. (29) and (30) with the value *O*(1) in the *C*-sense. Moreover, the asymptotic

the principal impossibility to describe explicitly neither the real shape of the waves at the time instant of the collision nor the real *ε*-size displacements of the trajectories after the interaction.

We now focus attention on another question: how to choose, from the set of all possible conservation and balance laws, those that allow to construct a multiphase asymptotic solution? It seems

of the interaction, *ν* > 0. Thus, it is small in the '

(*ε* <sup>2</sup>*l*+2), *<sup>u</sup>*˜ <sup>2</sup> <sup>=</sup> *<sup>ε</sup>* <sup>∑</sup> **Theorem 3.** *Let θ be sufficiently small. Then, the weak asymptotic solutions u*1, *k*<sup>1</sup> *and <sup>u</sup>*1,*<sup>k</sup>* 1 ′ *of the problem (2), (5) exist and they are asymptotically equivalent for all k* 1 , *k* 1 ′ ∈ {2, 3, 4}.

#### **3. Collision of distorted solitons**

We consider now the nonhomogeneous version of the gKdV equation (1). It is easy to verify that, in the case of rapidly varying solutions, the right-hand side *f* can be treated as a "small perturbation."

An approach to construct one-phase self-similar asymptotic solutions for (1) had been created in [6] (see also [17]). Let us generalize this approach to the multiphase case. From the beginning, we state that equation (1) is associated with balance laws, the first two of which are

$$\frac{\partial Q\_{\rangle}}{\partial t} + \frac{\partial P\_{j}}{\partial x} + K\_{\rangle} = O\_{\varpi}(\varepsilon^{2}), \quad j = 1, 2,\tag{44}$$

where *Qj* and *Pj* coincide with ones described in (12),

$$\mathbf{K}\_{\mathbf{\iota}} = -f(\boldsymbol{\mu}, \boldsymbol{\varepsilon} \, \boldsymbol{\mu}\_{\boldsymbol{\omega}}) \prime \quad \mathbf{K}\_{\mathbf{\iota}} = -uf(\boldsymbol{\mu}, \boldsymbol{\varepsilon} \, \boldsymbol{\mu}\_{\boldsymbol{\omega}}).\tag{45}$$

Note that, in contrast to *Kj* in (32), productions here are regularly degenerating functions with the value *O*(1) in the *C*-sense.

Let us first construct a two-phase version of self-similar asymptotics, which assumes a special initial data for (1) and discuss afterward how to treat it for more realistic initial data. By analogy with Definition 2, we write:

**Definition 5.** *Let a sequence u* = *u*(*t*, *x*, *ε*) *belong to the same functional space as in Definition 2. Then u is called a weak asymptotic mod <sup>O</sup>*' (*ε*¯¯*q*) *solution of (1) if the relation (44) hold uniformly in t* ∈ (0, *T*), ¯¯*<sup>q</sup>* <sup>=</sup> min{*μ*, 2}.

Generalizing one-phase asymptotics, we write the ansatz as

$$\boldsymbol{\mu} = \sum\_{i=1}^{2} \left[ \mathbf{G}\_{i}(\boldsymbol{\tau}, t) \boldsymbol{\omega}(\boldsymbol{\eta}\_{i}) + \varepsilon \{ \mathbf{z}\_{i}(\boldsymbol{x}, t) \mathbf{\hat{S}}(\boldsymbol{\eta}\_{i}) + \mathbf{\Theta}\_{i}(\boldsymbol{\tau}) \boldsymbol{\omega}(\boldsymbol{\eta}\_{i}) \} \right] \tag{46}$$

$$\mathbf{G}\_{\parallel}(\boldsymbol{\tau},t) = A\_{\parallel}(t) + \mathbf{S}\_{\parallel}(\boldsymbol{\tau}), \ \boldsymbol{\eta}\_{\parallel} = \beta\_{\parallel}(t)(\mathbf{x} - \boldsymbol{\varphi}\_{\parallel})/\boldsymbol{\varepsilon}, \ \boldsymbol{\varphi}\_{\parallel}(t,\boldsymbol{\tau},\boldsymbol{\varepsilon}) = \boldsymbol{\varphi}\_{\parallel} + \boldsymbol{\varepsilon}\boldsymbol{\varphi}\_{\parallel}.\tag{47}$$

Here *Ai* (*t*), *ϕ<sup>i</sup>*<sup>0</sup>  = *ϕ<sup>i</sup>*<sup>0</sup> (*t*), *β<sup>i</sup>* 2 (*t*) = *γ A <sup>μ</sup>*−1 (*t*), *ω*(*η*), *Si* (*τ*), *ϕ<sup>i</sup>*<sup>1</sup> (*τ*) are the same as in (13); *τ* = *ψ*<sup>0</sup> (*t*)/*ε* with *ψ*0 (*t*) = *ϕ*20(*t*) − *ϕ*10(*t*) denotes the "fast time" again; *<sup>z</sup> <sup>i</sup>* (*x*, *<sup>t</sup>*) <sup>∈</sup> <sup>∞</sup> ; and *<sup>i</sup>* , ℌ are smooth functions such that

$$\mathfrak{G}\_{](\tau)} \to 0 \quad \text{as} \quad \tau \to -\infty \quad , \quad \mathfrak{G}\_{](\tau)} \to \mathfrak{G}\_{]}^{\*} = \text{const}\_{\vert} \quad \text{as} \quad \tau \to +\infty \quad , \tag{48}$$

$$\mathfrak{H}(\mathfrak{q}) \to 1 \quad \text{as} \quad \eta \to -\infty \quad , \quad \mathfrak{H}(\mathfrak{q}) \to 0 \quad \text{as} \quad \eta \to +\infty \tag{49}$$

*a*<sup>2</sup> *Ai*

\_\_\_*<sup>d</sup>*

Next, we use (57), the equality *β<sup>i</sup>*

\_\_\_

 = 2/(*a*<sup>2</sup>

<sup>∂</sup>*<sup>z</sup>* \_\_\_*<sup>i</sup>*

In view of (58) *dϕ<sup>i</sup>*<sup>0</sup>

*zi*

 = *a*<sup>1</sup>

consistency condition

*zi*

*Ai*

(14), (15). In particular

Consequently, the condition *θ* ≤ *θ*\*

tion for (60)

where *c*<sup>2</sup>

will assume

Cauchy problem

where *c*<sup>1</sup>

<sup>2</sup> \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* <sup>=</sup> *aP*<sup>2</sup> ,*i* (0) , *a*<sup>1</sup> \_\_*d dt* \_\_ *Ai βi* + *zi* (*ϕi* , *t*) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* <sup>=</sup> *af*,*<sup>i</sup>* (0) \_\_\_ *βi*

equalities (4) and rewrite the model equation for *ω*(*η*) as follows:

*a*<sup>1</sup> = (*μ* + 1) *aμ*/2, *a*<sup>2</sup> = (*μ* + 3) *aμ*+<sup>1</sup> /4, *a*<sup>2</sup>

*dAi*

equalities in equations (54) and (55) are equivalent.

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *<sup>f</sup> u* ′(0, 0) *zi*

0

Furthermore, equating zero the coefficients of *δ*(*x* − *x*\*

*Ai* |*<sup>t</sup>*=<sup>0</sup> = *Ai*

(*x*, *t*) |*<sup>x</sup>*=*ϕ<sup>i</sup>*<sup>0</sup> (*t*) = √ \_\_ *γ af*,*<sup>i</sup>* (0) *Ai*

*zi*

(3 − *μ*)(1 + *μ*)/(2*a*<sup>2</sup>

(5 − *μ*)); *Ai*

*<sup>d</sup>*{−*<sup>ω</sup>* <sup>+</sup> *<sup>μ</sup>* <sup>+</sup> <sup>1</sup> \_\_\_\_

<sup>2</sup> <sup>=</sup> *<sup>γ</sup> Ai μ*−1

> ,*i* (0) *Ai* −1 , \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* <sup>=</sup> *aμ* \_\_ *a*1 *Ai μ*−1

*dt* <sup>=</sup> <sup>−</sup>*c*<sup>1</sup> *aK*<sup>2</sup>

Simple manipulations with (56) allow us to find relations between structural constants:

0

Next, equating zero the coefficients of the Heaviside functions, we obtain the equations

, *x* < *ϕ<sup>i</sup>*<sup>0</sup>

(*x*, *t*) |*<sup>t</sup>*=<sup>0</sup> = *zi* 0

0

\_\_ *γ af*,*<sup>i</sup>* (0) *Ai*

> (*x*, *t*) |*<sup>x</sup>*=*ϕ<sup>i</sup>*<sup>0</sup> (*t*)

(5 − *μ*)), *zi*

(*x*(*i*,0)) = {√

(*t*) > 0, *zi*

,  *<sup>ϕ</sup><sup>i</sup>*0|*<sup>t</sup>*=<sup>0</sup>

(3−4*μ*)/2(*t*) <sup>+</sup> *<sup>c</sup>*<sup>2</sup> *aK*<sup>2</sup>

(3−4*μ*)/2 + *c*<sup>2</sup> *aK*<sup>2</sup>

guaranties the existence of *Si*

We should note that the nonlinearity *uμ* in (1) can require the inequality *u* ≥ 0. To this end, we

= *x*(*i*,0)

/*dt* > 0, so we use the second equality in (55) to state the correct initial condi-

,*i* (0) *Ai*

,*i* (0) *Ai* (1−3*μ*)/2 }|*<sup>t</sup>*=<sup>0</sup>

) and *εδ*′

(*x*), *x* ≤ *x*(*i*,0)

<sup>0</sup> <sup>&</sup>gt; <sup>0</sup> and *x*(*i*, 0) are arbitrary numbers; and *i* = 1, 2. Note also that the first

<sup>2</sup> *<sup>ω</sup><sup>μ</sup>* <sup>+</sup> *<sup>d</sup>*\_\_\_\_ <sup>2</sup> *<sup>ω</sup>*

Equation (54) forms the closed system to define *Ai*

. (55)

143

(*t*). To simplify it, let us use the

http://dx.doi.org/10.5772/intechopen.71030

′ = (*μ* − 1) *aμ*+<sup>1</sup> /4. (57)

, *t* > 0, (58)

, (59)

(1−3*μ*)/2(*t*), *t* > 0, (61)

, (62)

. (63)

) yield (29), (30) again.

with the properties of

(*t*), *t* > 0, *i* = 1, 2. (60)

(*x*) is an arbitrary smooth function, which satisfies the

≥ 0 for *t* ≥ 0. (64)

(*x* − *x*\*

, *ϕ<sup>i</sup>*<sup>1</sup>

*<sup>d</sup> <sup>η</sup>*<sup>2</sup> } <sup>=</sup> <sup>0</sup>. (56)

, add the initial conditions, and obtain from (54) the

(*t*) and *ϕ<sup>i</sup>*<sup>0</sup>

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

with exponential rates. We assume also the intersection of the trajectories *x* = *ϕ<sup>i</sup>*<sup>0</sup> (*t*), *i* = 1, 2 at a point *x*\*  = *ϕ<sup>i</sup>*<sup>0</sup> (*t*\* ) namely,

$$\exists t^\* > 0 \text{ such that } \varphi\_{\mathbf{n}}(t^\*) = \varphi\_{\mathbf{n}}(t^\*), \psi\_{\mathbf{o}} \left. \frac{d\mathbb{M}}{dt} \{\varphi\_{\mathbf{n}}(t) - \varphi\_{\mathbf{n}}(t)\} \right|\_{t=t^\*} \neq 0. \tag{50}$$

It is easy to verify the weak representations with the precision *O*' (*<sup>ε</sup>* <sup>2</sup>):

$$\mu = \varepsilon \sum\_{\mu=1}^{2} \{a\_{\text{i}} \, K\_{\text{i}0}^{(1)} \, \delta(\mathbf{x} - \boldsymbol{\varphi}\_{\text{i}}) + a\_{\text{i}} \, K\_{\text{i}1}^{(1)\*} \, \delta(\mathbf{x} - \mathbf{x}^\*) + \mathbf{z}\_{\text{i}}(\mathbf{x}, t) \mathbf{H} \{\boldsymbol{\varphi}\_{\text{i}} - \mathbf{x} \} \}, \tag{51}$$

\_\_\_ ∂*u* <sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *<sup>a</sup>*<sup>1</sup> *<sup>ψ</sup>*̇ 0 \_\_∂ <sup>∂</sup>*<sup>τ</sup>* ∑ *i*=1 2 *Ki*1 (1)\* *δ*(*x* − *x*\*) + *ε* ∑ *i*=1 <sup>2</sup> ∂*z* \_\_\_*<sup>i</sup>* <sup>∂</sup>*<sup>t</sup> <sup>H</sup>*(*ϕ<sup>i</sup>* <sup>−</sup> *<sup>x</sup>*) +*ε a*<sup>1</sup> ∑ *i*=1 2 { *dKi*<sup>1</sup> (1) \_\_\_\_ *dt* <sup>+</sup> *zi* (*ϕi* , *t*) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* }*δ*(*<sup>x</sup>* <sup>−</sup> *<sup>ϕ</sup><sup>i</sup>* ) − *ε a*<sup>1</sup> ∑ *i*=1 2 *Ki*0 (1) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt <sup>δ</sup>*′ (*x* − *ϕ<sup>i</sup>* ) +*<sup>ε</sup>* <sup>∑</sup> *i*=1 2 {*ψ*̇ 0 \_\_∂ <sup>∂</sup>*t*(*a*<sup>1</sup> \_\_ *i βi* + *zi* (*x*\* , *<sup>t</sup>*) *<sup>ϕ</sup><sup>i</sup>*1) <sup>−</sup> *<sup>a</sup>* \_\_<sup>1</sup> *βi* 2 *Si d β* \_\_\_*<sup>i</sup> dt* }|*<sup>t</sup>*=*t*\* *δ*(*x* − *x*\*) −*ε a*<sup>1</sup> ∑ *i*=1 2 {*Ki*<sup>1</sup> (1) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* <sup>+</sup> *<sup>ψ</sup>*̇ <sup>0</sup>(*Ki* (1) \_\_\_\_ ∂*ϕ<sup>i</sup>*<sup>1</sup> <sup>∂</sup>*<sup>τ</sup>* <sup>+</sup> *χ*\_\_*i βi* ∂*Ki* (1) \_\_\_\_ <sup>∂</sup>*<sup>τ</sup>* )}|*<sup>t</sup>*=*t*\* *δ*′ (*x* − *x*\*), (52)

where *H*(*x*) is the Heaviside function, *H*(*x*) = 0 for *x* < 0 and *H*(*x*) = 1 for *x* > 0; *g*\* = def *g*(*τ*, *t*) |*<sup>t</sup>*=*<sup>t</sup>* \* , *χ<sup>i</sup>* <sup>=</sup> def *ϕi* (*Φ*(, *t* \* ), *τ*, *ε*) − *x*\* , and Φ(*ετ*, *t*\* ) is the solution of the equation *ϕ*20(*t*\*  + Φ) − *ϕ*10(*t* \*  + Φ) = *ετ*, which exists in accordance with (50).

Next, the existence of nonsoliton summands in (51) implying a correction of formula (22), namely

$$\begin{aligned} F(\boldsymbol{\mu}, \boldsymbol{\varepsilon}\boldsymbol{u}\_{\boldsymbol{\varepsilon}}) &= \boldsymbol{\varepsilon} \sum\_{i=1}^{2} a\_{\boldsymbol{\varepsilon},i}^{(0)} \boldsymbol{\beta}\_{i}^{-1} \delta(\boldsymbol{\chi} - \boldsymbol{\varphi}\_{i}) + \boldsymbol{\varepsilon} \boldsymbol{\rho}\_{2}^{-1} \boldsymbol{\Re} \boldsymbol{\xi}\_{\boldsymbol{\varepsilon}}^{(0)} \delta(\boldsymbol{\chi} - \boldsymbol{\chi}^{\*}) \\ &+ \boldsymbol{\varepsilon} \, F\_{\boldsymbol{u}}(\boldsymbol{0}, \boldsymbol{0}) \sum\_{i=1}^{2} z\_{i}(\boldsymbol{\chi}, t) H(\boldsymbol{\varphi}\_{i} - \boldsymbol{\chi}) + O\_{\boldsymbol{\varpi}}(\boldsymbol{\varepsilon}^{\neg}), \end{aligned} \tag{53}$$

where *aF*,*<sup>i</sup>* (0) and ℜ*<sup>F</sup>* (0) are defined in (23), (24), *Fu* ′ (0, 0) = ∂*F*(*u*, 0)/∂*u* | *<sup>u</sup>*=<sup>0</sup> .

Repeating the same calculations as above, we obtain linear combinations of *εδ*(*x* − *ϕ<sup>i</sup>* ), *εδ*′ (*x* − *ϕ<sup>i</sup>* ), *εH*(*ϕ<sup>i</sup>*  − *x*), *i* = 1, 2; *δ*(*x* − *x*\* ), *εδ*(*x* − *x*\* ), and *εδ*′ (*x* − *x*\* ). Equating zero, the coefficients of *εδ*(*x* − *ϕ<sup>i</sup>* ) and *εδ*′ (*x* − *ϕ<sup>i</sup>* ) yield

$$a\_1 A\_i \frac{d \, q\_{i0}}{dt} = \, a\_{p\_i \ell}^{(0)} \quad a\_2 \frac{d}{dt} \frac{A\_i^2}{\beta\_i} = \frac{a\_{k\_i \ell}^{(0)}}{\beta\_i \prime} \tag{54}$$

$$a\_2 A\_i^2 \frac{d\,\varphi\_{\rm{d}}}{dt} = a\_{p\_{r'}}^{\rm{q}0} \quad a\_1 \frac{d}{dt} \frac{A\_i}{\beta\_i} + z\_i \{\varphi\_{r'} t\} \frac{d\,\varphi\_{\rm{d}}}{dt} = \frac{a\_{j\bar{j}}^{\rm{q}0}}{\beta\_i}.\tag{55}$$

Equation (54) forms the closed system to define *Ai* (*t*) and *ϕ<sup>i</sup>*<sup>0</sup> (*t*). To simplify it, let us use the equalities (4) and rewrite the model equation for *ω*(*η*) as follows:

$$\frac{d}{d\eta} \Big| -\omega + \frac{\mu + 1}{2} \omega^{\mu} + \frac{d^2 \omega}{d|\eta^2|} \Big|\_{} = \ 0. \tag{56}$$

Simple manipulations with (56) allow us to find relations between structural constants:

Here *Ai*

such that

point *x*\*

*ϕi* (*Φ*(, *t* \* ), *τ*, *ε*) − *x*\*

namely

where *aF*,*<sup>i</sup>* (0) and ℜ*<sup>F</sup>* (0)

*εδ*(*x* − *ϕ<sup>i</sup>*

), *εH*(*ϕ<sup>i</sup>*

) and *εδ*′

*εδ*′ (*x* − *ϕ<sup>i</sup>*

*<sup>i</sup>*

 = *ϕ<sup>i</sup>*<sup>0</sup> (*t*\*

*u* = *ε* ∑

\_\_\_ ∂*u* <sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *<sup>a</sup>*<sup>1</sup> *<sup>ψ</sup>*̇

exists in accordance with (50).

*ψ*0

(*t*), *ϕ<sup>i</sup>*<sup>0</sup>

 = *ϕ<sup>i</sup>*<sup>0</sup> (*t*), *β<sup>i</sup>* 2

) namely,

∃*t*\* > 0 such that *ϕ*10(*t*\*) = *ϕ*20(*t*\*), *ψ*̇

*i*=1 2 {*a*<sup>1</sup> *Ki*<sup>0</sup> (1)

0 \_\_∂ <sup>∂</sup>*<sup>τ</sup>* ∑ *i*=1 2 *Ki*1

*i*=1 2 {*ψ*̇ 0 \_\_∂ <sup>∂</sup>*t*(*a*<sup>1</sup> \_\_ *i βi* + *zi* (*x*\*

*i*=1 2 { *dKi*<sup>1</sup> (1) \_\_\_\_ *dt* <sup>+</sup> *zi* (*ϕi* , *t*) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup>

*i*=1 2 {*Ki*<sup>1</sup> (1) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* <sup>+</sup> *<sup>ψ</sup>*̇

*F*(*u*, *εux*) = *ε* ∑

+ *ε Fu*

are defined in (23), (24), *Fu*

 − *x*), *i* = 1, 2; *δ*(*x* − *x*\*

) yield

(*x* − *ϕ<sup>i</sup>*

*a*<sup>1</sup> *Ai*

*i*=1 2 *aF*,*i* (0) *βi*

> ′ (0, 0) ∑ *i*=1 2 *zi*

+*ε a*<sup>1</sup> ∑

−*ε a*<sup>1</sup> ∑

, and Φ(*ετ*, *t*\*

(*t*) = *γ A <sup>μ</sup>*−1

142 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

(*τ*) → 0 *as τ* → −∞ , *<sup>i</sup>*

It is easy to verify the weak representations with the precision *O*'

*δ*(*x* − *ϕ<sup>i</sup>*

(1)\* *δ*(*x* − *x*\*) + *ε* ∑

<sup>0</sup>(*Ki*

where *H*(*x*) is the Heaviside function, *H*(*x*) = 0 for *x* < 0 and *H*(*x*) = 1 for *x* > 0; *g*\* =

) + *a*<sup>1</sup> *Ki*<sup>1</sup>

*i*=1 <sup>2</sup> ∂*z* \_\_\_*<sup>i</sup>*

*dt* }*δ*(*<sup>x</sup>* <sup>−</sup> *<sup>ϕ</sup><sup>i</sup>*

, *<sup>t</sup>*) *<sup>ϕ</sup><sup>i</sup>*1) <sup>−</sup> *<sup>a</sup>*

(1) \_\_\_\_ ∂*ϕ<sup>i</sup>*<sup>1</sup> <sup>∂</sup>*<sup>τ</sup>* <sup>+</sup> *χ*\_\_*i βi* ∂*Ki* (1) \_\_\_\_ <sup>∂</sup>*<sup>τ</sup>* )}|*<sup>t</sup>*=*t*\*

) is the solution of the equation *ϕ*20(*t*\*

Next, the existence of nonsoliton summands in (51) implying a correction of formula (22),

−1 *δ*(*x* − *ϕ<sup>i</sup>*

′

), *εδ*(*x* − *x*\*

\_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt* <sup>=</sup> *aP*<sup>1</sup> ,*i* (0) , *a*<sup>2</sup> \_\_*d dt Ai* 2 \_\_\_ *βi*

Repeating the same calculations as above, we obtain linear combinations of *εδ*(*x* − *ϕ<sup>i</sup>*

), and *εδ*′

) + <sup>2</sup> −1 ℜ*<sup>F</sup>* (0)

(*x*, *t*)*H*(*ϕ<sup>i</sup>* − *x*) + *O*'

(*x* − *x*\*

<sup>=</sup> *aK*<sup>2</sup> ,*i* (0) \_\_\_ *βi*


(0, 0) = ∂*F*(*u*, 0)/∂*u*

+*<sup>ε</sup>* <sup>∑</sup>

\_\_<sup>1</sup> *βi* 2 *Si d β* \_\_\_*<sup>i</sup> dt* }|*<sup>t</sup>*=*t*\*

(*t*) = *ϕ*20(*t*) − *ϕ*10(*t*) denotes the "fast time" again; *<sup>z</sup> <sup>i</sup>*

(*t*), *ω*(*η*), *Si*

with exponential rates. We assume also the intersection of the trajectories *x* = *ϕ<sup>i</sup>*<sup>0</sup>

(*τ*), *ϕ<sup>i</sup>*<sup>1</sup>

ℌ(*η*) → 1 *as η* → −∞ , ℌ(*η*) → 0 *as η* → +∞ (49)

(*τ*) → *<sup>i</sup>*

0 = def \_\_*d*

(1)\* *δ*(*x* − *x*\*) + *zi*

<sup>∂</sup>*<sup>t</sup> <sup>H</sup>*(*ϕ<sup>i</sup>* <sup>−</sup> *<sup>x</sup>*)

(*x*, *<sup>t</sup>*) <sup>∈</sup> <sup>∞</sup>

(*τ*) are the same as in (13); *τ* = *ψ*<sup>0</sup>

<sup>∞</sup> = *consti as τ* → +∞ , (48)


(*x*, *t*)*H*(*ϕ<sup>i</sup>* − *x*)}, (51)

(*x* − *ϕ<sup>i</sup>* )

> def *g*(*τ*, *t*) |*<sup>t</sup>*=*<sup>t</sup>* \* , *χ<sup>i</sup>* <sup>=</sup> def

(*ε*¯¯*<sup>q</sup>*), (53)

). Equating zero, the coefficients of

, (54)

 + Φ) = *ετ*, which

*δ*(*x* − *x*\*)

 + Φ) − *ϕ*10(*t*\*

*δ*′ (*x* − *x*\*),

*δ*(*x* − *x*\*)

; and *<sup>i</sup>*

*dt*(*ϕ*20(*t*) <sup>−</sup> *<sup>ϕ</sup>*10(*t*))

) − *ε a*<sup>1</sup> ∑ *i*=1 2 *Ki*0 (1) \_\_\_\_ *d ϕ<sup>i</sup>*<sup>0</sup> *dt <sup>δ</sup>*′

(*<sup>ε</sup>* <sup>2</sup>):

(*t*)/*ε* with

(*t*), *i* = 1, 2 at a

(52)

),

≠ 0. (50)

, ℌ are smooth functions

$$a\_1 = \left(\mu + 1\right) a\_{\mu}/2, \quad a\_2 = \left(\mu + 3\right) a\_{\mu + 1}/4, \quad a\_2^{'} = \left(\mu - 1\right) a\_{\mu + 1}/4. \tag{57}$$

Next, we use (57), the equality *β<sup>i</sup>* <sup>2</sup> <sup>=</sup> *<sup>γ</sup> Ai μ*−1 , add the initial conditions, and obtain from (54) the Cauchy problem

$$\frac{dA\_i}{dt} = -c\_1 A\_{k\_r l}^{00} A\_l^{-1}, \quad \frac{d\varphi\_{i0}}{dt} = \frac{a\_p}{a\_l} A\_l^{\mu - 1}, \quad t \ge 0,\tag{58}$$

$$\left.A\_{\right|}\right|\_{\nu=0} = \left.A\_{\right|\_{\nu}}^{0} \quad \left.\varphi\_{i0}\right|\_{\nu=0} = \left.\chi\_{(0)'}\tag{59}$$

where *c*<sup>1</sup>  = 2/(*a*<sup>2</sup> (5 − *μ*)); *Ai* <sup>0</sup> <sup>&</sup>gt; <sup>0</sup> and *x*(*i*, 0) are arbitrary numbers; and *i* = 1, 2. Note also that the first equalities in equations (54) and (55) are equivalent.

Next, equating zero the coefficients of the Heaviside functions, we obtain the equations

$$\frac{\partial z\_i}{\partial t} = f\_i(0,0)z\_f \quad \text{x} < \varphi\_{i0}(t), \quad t \ge 0, \quad i = 1,2. \tag{60}$$

In view of (58) *dϕ<sup>i</sup>*<sup>0</sup> /*dt* > 0, so we use the second equality in (55) to state the correct initial condition for (60)

$$\left. \Xi\_{\uparrow} (\mathbf{x}, t) \right|\_{x = \varphi\_a(t)} = \sqrt{\gamma} \left. a\_{j,l}^{(0)} A\_{\rceil}^{(3-4\mu)2} (t) + c\_2 \left. a\_{\mathbf{k}\_j}^{(0)} A\_{\rceil}^{(1-3\mu)2} (t) \right|\_{t} \quad t > 0,\tag{61}$$

$$\left. z\_{\uparrow}(\mathbf{x}, t) \right|\_{t=0} = \left. z\_{\downarrow}^{0}(\mathbf{x}), \quad \mathbf{x} \le \mathbf{x}\_{\downarrow (t)'} \tag{62}$$

where *c*<sup>2</sup>  = *a*<sup>1</sup> (3 − *μ*)(1 + *μ*)/(2*a*<sup>2</sup> (5 − *μ*)), *zi* 0 (*x*) is an arbitrary smooth function, which satisfies the consistency condition

$$\mathbf{z}\_{l}^{0}\{\mathbf{x}\_{(\downarrow0)}\} = \left\{\sqrt{\mathbf{y}}\ \mathbf{a}\_{f\downarrow}^{00}\ \mathbf{A}\_{l}^{(3-4\mu)2} + \mathbf{c}\_{2}\ \mathbf{a}\_{\mathbf{x}\_{f}l}^{(0)}\mathbf{A}\_{l}^{(1-3\mu)2}\right\}\Big|\_{t=0}.\tag{63}$$

We should note that the nonlinearity *uμ* in (1) can require the inequality *u* ≥ 0. To this end, we will assume

$$A\_{\restriction}(t) \ge 0, \quad \mathbf{z}\_{\restriction}(\mathbf{x}, t)\Big|\_{\mathbf{x}=\mathbf{e}\_{a}(0)} \ge 0 \quad \text{for} \quad t \ge 0. \tag{64}$$

Furthermore, equating zero the coefficients of *δ*(*x* − *x*\* ) and *εδ*′ (*x* − *x*\* ) yield (29), (30) again. Consequently, the condition *θ* ≤ *θ*\* guaranties the existence of *Si* , *ϕ<sup>i</sup>*<sup>1</sup> with the properties of (14), (15). In particular

$$S\_1 = -\theta \, S\_2 \quad S\_1 = \theta \theta\_1^\vee \, \lambda(\sigma) (1 + O(\theta \theta)),\tag{65}$$

where

$$q = \min\{1, \gamma\}, \quad \lambda(\sigma) = a\_2^{-1} \int\_{-\alpha}^{\omega} \omega(\eta)\omega(\eta\_{12})d\eta. \tag{66}$$

The last step of the construction is the determination of *<sup>i</sup>* (*τ*), *i* = 1, 2. By setting the coefficients of *εδ*(*x* − *x*\* ) zero, we obtain

$$\frac{\partial}{\partial \tau} \sum\_{i=1}^{2} \frac{\partial \mathbf{g}\_{i}}{\beta\_{i}^{\prime}} = \sf{\sf ff}\_{\mathbf{r}^{\prime}} \quad \left. \frac{\partial}{\partial \tau} \sum\_{i=1}^{2} \left( \frac{G\_{i}}{\beta\_{i}} + \frac{G\_{\gamma}}{\beta\_{2}} \lambda(\sigma) \right) \right|\_{\mathbf{r} = \mathbf{r}^{\prime}} \sf{\sf ff}\_{\mathbf{r}} = \sf{\sf ff}\_{\mathbf{r}^{\prime}} \tag{67}$$

where ¯¯*i* = 2 for *i* = 1 and ¯¯*i* = 1 for *i* = 2,

$$\mathfrak{H}\_1 = -\sum\_{i=1}^{2} \left| \frac{\varepsilon\_i(\mathbf{x}^\*, t)}{a\_i} \frac{d\,\varphi\_{\boldsymbol{\eta}}}{d\boldsymbol{\tau}} + \psi\_0^{-1} \frac{\partial}{\partial t} \frac{S\_i}{\beta\_i} \right| \Big|\_{\boldsymbol{\tau} = \boldsymbol{\zeta}^\*} + \frac{1}{\beta\_2} \mathfrak{R}\_{\boldsymbol{\gamma}'}^{(0)} \Big|\_{\boldsymbol{\tau} = \boldsymbol{\zeta}^\*} \tag{68}$$

$$\mathfrak{F}\_{2} = -\frac{\partial}{\partial \mathbf{r}} \sum\_{i=1}^{2} \left| a\_{H} \, z\_{i}(\mathbf{x}^{\*}, t) \frac{S\_{i}}{\beta\_{i}} + \mathbf{z}\_{i}(\mathbf{x}^{\*}, t) \frac{G\_{i}}{\beta\_{2}} \, \lambda\_{\mathrm{H},i}(\sigma) \right| \begin{vmatrix} \Re\_{k\_{i}}^{\mathrm{u}0} \\ \mu\_{i} \end{vmatrix}\_{\mathrm{\tinystackrel{\tiny{\tiny}{\mathrm{\tiny$$

Calculating the determinant *Δ* of the matrix in the left-hand part of (67) and using (65), we conclude

$$\Delta = \left\{ G\_2 - G\_1 + \lambda \{ G\_1 - \theta \ G\_2 \} \right\}|\_{\mathbb{M}^\*} = \left. \beta\_2^v (1 - \Theta^\circ - \lambda \{ \Theta - \Theta^\circ \} - \lambda^2 \left. \Theta^\circ (1 + O(\Theta^\circ)) \right\} \right|\_{\mathbb{M}^\*}.\tag{70}$$

Obviously, Δ ≠ 0 for sufficiently small *θ*. Since the right-hand sides *<sup>i</sup>* belong to the Schwartz space, the functions *<sup>i</sup>* exist and satisfy the assumption (48).

Henceforth, we pass to the final result:

**Theorem 4.** *Let θ be sufficiently small and let the assumptions (50), (63), and (64), if it is necessary, be fulfilled. Then, the self-similar two-wave weak asymptotic mod <sup>O</sup>*(*ε*¯¯*<sup>q</sup>*) *solution of the equation (1) exists and has the form (46).*

Let us finally stress that the self-similarity implies a special choice of the initial data: for the classical asymptotics in the *C*-sense, there appears a very restrictive condition for small correction of the soliton *A*(0)*ω*((*x* − *x*<sup>0</sup> )/*ε*) (see [6, 17]), and for weak asymptotics, there appears the restriction (63). If it is violated, then the perturbed soliton generates a rapidly oscillating tail of the amplitude *o*(1) ("radiation") instead of the smooth tails *εu*<sup>−</sup> (*x*, *t*) (see [21] and numerical results [14, 15, 17]). Nowadays, this radiation phenomenon can be described analytically only for integrable equations, so that we should use self-similar approximation for essentially nonintegrable equations. However, the smooth tail *εu*<sup>−</sup> (*x*, *t*), which can be treated

as an average of the radiation, describes sufficiently well the tendency of the radiation amplitude behavior, see graphics depicted in **Figures 3** and **4**, and other numerical results

t = 1.0

u

t = 2.0

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

10

http://dx.doi.org/10.5772/intechopen.71030

10

x

x

145


t


**Figure 3.** Example of noninteracting solitary waves, *μ* = 4, *f* = *u*(1 − *u*).

t

**Figure 4.** Example of interacting solitary waves, *μ* = 4, *f* = *u*(1 − *u*), *ε* = 0.1.

in [15, 17]).

A Perturbation Theory for Nonintegrable Equations with Small Dispersion http://dx.doi.org/10.5772/intechopen.71030 145

**Figure 3.** Example of noninteracting solitary waves, *μ* = 4, *f* = *u*(1 − *u*).

*S*<sup>1</sup> = −*θ S*<sup>2</sup> *S*<sup>1</sup> = <sup>1</sup>

*q* = *min*{1, *γ*}, *λ*(*σ*) = *a*<sup>2</sup>

<sup>∂</sup>*<sup>τ</sup>* ∑ *i*=1 <sup>2</sup> \_\_\_ *i βi* \* = <sup>1</sup>

) zero, we obtain

where ¯¯*i* = 2 for *i* = 1 and ¯¯*i* = 1 for *i* = 2,

<sup>1</sup> = −∑

<sup>2</sup> <sup>=</sup> <sup>−</sup>\_\_<sup>∂</sup>

space, the functions *<sup>i</sup>*

*and has the form (46).*

\_\_<sup>∂</sup>

The last step of the construction is the determination of *<sup>i</sup>*

144 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

*i*=1 2 { *zi* (*x*\* , *<sup>t</sup>*) \_\_\_\_\_\_ *<sup>a</sup>*<sup>1</sup>

Obviously, Δ ≠ 0 for sufficiently small *θ*. Since the right-hand sides *<sup>i</sup>*

*fulfilled. Then, the self-similar two-wave weak asymptotic mod O*(*ε*¯¯

exist and satisfy the assumption (48).

<sup>∂</sup>*<sup>τ</sup>* ∑ *i*=1 2 {*aH zi* (*x*\* , *t*) *S* \_\_*i βi* + *zi* ¯¯(*x*\* , *t*) \_\_ *Gi β*2 *λ<sup>H</sup>*,*<sup>i</sup>* (*σ*) } <sup>|</sup>*<sup>t</sup>*=*t*\* <sup>+</sup>

*aH* = *a*<sup>1</sup> /(2 *a*2), *cH* = 2 *a*<sup>2</sup> *ψ*̇

Δ = (*G*<sup>2</sup> − *G*<sup>1</sup> + *λ*(*G*<sup>1</sup> − *θ G*2)) |*<sup>t</sup>*=*t*\* = *β*<sup>2</sup>

Henceforth, we pass to the final result:

rection of the soliton *A*(0)*ω*((*x* − *x*<sup>0</sup>

,  \_\_<sup>∂</sup> <sup>∂</sup>*<sup>τ</sup>* ∑ *i*=1 2 (\_\_ *Gi βi* + *Gi* \_\_¯¯ *β*2 *λ*(*σ*) )|*<sup>t</sup>*=*t*\*

> \_\_\_\_ *d ϕ<sup>i</sup>*<sup>1</sup> *<sup>d</sup>* <sup>+</sup> *<sup>ψ</sup>*̇ 0 −1 \_\_∂ ∂*t S* \_\_*i βi*

> > <sup>0</sup> *β*<sup>2</sup> , *λ<sup>H</sup>*,*<sup>i</sup>*

Calculating the determinant *Δ* of the matrix in the left-hand part of (67) and using (65), we

**Theorem 4.** *Let θ be sufficiently small and let the assumptions (50), (63), and (64), if it is necessary, be* 

Let us finally stress that the self-similarity implies a special choice of the initial data: for the classical asymptotics in the *C*-sense, there appears a very restrictive condition for small cor-

the restriction (63). If it is violated, then the perturbed soliton generates a rapidly oscillat-

numerical results [14, 15, 17]). Nowadays, this radiation phenomenon can be described analytically only for integrable equations, so that we should use self-similar approximation for

ing tail of the amplitude *o*(1) ("radiation") instead of the smooth tails *εu*<sup>−</sup>

essentially nonintegrable equations. However, the smooth tail *εu*<sup>−</sup>

where

of *εδ*(*x* − *x*\*

conclude

*<sup>γ</sup> λ*(*σ*)(1 + *O*(*θ<sup>q</sup>*)), (65)

<sup>∞</sup> *<sup>ω</sup>*(*η*)*ω*(*η*12)*d*. (66)

*<sup>i</sup>* = <sup>2</sup>

ℜ*<sup>K</sup>*<sup>2</sup> (0) \_\_\_ *cH* |*<sup>t</sup>*=*t*\*

<sup>∞</sup> *ω*(*η<sup>i</sup>*2)ℌ(*η<sup>i</sup>*

} <sup>|</sup>*<sup>t</sup>*=*t*\* <sup>+</sup> \_\_<sup>1</sup> *β*2 ℜ*f* (0) |*<sup>t</sup>*=*t*\*

(*σ*) = *a*<sup>2</sup>

−1 ∫ −∞

*<sup>γ</sup>*(1 − *θγ* − *λ*(*θ* − *θγ*) − *λ*<sup>2</sup> *θγ*(1 + *O*(*θ<sup>q</sup>*)))

)/*ε*) (see [6, 17]), and for weak asymptotics, there appears

(*τ*), *i* = 1, 2. By setting the coefficients

, (67)

, (68)

¯¯2)*d*.

, (69)


belong to the Schwartz

(*x*, *t*) (see [21] and

(*x*, *t*), which can be treated

*<sup>q</sup>*) *solution of the equation (1) exists* 

. (70)

−1 ∫ −∞

**Figure 4.** Example of interacting solitary waves, *μ* = 4, *f* = *u*(1 − *u*), *ε* = 0.1.

as an average of the radiation, describes sufficiently well the tendency of the radiation amplitude behavior, see graphics depicted in **Figures 3** and **4**, and other numerical results in [15, 17]).

## **Acknowledgements**

The research was supported by SEP-CONACYT under grant no. 178690 (Mexico).

[10] Abreu E, Colombeau M, Panov E. Weak asymptotic methods for scalar equations and systems. Journal of Mathematical Analysis and Applications. 2016;**444**(2):1203-1232.

A Perturbation Theory for Nonintegrable Equations with Small Dispersion

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147

[11] Danilov V, Mitrovic D. Shock wave formation process for a multidimensional scalar conservation law. Quarterly of Applied Mathematics. 2011;**69**(4):613-634. Available from:

[12] Danilov V, Omel'yanov G, Shelkovich V. Weak asymptotics method and interaction of nonlinear waves. In: Karasev M, editor. Asymptotic Methods for Wave and Quantum Problems. Trans. Ser. 2. Vol. 208. Providence, RI: American Mathematical Society; 2003.

[13] Danilov V, Shelkovich V. Dynamics of propagation and interaction of delta-shock waves in conservation law systems. Journal of Differential Equations. 2005;**211**(2):333-381.

[14] Garcia M, Omel'yanov G. Interaction of solitary waves for the generalized KdV equation. Communications in Nonlinear Science and Numerical Simulation. 2012;**17**(8):3204-3218.

[15] Garcia M, Omel'yanov G. Interaction of solitons and the effect of radiation for the generalized KdV equation. Communications in Nonlinear Science and Numerical Simulation.

[16] Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2 × 2 system of conservation laws. Proceedings of the Edinburgh Mathematical Society II. 2012;**55**:711-729. DOI:

Nonlinear Phenomena in Complex Systems. 2014;**17**(2):106-115. Available from: http://

[18] Omel'yanov G. Soliton-type asymptotics for non-integrable equations: A survey. Mathematical Methods in The Applied Sciences. 2015;**38**(10):2062-2071. DOI: 10.1002/

[19] Omel'yanov G. Propagation and interaction of solitons for nonintegrable equations. Russian Journal of Mathematical Physics. 2016;**23**(2):225-243. DOI: 10.1134/S10619

[20] Omel'yanov G. Multi-soliton Collision for essentially nonintegrable equations. In: Oberguggenberger M et al., editors. Generalized Functions and Fourier Analysis, Series: Operator Theory: Advances and Applications. Birkhäuser, Cham. 2017;**260**:153-170. DOI:

[21] Kalyakin L. Perturbation of the Korteweg-de Vries soliton. Theoretical and Mathematical

Physics. 1992;**92**:736-747. Available from: https://goo.gl/aPPhqG


2014;**19**(8):2724-2733. Available from: https://doi.org/10.1016/j.cnsns.2013.06.024

Available from: https://doi.org/10.1016/j.jmaa.2016.06.047

doi: 10.1090/S0033-569X-2011-01234-9

Available from: https://goo.gl/Y6g4vv

https://doi.org/10.1017/S0013091512000065

https//doi.org/10.1007/978-3-319-51911-1\_9

elib.bsu.by/handle/123456789/116849

mma.3198 2014

20816020084

[17] Omel'yanov G, Valdez-Grijalva M. Asymptotics for a C1

p. 33-164. Available from: https://goo.gl/aPPhqG

Available from: https://doi.org/10.1016/j.cnsns.2011.12.001

### **Author details**

Georgy Omel'yanov

Address all correspondence to: omel@mat.uson.mx

University of Sonora, Hermosillo, Mexico

### **References**


[10] Abreu E, Colombeau M, Panov E. Weak asymptotic methods for scalar equations and systems. Journal of Mathematical Analysis and Applications. 2016;**444**(2):1203-1232. Available from: https://doi.org/10.1016/j.jmaa.2016.06.047

**Acknowledgements**

**Author details**

Georgy Omel'yanov

**References**

Address all correspondence to: omel@mat.uson.mx

146 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

springer.com/article/10.1007/BF01097030

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University of Sonora, Hermosillo, Mexico

10.1098/rspa.1987.0073

The research was supported by SEP-CONACYT under grant no. 178690 (Mexico).

[1] Bona J, Souganidis P, Strauss W. Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society of London A. 1987;**411**(1841):395-412. DOI:

[2] Faminskii A. Cauchy problem for the Korteweg-de Vries equation and its generalizations. Journal of Soviet Mathematics. 1990;**50**(1):1381-1420. Available from: https://link.

[3] Ablowitz M, Segur H. Solitons and the Inverse Scattering Transform. Philadelphia:

[4] Karpman V, Maslov E. Perturbation theory for solitons. Journal of Experimental and Theoretical Physics. 1977;**46**(2):281-291. Available from: http://jetp.ac.ru/cgi-bin/dn [5] Kaup D, Newell A. Solitons as particles, oscillators and in slowly changing media: A singular perturbation theory. Proceedings of the Royal Society of London Series A.

[6] Maslov V, Omel'yanov G. Asymptotic soliton-form solutions of equations with small dispersion. Russian Mathematical Surveys. 1981;**36**:73-149. Available from: http://iop-

[7] Gorshkov K, Ostrovsky L. Interaction of solitons in nonintegrable systems: Direct perturbation method and applications. Physica D. 1981;**3**:428-438. Available from: https://

[8] Danilov V, Shelkovich V. Generalized solutions of nonlinear differential equations and the Maslov algebras of distributions. Integral Transformations and Special Functions.

[9] Danilov V, Omel'yanov G. Weak asymptotics method and the interaction of infinitely narrow delta-solitons. Nonlinear Analysis: Theory Methods & Applications. 2003;**54**:773-799

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SIAM; 1981. Available from: http://dx.doi.org/10.1137/1.9781611970883.bm


**Chapter 9**

Provisional chapter

**Weakly Nonlinear Stability Analysis of a Nanofluid in a**

DOI: 10.5772/intechopen.71066

**Horizontal Porous Layer Using a Multidomain Spectral**

Horizontal Porous Layer Using a Multidomain Spectral

In this chapter, we present a weakly nonlinear stability analysis of the flow of a nanofluid in a porous medium with stress-free boundary conditions. Some previous studies have investigated cross-diffusion in a nanofluid layer although in most cases these studies mostly deal with linear stability analysis. It is important to study the nonlinear stability in flows subject to cross-diffusion due to the wide range of applications where such flows arise such as in hydrothermal growth, compact heat exchanges, the solidification of binary mixtures, geophysical systems, solar pond, etc. Here we consider flow between parallel plates with an applied magnetic field and zero nanoparticle flux at the boundaries. A truncated Fourier series is introduced reducing the flow equations to a Lorenz-type system of nonlinear evolution equations. The multidomain spectral method is used to solve the equations that describe the growth of the convection amplitudes. The solutions are obtained as sets of trajectories in the phase space. Some interesting spiral trajectories and

Keywords: nonlinear instability, nanofluid flow, porous medium, multidomain spectral

The enhancement of thermal conductivity of a fluid is a matter of supreme interest to engineers due to the important applications of fluids in heat transfer processes. Natural and forced convection plays an important role in heat transfer processes due to continuous molecular movements in fluid. Recent studies show that the suspension of solid nanoparticles in a fluid can substan-

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

tially improve the fluid's thermophysical properties, including thermal conductivity.

Weakly Nonlinear Stability Analysis of a Nanofluid in a

**Collocation Method**

Collocation Method

Sabyasachi Mondal

Sabyasachi Mondal

Abstract

collocation method

1. Introduction

Osman A.I. Noreldin, Precious Sibanda and

Osman A.I. Noreldin, Precious Sibanda and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

their sensitivity to the Rayleigh number are given.

http://dx.doi.org/10.5772/intechopen.71066

#### **Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral Collocation Method** Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral Collocation Method

DOI: 10.5772/intechopen.71066

Osman A.I. Noreldin, Precious Sibanda and Sabyasachi Mondal Osman A.I. Noreldin, Precious Sibanda and

Additional information is available at the end of the chapter Sabyasachi Mondal

http://dx.doi.org/10.5772/intechopen.71066 Additional information is available at the end of the chapter

#### Abstract

In this chapter, we present a weakly nonlinear stability analysis of the flow of a nanofluid in a porous medium with stress-free boundary conditions. Some previous studies have investigated cross-diffusion in a nanofluid layer although in most cases these studies mostly deal with linear stability analysis. It is important to study the nonlinear stability in flows subject to cross-diffusion due to the wide range of applications where such flows arise such as in hydrothermal growth, compact heat exchanges, the solidification of binary mixtures, geophysical systems, solar pond, etc. Here we consider flow between parallel plates with an applied magnetic field and zero nanoparticle flux at the boundaries. A truncated Fourier series is introduced reducing the flow equations to a Lorenz-type system of nonlinear evolution equations. The multidomain spectral method is used to solve the equations that describe the growth of the convection amplitudes. The solutions are obtained as sets of trajectories in the phase space. Some interesting spiral trajectories and their sensitivity to the Rayleigh number are given.

Keywords: nonlinear instability, nanofluid flow, porous medium, multidomain spectral collocation method

#### 1. Introduction

The enhancement of thermal conductivity of a fluid is a matter of supreme interest to engineers due to the important applications of fluids in heat transfer processes. Natural and forced convection plays an important role in heat transfer processes due to continuous molecular movements in fluid. Recent studies show that the suspension of solid nanoparticles in a fluid can substantially improve the fluid's thermophysical properties, including thermal conductivity.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

The term nanofluid describes a liquid containing a suspension of nanometer sized 1–100 nm solid particles [1]. Examples of commonly used nanoparticles include metallic particles such as Al, Cu and Ag, and oxides such as Al2O3 and CuO. The base fluid is often a common liquid such as water, ethylene, glycol, or oil. The enhancement of thermochemical properties of a fluid due to the addition of nanoparticles has been observed in experimental studies such as in [2, 3]. Researchers have investigated the influence of seven slip mechanisms, namely, inertia, Brownian diffusion, thermophoresis, diffusiophoresis, magnus effect, fluid drainage, and gravity in nanofluids. It has been shown that, in the absence of turbulence, the most significant among these mechanisms are the Brownian diffusion and thermophoresis.

2. Mathematical formulation

Da Pr ∂V

∂T

subject to the boundary conditions

Da <sup>¼</sup> <sup>μ</sup>~<sup>K</sup>

Rn ¼

<sup>μ</sup>h<sup>2</sup> , Pr <sup>¼</sup> <sup>μ</sup>

r<sup>p</sup> � r<sup>f</sup> � � <sup>ϕ</sup><sup>∗</sup>

r<sup>f</sup> α<sup>m</sup>

μα<sup>m</sup>

<sup>1</sup> � <sup>ϕ</sup><sup>∗</sup> 0 � �gKh

, Q <sup>¼</sup> <sup>δ</sup>B<sup>2</sup>

0K μ

, Rm <sup>¼</sup> <sup>r</sup>pϕ<sup>∗</sup>

<sup>∂</sup><sup>t</sup> ¼ �∇<sup>P</sup> <sup>þ</sup> Da∇<sup>2</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>V</sup> � <sup>∇</sup><sup>T</sup> <sup>¼</sup> <sup>∇</sup><sup>2</sup>

T þ NB

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>V</sup> � <sup>∇</sup><sup>C</sup> <sup>¼</sup> <sup>1</sup>

<sup>V</sup> � <sup>∇</sup><sup>ϕ</sup> <sup>¼</sup> <sup>1</sup>

∂ϕ ∂z þ NA

> ∂z þ NA

where V is the fluid velocity, T is the temperature, C is the solute concentration and ϕ is the volumetric fraction of nanoparticles. The dimensionless parameters are the Darcy number (modified by the viscosity ratio) Da, Prandtl number Pr, Hartmann-Darcy number Q, thermal Rayleigh-Darcy number Ra, nanoparticle Rayleigh number Rn and the basic density Rayleigh number Rm. The parameter NA is a modified diffusivity ratio, Le is the Lewis number, Rs is solutal Rayleigh number, NB is a modified nanoparticle density increment and Du is a modified Dufour parameter. The parameter Les is the thermo-nanofluid Lewis number, ν is the kinematic viscosity and Sr is a modified Soret parameter. These parameters have the form

∂C

V ¼ 0, T ¼ 1, C ¼ 1

<sup>V</sup> <sup>¼</sup> <sup>0</sup>, T <sup>¼</sup> <sup>0</sup>, C <sup>¼</sup> <sup>0</sup> <sup>∂</sup><sup>ϕ</sup>

1 σ ∂ϕ ∂t þ 1 ε Les <sup>∇</sup><sup>ϕ</sup> � <sup>∇</sup><sup>T</sup> <sup>þ</sup>

Le <sup>∇</sup><sup>2</sup>

Les <sup>∇</sup><sup>2</sup>

Consider viscous incompressible MHD nanofluid flow in an infinitely extended horizontal porous layer, confined between two boundaries at z ¼ 0 and z ¼ h. The layer is heated from below and cooled from above, see Figure 1. A Cartesian frame of reference is chosen in which the z-axis is vertically upward. The boundaries are perfectly conducting. The temperature at the lower and upper walls is Tc and Th, respectively with Th > Tc. The Oberbeck-Boussinesq approximation and the Darcy law are assumed to be applicable. The continuity equation, momentum equation, energy equation, concentration equation and volumetric fraction nanoparticle equation, which describe the above configuration in dimensionless form, are given as

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

∇ � V ¼ 0, (1)

C, (3)

T, (4)

http://dx.doi.org/10.5772/intechopen.71066

151

T, (5)

<sup>∂</sup><sup>z</sup> <sup>¼</sup> 0 at <sup>z</sup> <sup>¼</sup> <sup>0</sup>, (6)

<sup>∂</sup><sup>z</sup> <sup>¼</sup> 0 at <sup>z</sup> <sup>¼</sup> <sup>1</sup>, (7)

, Les <sup>¼</sup> <sup>α</sup><sup>m</sup>

, Le <sup>¼</sup> <sup>α</sup><sup>m</sup> DS

DB

, (8)

, (9)

<sup>V</sup> � <sup>V</sup> <sup>þ</sup> QVbez � Rmbez <sup>þ</sup> RaTbez <sup>þ</sup> RsCbez � Rnϕbez, (2)

Les <sup>∇</sup><sup>T</sup> � <sup>∇</sup><sup>T</sup> <sup>þ</sup> Du∇<sup>2</sup>

NANB

<sup>C</sup> <sup>þ</sup> Sr∇<sup>2</sup>

ϕ þ NA Les <sup>∇</sup><sup>2</sup>

∂T

∂T

, Ra <sup>¼</sup> <sup>r</sup><sup>f</sup> <sup>β</sup>Khg T<sup>∗</sup>

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>ϕ</sup><sup>∗</sup>

μα<sup>m</sup>

<sup>h</sup> � <sup>T</sup><sup>∗</sup> c � �

μα<sup>m</sup>

0 � �r<sup>f</sup> gKh

The classical Rayleigh-Benard convection problem in a heated horizontal layer has been extensively studied in the literature. Among recent studies on nanofluids, Tzou [4] studied the thermal instability and natural convection in nanofluid flow using an eigenfunction expansion method. Narayana et al. [5, 6] studied convection and the stability of a Maxwell fluid in a porous medium. Yadav et al. [7] investigated thermal instability of a rotating nanofluid layer. The studies by Kuznetsov and Nield [8–11] focused on thermal instability in a porous layer saturated with a nanofluid. They investigated the onset of instability in a horizontal porous layer using a model for the nanofluid that incorporated particle Brownian motion and thermophoresis. Related studies with various assumptions on the geometry and flow structure have been made by [12–15]. In the last few decades, researchers have also investigated thermal instability in a horizontal nanofluid layer subject to an applied magnetic field [16, 17]. The effects of a magnetic field on convection and the onset of instability have important applications in problems such as in cooling systems, pumps, magnetohydrodynamics and generators. The experimental study by Heris et al. [18] showed that thermal efficiency could be achieved by subjecting the flow to a magnetic field. The studies by Ghasemi et al. [19] and Hamad et al. [20] focused on the flow behavior and heat transfer in an electrically conducting nanofluid under the influence of a magnetic field and subject to Brownian diffusion and thermophoresis. They used a water-based nanofluid containing different types of nanoparticles such as copper, alumina and silver in their numerical simulations. Related studies of interest include [21–24]. Rana et al. [25] studied thermal convection in a Walters (Model B) fluid in a porous medium. They showed that a magnetic field may introduce oscillatory instability modes and acts to stabilize the system.

In this chapter, we give a weakly nonlinear stability analysis of a nanofluid layer with an applied magnetic field, stress free boundary conditions and under the assumption of zero nanoparticle flux at the boundary. The studies by Kuznetsov and Nield [9] and Nield and Kuznetsov [10, 11] investigated cross-diffusion in a nanofluid layer. However, these studies mostly presented a linear stability analysis. It is important to study the nonlinear regime for a nanofluid flow subject to cross-diffusion due to the wide range of applications where such flows may arise. Typical examples may be found in hydrothermal growth, compact heat exchanges, solidification of binary mixtures, geophysical systems, and so on. Hence, with this in mind, we studied the finite amplitude convection in a nanofluid flows subject to crossdiffusion. By introducing a truncated Fourier series, a Lorenz-type system of seven nonlinear differential equations is obtained. The recent multidomain spectral method is used to solve the nonlinear equations. This method is accurate and very easy to implement compared to older methods such as finite difference methods. An analysis of heat and mass transfer for different parameters such as the Prandtl number, the Dufour and thermophoresis is presented.

#### 2. Mathematical formulation

The term nanofluid describes a liquid containing a suspension of nanometer sized 1–100 nm solid particles [1]. Examples of commonly used nanoparticles include metallic particles such as Al, Cu and Ag, and oxides such as Al2O3 and CuO. The base fluid is often a common liquid such as water, ethylene, glycol, or oil. The enhancement of thermochemical properties of a fluid due to the addition of nanoparticles has been observed in experimental studies such as in [2, 3]. Researchers have investigated the influence of seven slip mechanisms, namely, inertia, Brownian diffusion, thermophoresis, diffusiophoresis, magnus effect, fluid drainage, and gravity in nanofluids. It has been shown that, in the absence of turbulence, the most significant

The classical Rayleigh-Benard convection problem in a heated horizontal layer has been extensively studied in the literature. Among recent studies on nanofluids, Tzou [4] studied the thermal instability and natural convection in nanofluid flow using an eigenfunction expansion method. Narayana et al. [5, 6] studied convection and the stability of a Maxwell fluid in a porous medium. Yadav et al. [7] investigated thermal instability of a rotating nanofluid layer. The studies by Kuznetsov and Nield [8–11] focused on thermal instability in a porous layer saturated with a nanofluid. They investigated the onset of instability in a horizontal porous layer using a model for the nanofluid that incorporated particle Brownian motion and thermophoresis. Related studies with various assumptions on the geometry and flow structure have been made by [12–15]. In the last few decades, researchers have also investigated thermal instability in a horizontal nanofluid layer subject to an applied magnetic field [16, 17]. The effects of a magnetic field on convection and the onset of instability have important applications in problems such as in cooling systems, pumps, magnetohydrodynamics and generators. The experimental study by Heris et al. [18] showed that thermal efficiency could be achieved by subjecting the flow to a magnetic field. The studies by Ghasemi et al. [19] and Hamad et al. [20] focused on the flow behavior and heat transfer in an electrically conducting nanofluid under the influence of a magnetic field and subject to Brownian diffusion and thermophoresis. They used a water-based nanofluid containing different types of nanoparticles such as copper, alumina and silver in their numerical simulations. Related studies of interest include [21–24]. Rana et al. [25] studied thermal convection in a Walters (Model B) fluid in a porous medium. They showed that a magnetic field may introduce oscillatory instability modes and acts to stabilize the system.

In this chapter, we give a weakly nonlinear stability analysis of a nanofluid layer with an applied magnetic field, stress free boundary conditions and under the assumption of zero nanoparticle flux at the boundary. The studies by Kuznetsov and Nield [9] and Nield and Kuznetsov [10, 11] investigated cross-diffusion in a nanofluid layer. However, these studies mostly presented a linear stability analysis. It is important to study the nonlinear regime for a nanofluid flow subject to cross-diffusion due to the wide range of applications where such flows may arise. Typical examples may be found in hydrothermal growth, compact heat exchanges, solidification of binary mixtures, geophysical systems, and so on. Hence, with this in mind, we studied the finite amplitude convection in a nanofluid flows subject to crossdiffusion. By introducing a truncated Fourier series, a Lorenz-type system of seven nonlinear differential equations is obtained. The recent multidomain spectral method is used to solve the nonlinear equations. This method is accurate and very easy to implement compared to older methods such as finite difference methods. An analysis of heat and mass transfer for different

parameters such as the Prandtl number, the Dufour and thermophoresis is presented.

among these mechanisms are the Brownian diffusion and thermophoresis.

150 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Consider viscous incompressible MHD nanofluid flow in an infinitely extended horizontal porous layer, confined between two boundaries at z ¼ 0 and z ¼ h. The layer is heated from below and cooled from above, see Figure 1. A Cartesian frame of reference is chosen in which the z-axis is vertically upward. The boundaries are perfectly conducting. The temperature at the lower and upper walls is Tc and Th, respectively with Th > Tc. The Oberbeck-Boussinesq approximation and the Darcy law are assumed to be applicable. The continuity equation, momentum equation, energy equation, concentration equation and volumetric fraction nanoparticle equation, which describe the above configuration in dimensionless form, are given as

$$\nabla \cdot V = 0,\tag{1}$$

$$\frac{D a \partial V}{\text{Pr } \partial t} = -\nabla P + D a \nabla^2 V - V + Q V \widehat{e}\_z - R m \widehat{e}\_z + R a T \widehat{e}\_z + R s \mathbb{C} \widehat{e}\_z - R n \phi \widehat{e}\_z \tag{2}$$

$$\frac{\partial T}{\partial t} + V \cdot \nabla T = \nabla^2 T + \frac{N\_B}{\text{Les}} \nabla \phi \cdot \nabla T + \frac{N\_A N\_B}{\text{Les}} \nabla T \cdot \nabla T + Du \nabla^2 \mathbf{C},\tag{3}$$

$$\frac{\partial \mathbf{C}}{\partial t} + V \cdot \nabla \mathbf{C} = \frac{1}{Le} \nabla^2 \mathbf{C} + Sr \nabla^2 T,\tag{4}$$

$$\frac{1}{\sigma} \frac{\partial \phi}{\partial t} + \frac{1}{\varepsilon} V \cdot \nabla \phi = \frac{1}{L \text{cs}} \nabla^2 \phi + \frac{N\_A}{L \text{cs}} \nabla^2 T,\tag{5}$$

subject to the boundary conditions

$$V = 0, T = 1, \mathcal{C} = 1 \frac{\partial \phi}{\partial z} + N\_A \frac{\partial T}{\partial z} = 0 \text{ at } z = 0,\tag{6}$$

$$V = 0, \quad T = 0, \quad \mathcal{C} = 0 \quad \quad \frac{\partial \phi}{\partial z} + N\_A \frac{\partial T}{\partial z} = 0 \quad \text{at} \quad z = 1,\tag{7}$$

where V is the fluid velocity, T is the temperature, C is the solute concentration and ϕ is the volumetric fraction of nanoparticles. The dimensionless parameters are the Darcy number (modified by the viscosity ratio) Da, Prandtl number Pr, Hartmann-Darcy number Q, thermal Rayleigh-Darcy number Ra, nanoparticle Rayleigh number Rn and the basic density Rayleigh number Rm. The parameter NA is a modified diffusivity ratio, Le is the Lewis number, Rs is solutal Rayleigh number, NB is a modified nanoparticle density increment and Du is a modified Dufour parameter. The parameter Les is the thermo-nanofluid Lewis number, ν is the kinematic viscosity and Sr is a modified Soret parameter. These parameters have the form

$$\text{Da} = \frac{\tilde{\mu}K}{\mu h^2}, \quad \text{Pr} = \frac{\mu}{\rho\_f \alpha\_m}, \quad \text{Q} = \frac{\delta \mathbf{B}\_0^2 K}{\mu}, \quad \text{Ra} = \frac{\rho\_f \beta \text{Klg} \left(T\_h^\* - T\_c^\*\right)}{\mu \alpha\_m}, \quad \text{Les} = \frac{\alpha\_m}{D\_\text{B}}.\tag{8}$$

$$R\text{tr} = \frac{\left(\rho\_p - \rho\_f\right)\left(\phi\_1^\* - \phi\_0^\*\right)\text{g}\text{K}\text{h}}{\mu\alpha\_m}, \quad R\text{m} = \frac{\rho\_p\phi\_0^\* + (1 - \phi\_0^\*)\rho\_f\text{g}\text{K}\text{h}}{\mu\alpha\_m}, \quad \text{L}\sigma = \frac{\alpha\_m}{D\_S}, \tag{9}$$

Figure 1. A schematic diagram of the problem.

$$N\_B = \frac{\varepsilon (\rho c)\_p}{\left(\rho c\right)\_f} \left(\phi\_1^\* - \phi\_0^\*\right)\_\prime \quad N\_A = \frac{D\_\Gamma \left(T\_h^\* - T\_c^\*\right)}{D\_\mathcal{B} T\_c^\* \left(\phi\_1^\* - \phi\_0^\*\right)}, \quad \text{Rs} = \frac{\rho\_f \beta \text{Khg}\left(\mathcal{C}\_h^\* - \mathcal{C}\_c^\*\right)}{\mu \alpha\_m},\tag{10}$$

$$Du = \frac{\sigma D\_{\rm TC} \left(\mathsf{C}\_{h}^{\*} - \mathsf{C}\_{c}^{\*}\right)}{\alpha\_{m} \left(T\_{h}^{\*} - T\_{c}^{\*}\right)}, \quad Sr = \frac{\sigma D\_{\rm CT} \left(T\_{h}^{\*} - T\_{c}^{\*}\right)}{\alpha\_{m} \left(\mathsf{C}\_{h}^{\*} - \mathsf{C}\_{c}^{\*}\right)},\tag{11}$$

<sup>u</sup> <sup>¼</sup> <sup>∂</sup><sup>Ψ</sup> ∂z

Eqs. (1)–(5) may now be simplified by introducing the truncated Fourier series

Lorenz-type system of nonlinear ordinary differential equations

<sup>Y</sup>\_ <sup>3</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>Y</sup>\_ <sup>5</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>Y</sup>\_ <sup>6</sup> <sup>¼</sup> <sup>σ</sup><sup>R</sup> ε

<sup>Y</sup><sup>6</sup> ¼ �πRD11, Y<sup>7</sup> ¼ �πRD20, t<sup>∗</sup> <sup>¼</sup> <sup>γ</sup>t, R <sup>¼</sup> <sup>α</sup><sup>2</sup>

subject to Ynð Þ¼ <sup>0</sup> <sup>Y</sup><sup>0</sup>

<sup>Y</sup><sup>1</sup> <sup>¼</sup> απ γ

the phase space. We can easily show that

equations above:

<sup>Y</sup>\_ <sup>4</sup> <sup>¼</sup> RY<sup>1</sup> � <sup>1</sup>

<sup>Y</sup><sup>1</sup> � <sup>σ</sup>

<sup>Y</sup>\_ <sup>7</sup> <sup>¼</sup> NA<sup>σ</sup> 2ε

<sup>Y</sup>\_ <sup>1</sup> <sup>¼</sup> Pr

, w ¼ � <sup>∂</sup><sup>Ψ</sup>

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

C<sup>0</sup> ¼ C<sup>11</sup> cos αx sin πz þ C<sup>02</sup> sin 2πz, ϕ<sup>0</sup> ¼ �NAð Þ D<sup>11</sup> cos αx sin πz þ D<sup>02</sup> sin 2πz , (14)

where A11, B11, B02, C11, C02, D<sup>11</sup> and D<sup>02</sup> are amplitudes that depend on time. This leads to the

Da �BY<sup>1</sup> � N Yð Þþ <sup>2</sup> <sup>þ</sup> <sup>Y</sup><sup>4</sup>

<sup>Y</sup>1Y<sup>2</sup> � <sup>G</sup> <sup>1</sup>

Leð Þ� <sup>Y</sup><sup>6</sup> � <sup>Y</sup><sup>2</sup>

<sup>Y</sup>1Y<sup>7</sup> � <sup>G</sup>

<sup>B</sup> <sup>¼</sup> Daγ<sup>2</sup> <sup>þ</sup> <sup>γ</sup> � <sup>α</sup><sup>2</sup><sup>Q</sup>

Eqs. (15)–(21) give an approximate description of the full dimensional nonlinear system. An analytical solution of the system of nonlinear ordinary differential Eqs. (15)–(21) is not possible for the general time variable t. However, it is possible to discuss the stability of the nonlinear system of equations. The system of equations is uniformly bounded in time and dissipative in

∂x :

Ψ<sup>0</sup> ¼ A<sup>11</sup> sin αx sin πz, T<sup>0</sup> ¼ B<sup>11</sup> cos αx sin πz þ B<sup>02</sup> sin 2πz, (13)

Les <sup>Y</sup><sup>5</sup> <sup>þ</sup> SrY<sup>3</sup>

NAσ ε

<sup>n</sup> for n ¼ 1, 2, ⋯, 7: The following variables have been introduced in the

<sup>γ</sup><sup>3</sup> Ra, G <sup>¼</sup> <sup>4</sup>π<sup>2</sup>

γ

A11, Y<sup>2</sup> ¼ �πRB11, Y<sup>3</sup> ¼ �πRB02, Y<sup>4</sup> ¼ �πRC11, Y<sup>5</sup> ¼ �πRC20,

<sup>γ</sup><sup>2</sup> :

NARn <sup>R</sup> <sup>Y</sup><sup>6</sup> (15)

<sup>Y</sup>\_ <sup>2</sup> <sup>¼</sup> RY<sup>1</sup> � <sup>Y</sup><sup>2</sup> � DuY<sup>4</sup> � <sup>Y</sup>1Y<sup>3</sup> (16)

Y1Y<sup>2</sup> � G Yð Þ <sup>3</sup> þ DuY<sup>5</sup> (17)

LesY<sup>4</sup> � SrY<sup>2</sup> � <sup>Y</sup>1Y<sup>5</sup> (18)

(19)

Leð Þ <sup>Y</sup><sup>7</sup> � <sup>Y</sup><sup>3</sup> (21)

Y1Y<sup>7</sup> (20)

http://dx.doi.org/10.5772/intechopen.71066

153

and <sup>N</sup> <sup>¼</sup> Rs

Ra ,

where r<sup>f</sup> , rp, μ~, β1, β2, κm, δ, ε and K are the fluid density, nanoparticle density, effective viscosity of porous medium, thermal volumetric expansion coefficient of the fluid, solutal volumetric expansion coefficient, the thermal conductivity of porous medium, the electrical conductivity, the porosity, and permeability of porous medium, respectively. The gravitational acceleration is denoted by g and DB is the Brownian diffusion coefficient, DT is the thermophoresis diffusion coefficient, DS is the solutal diffusion coefficient, DTC is the Dufour parameter and DCT is the Soret parameter. The heat capacity of the fluid is ð Þ rc <sup>f</sup> , ð Þ rc <sup>p</sup> is the effective heat capacity of the nanoparticle, ð Þ rc <sup>m</sup> is the effective heat capacity of the porous medium and B<sup>0</sup> is the uniform magnetic field strength.

The basic state is the time independent solution of Eqs. (1)–(5). Solving these equations with boundary conditions, we obtain

$$T\_b = 1 - z, \quad \mathbb{C}\_b = 1 - z, \quad \phi\_b = \phi\_0 + N\_A z. \tag{12}$$

#### 3. Weakly nonlinear stability analysis

In this section, we restrict the analysis to the case of two-dimensional disturbances. We define the stream function Ψ by the equations

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral… http://dx.doi.org/10.5772/intechopen.71066 153

$$
\mu = \frac{\partial \Psi}{\partial z}, \quad w = -\frac{\partial \Psi}{\partial x}.
$$

Eqs. (1)–(5) may now be simplified by introducing the truncated Fourier series

NB <sup>¼</sup> <sup>ε</sup>ð Þ <sup>r</sup><sup>c</sup> <sup>p</sup> ð Þ rc <sup>f</sup>

Figure 1. A schematic diagram of the problem.

uniform magnetic field strength.

boundary conditions, we obtain

3. Weakly nonlinear stability analysis

the stream function Ψ by the equations

ϕ∗ <sup>1</sup> � <sup>ϕ</sup><sup>∗</sup> 0

, NA <sup>¼</sup> DT <sup>T</sup><sup>∗</sup>

Du <sup>¼</sup> <sup>σ</sup>DTC <sup>C</sup><sup>∗</sup>

152 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

α<sup>m</sup> T<sup>∗</sup>

<sup>h</sup> � <sup>T</sup><sup>∗</sup> c 

, Sr <sup>¼</sup> <sup>σ</sup>DCT <sup>T</sup><sup>∗</sup>

where r<sup>f</sup> , rp, μ~, β1, β2, κm, δ, ε and K are the fluid density, nanoparticle density, effective viscosity of porous medium, thermal volumetric expansion coefficient of the fluid, solutal volumetric expansion coefficient, the thermal conductivity of porous medium, the electrical conductivity, the porosity, and permeability of porous medium, respectively. The gravitational acceleration is denoted by g and DB is the Brownian diffusion coefficient, DT is the thermophoresis diffusion coefficient, DS is the solutal diffusion coefficient, DTC is the Dufour parameter and DCT is the Soret parameter. The heat capacity of the fluid is ð Þ rc <sup>f</sup> , ð Þ rc <sup>p</sup> is the effective heat capacity of the nanoparticle, ð Þ rc <sup>m</sup> is the effective heat capacity of the porous medium and B<sup>0</sup> is the

The basic state is the time independent solution of Eqs. (1)–(5). Solving these equations with

In this section, we restrict the analysis to the case of two-dimensional disturbances. We define

, Rs <sup>¼</sup> <sup>r</sup><sup>f</sup> <sup>β</sup>Khg C<sup>∗</sup>

α<sup>m</sup> C<sup>∗</sup>

Tb ¼ 1 � z, Cb ¼ 1 � z, ϕ<sup>b</sup> ¼ ϕ<sup>0</sup> þ NAz: (12)

<sup>h</sup> � <sup>T</sup><sup>∗</sup> c 

<sup>h</sup> � <sup>C</sup><sup>∗</sup> c

<sup>h</sup> � <sup>C</sup><sup>∗</sup> c 

, (10)

μα<sup>m</sup>

, (11)

DBT<sup>∗</sup> <sup>c</sup> ϕ<sup>∗</sup> <sup>1</sup> � <sup>ϕ</sup><sup>∗</sup> 0

<sup>h</sup> � <sup>C</sup><sup>∗</sup> c 

<sup>h</sup> � <sup>T</sup><sup>∗</sup> c

$$\Psi' = A\_{11}\sin\alpha x \sin\pi z, \quad T' = B\_{11}\cos\alpha x \sin\pi z + B\_{02}\sin2\pi z,\tag{13}$$

$$\mathbf{C}' = \mathbf{C}\_{11}\cos\alpha x \sin\pi z + \mathbf{C}\_{02}\sin 2\pi z, \quad \phi' = -N\_A(D\_{11}\cos\alpha x \sin\pi z + D\_{02}\sin 2\pi z), \tag{14}$$

where A11, B11, B02, C11, C02, D<sup>11</sup> and D<sup>02</sup> are amplitudes that depend on time. This leads to the Lorenz-type system of nonlinear ordinary differential equations

$$\dot{Y}\_1 = \frac{\text{Pr}}{\text{Da}} \left\{-BY\_1 - N(Y\_2 + Y\_4) + \frac{N\_ARn}{R}Y\_6 \right\} \tag{15}$$

$$
\dot{Y}\_2 = RY\_1 - Y\_2 - DuY\_4 - Y\_1Y\_3 \tag{16}
$$

$$
\dot{Y}\_3 = \frac{1}{2} Y\_1 Y\_2 - G(Y\_3 + DuY\_5) \tag{17}
$$

$$
\dot{Y}\_4 = RY\_1 - \frac{1}{Les}Y\_4 - SrY\_2 - Y\_1Y\_5 \tag{18}
$$

$$\dot{Y}\_5 = \frac{1}{2} Y\_1 Y\_2 - G \left(\frac{1}{\text{Les}} Y\_5 + Sr Y\_3\right) \tag{19}$$

$$\dot{Y}\_{\delta} = \frac{\sigma R}{\varepsilon} Y\_1 - \frac{\sigma}{L\varepsilon} (Y\_{\delta} - Y\_2) - \frac{N\_A \sigma}{\varepsilon} Y\_1 Y\_7 \tag{20}$$

$$
\dot{Y}\_7 = \frac{N\_A \sigma}{2\varepsilon} Y\_1 Y\_7 - \frac{G}{L\varepsilon} (Y\_7 - Y\_3) \tag{21}
$$

subject to Ynð Þ¼ <sup>0</sup> <sup>Y</sup><sup>0</sup> <sup>n</sup> for n ¼ 1, 2, ⋯, 7: The following variables have been introduced in the equations above:

$$Y\_1 = \frac{a\pi}{\mathcal{Y}} A\_{11\prime} \quad Y\_2 = -\pi R B\_{11\prime} \quad Y\_3 = -\pi R B\_{02\prime} \quad Y\_4 = -\pi R C\_{11\prime} \quad Y\_5 = -\pi R C\_{20\prime}$$

$$Y\_6 = -\pi R D\_{11\prime} \quad Y\_7 = -\pi R D\_{20\prime} \quad t^\* = \gamma t \quad R = \frac{a^2}{\mathcal{Y}^3} R a \quad G = \frac{4\pi^2}{\mathcal{Y}} \quad \text{and} \quad N = \frac{R s}{Ra},$$

$$B = \frac{Da\gamma^2 + \gamma - a^2 Q}{\mathcal{Y}^2}.$$

Eqs. (15)–(21) give an approximate description of the full dimensional nonlinear system. An analytical solution of the system of nonlinear ordinary differential Eqs. (15)–(21) is not possible for the general time variable t. However, it is possible to discuss the stability of the nonlinear system of equations. The system of equations is uniformly bounded in time and dissipative in the phase space. We can easily show that

$$\sum\_{i=1}^{7} \frac{\partial \dot{Y}\_i}{\partial Y\_i} = -\left[\frac{DaB}{\Pr} + 1 + G + Les^{-1} + GLes^{-1} + \sigma Le^{-1} + GLe^{-1}\right].\tag{22}$$

<sup>T</sup> ¼ � DaB

Le � DuNRSr ε

þ

BDuSr Le þ

The trace is always negative, but the sign of determinant depends on the parameter values. If

To study the influence of various physical parameters on the average Nusselt and Sherwood numbers, we solved the nonlinear system of Eqs. (15)–(21) numerically using the multidomain spectral collocation method. This is a novel technique for solving nonlinear initial value problems and parabolic equations with large time domains. It has been suggested in the literature that the method gives better accuracy compared to other methods such as finite difference and Runge-Kutta methods [26]. To apply the multidomain spectral collocation to the nonlinear system of equations, we first divide the interval 0½ � ; T into subintervals Ω<sup>i</sup> ¼ ti�<sup>1</sup>; ti ½ � for i ¼ 1, 2, ⋯, p. The

<sup>t</sup> <sup>¼</sup> ti � ti�<sup>1</sup>

Da �BY<sup>i</sup>

dY<sup>i</sup> 2 dt <sup>¼</sup> RY<sup>i</sup>

dY<sup>i</sup> 3 dt <sup>¼</sup> <sup>1</sup> 2 Yi 1Yi

dY<sup>i</sup> 4 dt <sup>¼</sup> RY<sup>i</sup>

dY<sup>i</sup> 5 dt <sup>¼</sup> <sup>1</sup> 2 Yi 1Yi <sup>2</sup> <sup>τ</sup> <sup>þ</sup>

is used to transform each subinterval Ω<sup>i</sup> into the interval ½ � �1; 1 . The system of Eqs. (15)–(21)

<sup>2</sup> þ Y<sup>4</sup> <sup>þ</sup>

<sup>2</sup> � DuY<sup>i</sup>

<sup>4</sup> � SrY<sup>i</sup>

Les <sup>Y</sup><sup>i</sup>

<sup>2</sup> � G Y<sup>i</sup>

<sup>2</sup> � <sup>G</sup> <sup>1</sup>

<sup>1</sup> � N Y<sup>i</sup>

<sup>1</sup> � <sup>Y</sup><sup>i</sup>

<sup>1</sup> � <sup>1</sup> LesY<sup>i</sup> ti þ ti�<sup>1</sup>

NARn Ra <sup>Y</sup><sup>i</sup> 6 , (30)

> <sup>4</sup> � <sup>Y</sup><sup>i</sup> 1Yi

<sup>3</sup> <sup>þ</sup> DuY<sup>i</sup> 5

> <sup>2</sup> � <sup>Y</sup><sup>i</sup> 1Yi

<sup>5</sup> <sup>þ</sup> SrY<sup>i</sup> 3

and

<sup>d</sup> <sup>¼</sup> <sup>σ</sup>PrG<sup>3</sup>

d < 0 then

transformation

can be written in the form

dY<sup>i</sup> 1 dt <sup>¼</sup> Pr

DaLeLes � DuNRSr

suggesting a saddle point.

4. Method of solution

Pr <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>G</sup> <sup>þ</sup> Les�<sup>1</sup> <sup>þ</sup> GLes�<sup>1</sup> <sup>þ</sup> <sup>σ</sup>Le�<sup>1</sup> <sup>þ</sup> GLe�<sup>1</sup>

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

DuRSr Le þ

ð Þ 1 � N εDuSrRLes þ ð Þ εB � NRLe DuSrLes þ ð Þ ε þ Le NLe < ð Þ B þ R ε, (28)

, (26)

NR LeLes þ

NR Les<sup>ε</sup> � <sup>B</sup>

LeLes � <sup>R</sup>

http://dx.doi.org/10.5772/intechopen.71066

<sup>2</sup> (29)

<sup>3</sup>, (31)

<sup>5</sup>, (33)

, (32)

, (34)

LeLes :

(27)

155

This is always true if B ≥ 0. As has been shown in previous studies, the trajectories may be attracted to a fixed point, limit cycle or other attractor. For a set of initial points in the phase space occupying a region Vð Þ0 at time t ¼ 0, after a time t > 0, the end point of the corresponding trajectories fills a volume

$$V(t) = V(0)\exp\left\{-\left[\frac{DaB}{\Pr} + 1 + G + L\varepsilon\mathrm{s}^{-1} + GL\varepsilon\mathrm{s}^{-1} + \sigma L\varepsilon^{-1} + GL\varepsilon^{-1}\right]t\right\}.\tag{23}$$

Eq. (23) shows that the volume decays exponentially with time. Further, it can be noted that the system of Eqs. (15)–(21) are invariant under the transformation

$$S(Y\_1, Y\_2, Y\_3, Y\_4, Y\_5, Y\_6, Y\_7) \to -S(Y\_1, Y\_2, Y\_3, Y\_4, Y\_5, Y\_6, Y\_7). \tag{24}$$

We obtain the possible stationary points of the nonlinear system of equations by setting <sup>Y</sup>\_ <sup>i</sup> <sup>¼</sup> <sup>0</sup> for i ¼ 1, 2, ⋯, 7. One of these stationary points is Yi ¼ 0 and by linearizing about this point, we obtain the Jacobian matrix

$$A = \begin{pmatrix} -\frac{PrB}{Da} & -\frac{Pr}{Da} & 0 & \frac{PrN}{Da} & 0 & \frac{PrN\_A Rn}{DaR} & 0\\ R & -1 & 0 & -Du & 0 & 0 & 0\\ 0 & 0 & -G & 0 & -GDu & 0 & 0\\ R & -Sr & 0 & -L\varepsilon^{-1} & 0 & 0 & 0\\ 0 & 0 & -GSr & 0 & -GL\varepsilon^{-1} & 0 & 0\\ \frac{\sigma R}{\varepsilon} & \sigma L\varepsilon^{-1} & 0 & 0 & 0 & -\sigma L\varepsilon^{-1} & 0\\ 0 & 0 & GL\varepsilon^{-1} & 0 & 0 & 0 & -GL\varepsilon^{-1} \end{pmatrix}.\tag{25}$$

The eigenvalues of the above matrix depend on the various parameters. For the specific parameters <sup>R</sup> <sup>¼</sup> 103 , Da ¼ 20, Pr ¼ 10, N ¼ 25, Du ¼ 0:2, Sr ¼ 3, Les ¼ 10, Le ¼ 5, σ ¼ 0:05, G ¼ 3 and ε ¼ 0:04, the characteristic polynomial is

$$P(\lambda) = \lambda^7 + 21.41\lambda^6 + 651.5\lambda^5 + 5391.7\lambda^4 + 12772.232\lambda^3 - 370.962\lambda^2 - 2996.712\lambda + 545.87$$

with eigenvalues

$$\begin{aligned} \lambda\_1 &= 0.2955056985, \ \lambda\_2 = 0.2402382976, \ \lambda\_3 = -0.6139990637, \ \lambda\_4 = -4.886000936, \\\ \lambda\_5 &= -5, \ \lambda\_6 = -5.7228719981 - 21.9033954659i, \ \lambda\_7 = -5.7228719981 + 21.9033954659i. \end{aligned}$$

This stationary point is a saddle point. Nonetheless, because the eigenvalues depend on various parameters, we cannot make general conclusions as to the stability of the system. We note, however, that if we denote the trace of the matrix A by T and the determinant d, then

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral… http://dx.doi.org/10.5772/intechopen.71066 155

$$T = -\left(\frac{\text{DaB}}{\text{Pr}} + 1 + \text{G} + \text{Les}^{-1} + \text{GLes}^{-1} + \sigma \text{L} \varepsilon^{-1} + \text{GL} \varepsilon^{-1}\right),\tag{26}$$

and

X 7

∂Y\_ <sup>i</sup> ∂Yi

V tðÞ¼ <sup>V</sup>ð Þ<sup>0</sup> exp � DaB

the system of Eqs. (15)–(21) are invariant under the transformation

Da <sup>0</sup> PrN

¼ � DaB

154 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Pr <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>G</sup> <sup>þ</sup> Les�<sup>1</sup> <sup>þ</sup> GLes�<sup>1</sup> <sup>þ</sup> <sup>σ</sup>Le�<sup>1</sup> <sup>þ</sup> GLe�<sup>1</sup> � �

Pr <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>G</sup> <sup>þ</sup> Les�<sup>1</sup> <sup>þ</sup> GLes�<sup>1</sup> <sup>þ</sup> <sup>σ</sup>Le�<sup>1</sup> <sup>þ</sup> GLe�<sup>1</sup> � �

S Yð <sup>1</sup>; Y2;Y3; Y4;Y5;Y6;Y7Þ!�S Yð Þ <sup>1</sup>;Y2;Y3;Y4;Y5; Y6;Y<sup>7</sup> : (24)

Da <sup>0</sup> PrNARn

DaR <sup>0</sup>

� �

This is always true if B ≥ 0. As has been shown in previous studies, the trajectories may be attracted to a fixed point, limit cycle or other attractor. For a set of initial points in the phase space occupying a region Vð Þ0 at time t ¼ 0, after a time t > 0, the end point of the corresponding

Eq. (23) shows that the volume decays exponentially with time. Further, it can be noted that

We obtain the possible stationary points of the nonlinear system of equations by setting <sup>Y</sup>\_ <sup>i</sup> <sup>¼</sup> <sup>0</sup> for i ¼ 1, 2, ⋯, 7. One of these stationary points is Yi ¼ 0 and by linearizing about this point, we

> R �1 0 �Du 0 00 0 0 �G 0 �GDu 0 0 <sup>R</sup> �Sr <sup>0</sup> �Les�<sup>1</sup> 0 00 0 0 �GSr <sup>0</sup> �GLes�<sup>1</sup> 0 0

> > <sup>σ</sup>Le�<sup>1</sup> 00 0 �σLe�<sup>1</sup> <sup>0</sup>

, Da ¼ 20, Pr ¼ 10, N ¼ 25, Du ¼ 0:2, Sr ¼ 3, Les ¼ 10, Le ¼ 5, σ ¼ 0:05, G ¼ 3 and

0 0 GLe�<sup>1</sup> 00 0 �GLe�<sup>1</sup>

The eigenvalues of the above matrix depend on the various parameters. For the specific param-

<sup>P</sup>ð Þ¼ <sup>λ</sup> <sup>λ</sup><sup>7</sup> <sup>þ</sup> <sup>21</sup>:41λ<sup>6</sup> <sup>þ</sup> <sup>651</sup>:5λ<sup>5</sup> <sup>þ</sup> <sup>5391</sup>:7λ<sup>4</sup> <sup>þ</sup> <sup>12772</sup>:232λ<sup>3</sup> � <sup>370</sup>:962λ<sup>2</sup> � <sup>2996</sup>:712<sup>λ</sup> <sup>þ</sup> <sup>545</sup>:<sup>8</sup>

This stationary point is a saddle point. Nonetheless, because the eigenvalues depend on various parameters, we cannot make general conclusions as to the stability of the system. We note, however, that if we denote the trace of the matrix A by T and the determinant d, then

λ<sup>1</sup> ¼ 0:2955056985, λ<sup>2</sup> ¼ 0:2402382976, λ<sup>3</sup> ¼ �0:6139990637, λ<sup>4</sup> ¼ �4:886000936, λ<sup>5</sup> ¼ �5, λ<sup>6</sup> ¼ �5:7228719981 � 21:9033954659i, λ<sup>7</sup> ¼ �5:7228719981 þ 21:9033954659i:

: (22)

: (23)

t

1

CCCCCCCCCCCCCCA

: (25)

i¼1

trajectories fills a volume

obtain the Jacobian matrix

0

BBBBBBBBBBBBBB@

� PrB

σR ε

ε ¼ 0:04, the characteristic polynomial is

Da � Pr

A ¼

eters <sup>R</sup> <sup>¼</sup> 103

with eigenvalues

$$d = \frac{\sigma \text{PrG}^3}{Da Le Le} \left( -\frac{Du NRSr}{L\varepsilon} - \frac{Du NRSr}{\varepsilon} + \frac{BD u Sr}{L\varepsilon} + \frac{Du RSr}{L\varepsilon} + \frac{NR}{Le L\varepsilon\varepsilon} + \frac{NR}{L\varepsilon\varepsilon} - \frac{B}{Le L\varepsilon\varepsilon} - \frac{R}{L\varepsilon\varepsilon\varepsilon} \right). \tag{27}$$

The trace is always negative, but the sign of determinant depends on the parameter values. If d < 0 then

$$(1 - N)\varepsilon DuSr\text{Les} + (\varepsilon B - \text{NRL}\varepsilon)DuSr\text{Les} + (\varepsilon + L\varepsilon)\text{NL}\varepsilon < (B + R)\varepsilon,\tag{28}$$

suggesting a saddle point.

#### 4. Method of solution

To study the influence of various physical parameters on the average Nusselt and Sherwood numbers, we solved the nonlinear system of Eqs. (15)–(21) numerically using the multidomain spectral collocation method. This is a novel technique for solving nonlinear initial value problems and parabolic equations with large time domains. It has been suggested in the literature that the method gives better accuracy compared to other methods such as finite difference and Runge-Kutta methods [26]. To apply the multidomain spectral collocation to the nonlinear system of equations, we first divide the interval 0½ � ; T into subintervals Ω<sup>i</sup> ¼ ti�<sup>1</sup>; ti ½ � for i ¼ 1, 2, ⋯, p. The transformation

$$t = \frac{t\_i - t\_{i-1}}{2}\tau + \frac{t\_i + t\_{i-1}}{2} \tag{29}$$

is used to transform each subinterval Ω<sup>i</sup> into the interval ½ � �1; 1 . The system of Eqs. (15)–(21) can be written in the form

$$\frac{dY\_1^i}{dt} = \frac{\text{Pr}}{Da} \left\{-BY\_1^i - N\left(Y\_2^i + Y\_4\right) + \frac{N\_ARn}{Ra}Y\_6^i\right\},\tag{30}$$

$$\frac{dY\_2^i}{dt} = RY\_1^i - Y\_2^i - DuY\_4^i - Y\_1^i Y\_{3'}^i \tag{31}$$

$$\frac{dY\_3^i}{dt} = \frac{1}{2} Y\_1^i Y\_2^i - G \left( Y\_3^i + Du Y\_5^i \right) \tag{32}$$

$$\frac{dY\_4^i}{dt} = RY\_1^i - \frac{1}{Les}Y\_4^i - SrY\_2^i - Y\_1^i Y\_{5'}^i \tag{33}$$

$$\frac{dY\_5^i}{dt} = \frac{1}{2} Y\_1^i Y\_2^i - G \left(\frac{1}{L\text{cs}} Y\_5^i + \text{Sr} Y\_3^i\right),\tag{34}$$

$$\frac{dY\_6^i}{dt} = \frac{\sigma R}{\varepsilon} Y\_1^i - \frac{\sigma}{Le} \left( Y\_6^i - Y\_2^i \right) - \frac{N\_A \sigma}{\varepsilon} Y\_1^i Y\_{7\prime}^i \tag{35}$$

$$\frac{dY\_7^i}{dt} = \frac{N\_A \sigma}{2\varepsilon} Y\_1^i Y\_7^i - \frac{G}{L\varepsilon} (Y\_7^i - Y\_3^i),\tag{36}$$

5. Heat and mass transfer

6. Results and discussion

critical Rayleigh number for low Darcy numbers.

by the average Nusselt number Nu tð Þ where

Nu tðÞ¼ 1 þ

Substituting Eqs. (12) and (13) into Eq. (43), we obtain

α 2π Ð 2π α 0 ∂T <sup>∂</sup><sup>z</sup> dx

2 4

α 2π Ð 2π α 0 ∂Tb <sup>∂</sup><sup>z</sup> dx

Nu tðÞ¼ 1 þ

Sh tðÞ¼ 1 þ

has very little effect on the heat and mass transfer for this type of flow.

The study of heat and mass transfer in a horizontal nanofluid layer heated from below and cooled from above has important engineering applications. We define the rate of heat transfer

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

þ Du 1 þ

<sup>R</sup> <sup>Y</sup><sup>3</sup> <sup>þ</sup> Du <sup>1</sup> <sup>þ</sup>

<sup>R</sup> <sup>Y</sup><sup>5</sup> <sup>þ</sup> Sr <sup>1</sup> <sup>þ</sup>

We have studied the weakly nonlinear instability of nanofluid flow in a horizontal layer with stress free boundary conditions. For numerical simulations, the parameter values were chosen from the literature on nanofluid flow such as [4, 7]. In the literature, the critical Rayleigh number is found when the Darcy number is very large. In this study, we investigated the

The method of solution described in Section 4 was used to solve Eqs. (15)–(21). All computations are carried out up to a value of maximum time tmax ¼ 1, and solutions are obtained using initial conditions selected in the neighborhood of stationary points. Periodic solution sets were obtained for the system of nonlinear equations. We determined the rate of heat and mass transfer as functions of time for different parameter values. The results are shown in Figures 2–4. Figure 2 shows the effect of the Dufour and Soret parameters on the Nusselt and Sherwood numbers with time t. Figure 2(a) shows how the heat transfer coefficient changes with both the Dufour parameter and time. The heat transfer coefficient increases with the Dufour parameter but eventually settles to a steady value with time. In Figure 2(b), the Soret parameter is similarly shown to enhance the mass transfer coefficient. We investigated the effect of the Prandtl and Lewis numbers (see Figures 3 and 4). An increase in the Lewis number enhances both heat and mass transfer in a nanofluid layer heated from below. However, Figure 3 shows that increasing the Prandtl number reduces the amplitude of oscillatory heat and mass transfer. The Prandtl number can lead to both positive and negative contributions to the Nusselt and Sherwood numbers. It is interesting to note that our investigation shows that the magnetic field parameter

8 < :

α 2π Ð 2π α 0 ∂C <sup>∂</sup><sup>z</sup> dx 3 5 z¼0 9 =

http://dx.doi.org/10.5772/intechopen.71066

� �: (44)

� � (45)

;: (43)

157

2 4

2 <sup>R</sup>Y<sup>5</sup>

2 <sup>R</sup> <sup>Y</sup><sup>3</sup>

α 2π Ð 2π α 0 ∂Cb <sup>∂</sup><sup>z</sup> dx

3 5 z¼0

2

Similarly, the rate of mass transfer stated in terms of the average Sherwood number is

2

subject to

$$Y\_n^i(t\_{i-1}) = Y\_n^{i-1}(t\_{i-1}) \quad \text{for} \quad n = 1, 2, \cdots, 7. \tag{37}$$

The first step in using the multidomain spectral collocation method (MDSCM) concerns the quasilinearization of Eqs. (30)–(36) leading to a system of equations in the form

$$\sum\_{n=1}^{7} \sigma\_{(j,n)r}^{i} Y\_{n,r+1}^{i} - \frac{dY\_{j,r+1}^{i}}{dt} = R\_{jr'}^{i} \tag{38}$$

subject to

$$Y\_{n,r+1}^{i}(t\_{i-1}) = Y\_{n,r+1}^{i-1}(t\_{i-1}) \quad \text{for} \quad n = 1, 2, \cdots, 7. \tag{39}$$

where ai ð Þ <sup>j</sup>;<sup>n</sup> <sup>r</sup> and Ri jr for j ¼ 1, 2, …, 7 are given in the Appendix. Having linearized the equations, the second step is to integrate Eqs. (30)–(36). To this end, we use the Gauss-Lobatto nodes

$$
\pi^i\_j = \cos \frac{\pi j}{\text{Nc}'} \quad \text{for } j = 0, 1, \cdots, \text{Nc}. \tag{40}
$$

We approximate the derivatives of the unknown functions Y<sup>i</sup> n,rþ<sup>1</sup>ð Þ<sup>t</sup> at the collocation points by

$$\frac{dY^i\_{\boldsymbol{n},r+1}}{dt}\left(\tau^i\_{\boldsymbol{\beta}}\right) = \sum\_{k=0}^{Nc} \mathbf{D}\_{\boldsymbol{\beta}k} Y^i\_{\boldsymbol{n},r+1} \left(\tau^i\_{\boldsymbol{\beta}}\right) = \left[\mathbf{D} \mathbf{U}^i\_{\boldsymbol{n},r+1}\right]\_{\boldsymbol{\beta}}\tag{41}$$

where D ¼ 2D=ð Þ ti � ti�<sup>1</sup> , D is the Chebyshev differentiation matrix and

$$\mathbf{U}\_{n,r+1}^{i} = \left(Y\_{n,r+1}^{i}\left(\tau\_{0}^{i}\right), \cdots Y\_{n,r+1}^{i}\left(\tau\_{Nc}^{i}\right)\right)^{T}\mathbf{f}$$

is a vector of the unknown functions at the collocation points. Substituting Eq. (41) into Eqs. (38) and reducing the result into matrix form, we obtain

$$\begin{array}{rcl} \mathbf{A} \mathbf{U}\_{n,r+1}^{i} &=& \mathbf{R}\_{n\prime}^{i} \\ \mathbf{U}\_{n,r+1}^{i} \left( \boldsymbol{\pi}\_{N\_{\varepsilon}}^{i-1} \right) &=& \mathbf{U}\_{n}^{i} \left( \boldsymbol{\pi}\_{N\_{\varepsilon}}^{i-1} \right) \\ \end{array} \quad \begin{array}{rcl} \mathbf{u} = \mathbf{0} \\ \end{array} \tag{42}$$

where the matrices <sup>A</sup> <sup>¼</sup> Aij � � and <sup>R</sup><sup>i</sup> <sup>n</sup> are given in the Appendix.

#### 5. Heat and mass transfer

dY<sup>i</sup> 6 dt <sup>¼</sup> <sup>σ</sup><sup>R</sup> ε Yi <sup>1</sup> � <sup>σ</sup> Le <sup>Y</sup><sup>i</sup>

156 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Yi

Yi

subject to

subject to

where ai

ð Þ <sup>j</sup>;<sup>n</sup> <sup>r</sup> and Ri

dY<sup>i</sup> 7 dt <sup>¼</sup> NA<sup>σ</sup> 2ε Yi 1Yi <sup>7</sup> � <sup>G</sup> Le <sup>Y</sup><sup>i</sup>

<sup>n</sup>ð Þ¼ ti�<sup>1</sup> <sup>Y</sup><sup>i</sup>�<sup>1</sup>

X 7

n¼1 ai ð Þ <sup>j</sup>;<sup>n</sup> <sup>r</sup>Y<sup>i</sup>

n,rþ<sup>1</sup>ð Þ¼ ti�<sup>1</sup> <sup>Y</sup><sup>i</sup>�<sup>1</sup>

τi <sup>j</sup> ¼ cos

We approximate the derivatives of the unknown functions Y<sup>i</sup>

Ui

Eqs. (38) and reducing the result into matrix form, we obtain

n,rþ<sup>1</sup> <sup>τ</sup><sup>i</sup>�<sup>1</sup> Nc � �

� � and R<sup>i</sup>

AU<sup>i</sup>

Ui

where the matrices A ¼ Aij

dY<sup>i</sup> n,rþ1 dt <sup>τ</sup><sup>i</sup> j � �

quasilinearization of Eqs. (30)–(36) leading to a system of equations in the form

<sup>6</sup> � <sup>Y</sup><sup>i</sup> 2 � � � NA<sup>σ</sup>

The first step in using the multidomain spectral collocation method (MDSCM) concerns the

n,rþ<sup>1</sup> � dY<sup>i</sup>

tions, the second step is to integrate Eqs. (30)–(36). To this end, we use the Gauss-Lobatto nodes

n,rþ<sup>1</sup> <sup>τ</sup> j j � �

� � � �<sup>T</sup>

n,rþ<sup>1</sup> <sup>τ</sup><sup>i</sup> Nc

πj

<sup>¼</sup> <sup>X</sup>Nc k¼0 DjkY<sup>i</sup>

> n,rþ<sup>1</sup> <sup>τ</sup><sup>i</sup> 0 � �; ⋯Y<sup>i</sup>

is a vector of the unknown functions at the collocation points. Substituting Eq. (41) into

n,

<sup>n</sup> τ<sup>i</sup>�<sup>1</sup> Nc � �

<sup>n</sup> are given in the Appendix.

<sup>¼</sup> <sup>U</sup><sup>i</sup>

where D ¼ 2D=ð Þ ti � ti�<sup>1</sup> , D is the Chebyshev differentiation matrix and

n,rþ<sup>1</sup> <sup>¼</sup> <sup>Y</sup><sup>i</sup>

n,rþ<sup>1</sup> <sup>¼</sup> <sup>R</sup><sup>i</sup>

j,rþ1 dt <sup>¼</sup> Ri

jr for j ¼ 1, 2, …, 7 are given in the Appendix. Having linearized the equa-

ε Yi 1Yi

<sup>7</sup> � <sup>Y</sup><sup>i</sup> 3 <sup>7</sup>, (35)

� �, (36)

jr, (38)

n,rþ<sup>1</sup>ð Þ<sup>t</sup> at the collocation points by

, (41)

<sup>n</sup> ð Þ ti�<sup>1</sup> for n ¼ 1, 2, ⋯, 7: (37)

n,rþ<sup>1</sup>ð Þ ti�<sup>1</sup> for <sup>n</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, <sup>7</sup>: (39)

Nc , for <sup>j</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>⋯</sup>, Nc: (40)

n,rþ1 h i

,

j

, n <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, <sup>7</sup>: (42)

<sup>¼</sup> DU<sup>i</sup>

The study of heat and mass transfer in a horizontal nanofluid layer heated from below and cooled from above has important engineering applications. We define the rate of heat transfer by the average Nusselt number Nu tð Þ where

$$Nu(t) = 1 + \left[\frac{\frac{\alpha}{2\pi} \int\_0^{\frac{2\pi}{\alpha}} \frac{\partial T}{\partial \mathbf{z}} d\mathbf{x}}{\frac{\alpha}{2\pi} \int\_0^{\frac{2\pi}{\alpha}} \frac{\partial T\_b}{\partial \mathbf{z}} d\mathbf{x}}\right]\_{z=0} + Du \left\{ 1 + \left[\frac{\frac{\alpha}{2\pi} \int\_0^{\frac{2\pi}{\alpha}} \frac{\partial C}{\partial \mathbf{z}} d\mathbf{x}}{\frac{\alpha}{2\pi} \int\_0^{\frac{2\pi}{\alpha}} \frac{\partial C\_b}{\partial \mathbf{z}} d\mathbf{x}}\right]\_{z=0} \right\}. \tag{43}$$

Substituting Eqs. (12) and (13) into Eq. (43), we obtain

$$Nu(t) = 1 + \frac{2}{R}Y\_3 + Du\left(1 + \frac{2}{R}Y\_5\right). \tag{44}$$

Similarly, the rate of mass transfer stated in terms of the average Sherwood number is

$$Sh(t) = 1 + \frac{2}{R}Y\_5 + Sr\left(1 + \frac{2}{R}Y\_3\right) \tag{45}$$

#### 6. Results and discussion

We have studied the weakly nonlinear instability of nanofluid flow in a horizontal layer with stress free boundary conditions. For numerical simulations, the parameter values were chosen from the literature on nanofluid flow such as [4, 7]. In the literature, the critical Rayleigh number is found when the Darcy number is very large. In this study, we investigated the critical Rayleigh number for low Darcy numbers.

The method of solution described in Section 4 was used to solve Eqs. (15)–(21). All computations are carried out up to a value of maximum time tmax ¼ 1, and solutions are obtained using initial conditions selected in the neighborhood of stationary points. Periodic solution sets were obtained for the system of nonlinear equations. We determined the rate of heat and mass transfer as functions of time for different parameter values. The results are shown in Figures 2–4. Figure 2 shows the effect of the Dufour and Soret parameters on the Nusselt and Sherwood numbers with time t. Figure 2(a) shows how the heat transfer coefficient changes with both the Dufour parameter and time. The heat transfer coefficient increases with the Dufour parameter but eventually settles to a steady value with time. In Figure 2(b), the Soret parameter is similarly shown to enhance the mass transfer coefficient. We investigated the effect of the Prandtl and Lewis numbers (see Figures 3 and 4). An increase in the Lewis number enhances both heat and mass transfer in a nanofluid layer heated from below. However, Figure 3 shows that increasing the Prandtl number reduces the amplitude of oscillatory heat and mass transfer. The Prandtl number can lead to both positive and negative contributions to the Nusselt and Sherwood numbers. It is interesting to note that our investigation shows that the magnetic field parameter has very little effect on the heat and mass transfer for this type of flow.

increases leading to certain chaotic forms. Figures 8–11 show the trajectories in the threedimensional phase space. Here, we observe similar solution sets as in the two-dimensional

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

http://dx.doi.org/10.5772/intechopen.71066

159

Figure 5. The trajectories of the system of nonlinear equations projected on the Y1, Y2-plane when revised Rayleigh number (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104 when Da <sup>¼</sup> <sup>0</sup>:05, Les <sup>¼</sup> 100, <sup>σ</sup> <sup>¼</sup> <sup>0</sup>:05, <sup>ε</sup> <sup>¼</sup> <sup>0</sup>:04, <sup>Q</sup> <sup>¼</sup> 10, Du <sup>¼</sup> <sup>0</sup>:2,

Figure 6. Trajectories of the system of nonlinear equations projected on the ð Þ Y1; Y<sup>3</sup> plane when the revised Rayleigh number (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104 when Da <sup>¼</sup> <sup>0</sup>:05, Les <sup>¼</sup> 100, <sup>σ</sup> <sup>¼</sup> <sup>0</sup>:05, <sup>ε</sup> <sup>¼</sup> <sup>0</sup>:04, <sup>Q</sup> <sup>¼</sup> 10, Du <sup>¼</sup> <sup>0</sup>:2, Sr <sup>¼</sup> <sup>0</sup>:3,

Figure 4. The effect of Lewis number on (a) the Nusselt number Nu and (b) the Sherwood number Sh when Da ¼ 0:05,

phase portraits.

Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

Le ¼ 2, N ¼ 25 and Rn ¼ 5.

Du ¼ 0:2, ε ¼ 0:04, σ ¼ 0:05, Les ¼ 100, Rn ¼ 5 and Ra ¼ 1000.

Figure 2. The effect of cross-diffusive parameters on (a) the Nusselt number Nu and (b) the Sherwood number Sh for Da ¼ 0:05, Le ¼ 2, Du ¼ 0:2, ε ¼ 0:04, σ ¼ 0:05, Les ¼ 100, Rn ¼ 5, Ra ¼ 1000 and various values of the Dufour and Soret parameters.

Figure 3. The effect of Prandtl number Pr on (a) the Nusselt number Nu and (b) the Sherwood number Sh when Da ¼ 0:05, Le ¼ 2, Du ¼ 0:2, ε ¼ 0:04, σ ¼ 0:05, Les ¼ 100, Rn ¼ 5 and Ra ¼ 1000.

Figures 5–11 show the effect of the Rayleigh number on the trajectories projected onto the Yi;Yj phase planes. The solution sets provide a visual representation of the system's behavior with every phase point on the phase space representing the physical state of the system. The convective solution sets for different values of R have been presented with the trajectories projected onto the Yi;Yj phase planes. These trajectories spiral toward the fixed point for Rayleigh numbers from 10<sup>2</sup> to 104 . The solution sets give spiral phase portraits as R increases and for the high Rayleigh numbers, the trajectories spiral many times before they reach a fixed point.

Figures 5–8 show the phase portraits projected onto the Yi;Yj - plane correspond to a simple spiral for <sup>R</sup> <sup>¼</sup> 100. As <sup>R</sup> is increased to 10<sup>4</sup> , the complexity of the trajectories increases leading to certain chaotic forms. Figures 8–11 show the trajectories in the threedimensional phase space. Here, we observe similar solution sets as in the two-dimensional phase portraits.

Figure 4. The effect of Lewis number on (a) the Nusselt number Nu and (b) the Sherwood number Sh when Da ¼ 0:05, Du ¼ 0:2, ε ¼ 0:04, σ ¼ 0:05, Les ¼ 100, Rn ¼ 5 and Ra ¼ 1000.

Figure 5. The trajectories of the system of nonlinear equations projected on the Y1, Y2-plane when revised Rayleigh number (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104 when Da <sup>¼</sup> <sup>0</sup>:05, Les <sup>¼</sup> 100, <sup>σ</sup> <sup>¼</sup> <sup>0</sup>:05, <sup>ε</sup> <sup>¼</sup> <sup>0</sup>:04, <sup>Q</sup> <sup>¼</sup> 10, Du <sup>¼</sup> <sup>0</sup>:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

Figures 5–11 show the effect of the Rayleigh number on the trajectories projected onto the Yi;Yj

Figure 3. The effect of Prandtl number Pr on (a) the Nusselt number Nu and (b) the Sherwood number Sh when

Figure 2. The effect of cross-diffusive parameters on (a) the Nusselt number Nu and (b) the Sherwood number Sh for Da ¼ 0:05, Le ¼ 2, Du ¼ 0:2, ε ¼ 0:04, σ ¼ 0:05, Les ¼ 100, Rn ¼ 5, Ra ¼ 1000 and various values of the Dufour and Soret

phase planes. The solution sets provide a visual representation of the system's behavior with every phase point on the phase space representing the physical state of the system. The convective solution sets for different values of R have been presented with the trajectories projected onto the

phase planes. These trajectories spiral toward the fixed point for Rayleigh numbers from

numbers, the trajectories spiral many times before they reach a fixed point.

Figures 5–8 show the phase portraits projected onto the Yi;Yj

Da ¼ 0:05, Le ¼ 2, Du ¼ 0:2, ε ¼ 0:04, σ ¼ 0:05, Les ¼ 100, Rn ¼ 5 and Ra ¼ 1000.

158 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

simple spiral for <sup>R</sup> <sup>¼</sup> 100. As <sup>R</sup> is increased to 10<sup>4</sup>

. The solution sets give spiral phase portraits as R increases and for the high Rayleigh

Yi;Yj

10<sup>2</sup> to 104

parameters.


, the complexity of the trajectories

Figure 6. Trajectories of the system of nonlinear equations projected on the ð Þ Y1; Y<sup>3</sup> plane when the revised Rayleigh number (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104 when Da <sup>¼</sup> <sup>0</sup>:05, Les <sup>¼</sup> 100, <sup>σ</sup> <sup>¼</sup> <sup>0</sup>:05, <sup>ε</sup> <sup>¼</sup> <sup>0</sup>:04, <sup>Q</sup> <sup>¼</sup> 10, Du <sup>¼</sup> <sup>0</sup>:2, Sr <sup>¼</sup> <sup>0</sup>:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

Figure 7. Trajectories of the system of nonlinear equations projected on the ð Þ Y1; Y<sup>5</sup> -plane showing the sensitive dependence of the trajectories on the revised Rayleigh number for (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>3</sup> and (c) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>4</sup> when Da <sup>¼</sup> <sup>0</sup>:05, Les ¼ 100, σ ¼ 0:05, ε ¼ 0:04, Q ¼ 10, Du ¼ 0:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

Figure 8. Trajectories of the system of nonlinear equations projected on the ð Þ Y1; Y<sup>6</sup> -plane showing the sensitive dependence of the trajectories on the revised Rayleigh number for (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>3</sup> and (c) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>4</sup> when Da <sup>¼</sup> <sup>0</sup>:05, Les ¼ 100, σ ¼ 0:05, ε ¼ 0:04, Q ¼ 10, Du ¼ 0:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5:

become thinner. Also, a high buoyancy ratio changes the flow structure, and this impacts significantly on the concentration field, which builds up a vertical stratification in the enclosure. It is interesting to note that for N ¼ �25, the effect of the solutal buoyancy force is in the opposite direction of the thermal buoyancy force. The isothermal and isoconcentration profiles are situated toward the left wall, while for N ¼ 1, the thermal and solutal buoyancy forces are equal. For N ¼ 25, the effect of solutal buoyancy force is in the same direction as the thermal buoyancy force. In such cases, the isothermal and isoconcentration contours are mostly toward the right wall.

Figure 11. Flow trajectories and bifurcations in the three-dimensional space ð Þ Y1; Y6; Y<sup>7</sup> for Rayleigh numbers (a)

Figure 10. Flow trajectories and bifurcations in the three-dimensional space ð Þ Y1; Y2; Y<sup>6</sup> for Rayleigh numbers (a)

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

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161

<sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104.

<sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104.

We observe that when N ¼ �25, the stream function values in the central eddies increase because the thickness of the boundary layer increases with the buoyancy ratio. The streamlines and the flow behavior are affected by the change in the buoyancy ratio, but the flow pattern remains unaltered. As N decreases from 1 to �25, the streamlines become very dense to the left side of nanofluid layer while when N increases from 1 to 25, the streamlines are less so. The buoyancy forces that drive the nanofluid motion are mainly due to the temperature gradient.

Three different types of eddies are observed for the isoconcentration contours when N ¼ 25. Of these, two have a clockwise rotation and one is anticlockwise. It is seen that the small eddy at the right bottom edge is diminished as N decreases from 1 to �25. Here, the concentration boundary layer decreases due to increasing N values, hence the buoyancy ratio has a significant influence on the concentration gradient. As the buoyancy ratio N increases from 1 to 25 the isoconcen-

trations become very dense at the bottom of nanofluid layer.

Figure 9. The bifurcations in the three-dimension solution space ð Þ <sup>Y</sup>1; <sup>Y</sup>2; <sup>Y</sup><sup>3</sup> for (a) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>2</sup> , (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104 when Da ¼ 0:05, Les ¼ 100, σ ¼ 0:05, ε ¼ 0:04, Q ¼ 10, Du ¼ 0:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

Figures 12 and 13 show the streamline, isotherm and isoconcentration contours in the nanofluid flow for different values of the Darcy number and buoyancy ratio. Figure 12 displays the streamlines for various values of the buoyancy ratio term. Two different eddies are observed. The clockwise and anticlockwise flows are shown via negative and positive stream function values, respectively. The anticlockwise rotating flow occupies the largest area of the nanofluid layer.

For low buoyancy ratio parameters, the flow structure is significantly influenced by the buoyancy within the whole enclosure. Increasing the buoyancy ratio causes the boundary layer thickness to

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral… http://dx.doi.org/10.5772/intechopen.71066 161

Figure 10. Flow trajectories and bifurcations in the three-dimensional space ð Þ Y1; Y2; Y<sup>6</sup> for Rayleigh numbers (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104.

Figure 11. Flow trajectories and bifurcations in the three-dimensional space ð Þ Y1; Y6; Y<sup>7</sup> for Rayleigh numbers (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104.

become thinner. Also, a high buoyancy ratio changes the flow structure, and this impacts significantly on the concentration field, which builds up a vertical stratification in the enclosure. It is interesting to note that for N ¼ �25, the effect of the solutal buoyancy force is in the opposite direction of the thermal buoyancy force. The isothermal and isoconcentration profiles are situated toward the left wall, while for N ¼ 1, the thermal and solutal buoyancy forces are equal. For N ¼ 25, the effect of solutal buoyancy force is in the same direction as the thermal buoyancy force. In such cases, the isothermal and isoconcentration contours are mostly toward the right wall.

We observe that when N ¼ �25, the stream function values in the central eddies increase because the thickness of the boundary layer increases with the buoyancy ratio. The streamlines and the flow behavior are affected by the change in the buoyancy ratio, but the flow pattern remains unaltered. As N decreases from 1 to �25, the streamlines become very dense to the left side of nanofluid layer while when N increases from 1 to 25, the streamlines are less so. The buoyancy forces that drive the nanofluid motion are mainly due to the temperature gradient.

Figures 12 and 13 show the streamline, isotherm and isoconcentration contours in the nanofluid flow for different values of the Darcy number and buoyancy ratio. Figure 12 displays the streamlines for various values of the buoyancy ratio term. Two different eddies are observed. The clockwise and anticlockwise flows are shown via negative and positive stream function values, respectively. The anticlockwise rotating flow occupies the largest area of the nanofluid layer.

, (b) <sup>R</sup> <sup>¼</sup> 103 and (c) <sup>R</sup> <sup>¼</sup> 104

Figure 7. Trajectories of the system of nonlinear equations projected on the ð Þ Y1; Y<sup>5</sup> -plane showing the sensitive dependence of the trajectories on the revised Rayleigh number for (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>3</sup> and (c) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>4</sup> when Da <sup>¼</sup> <sup>0</sup>:05,

Figure 8. Trajectories of the system of nonlinear equations projected on the ð Þ Y1; Y<sup>6</sup> -plane showing the sensitive dependence of the trajectories on the revised Rayleigh number for (a) <sup>R</sup> <sup>¼</sup> 102, (b) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>3</sup> and (c) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>4</sup> when Da <sup>¼</sup> <sup>0</sup>:05,

Les ¼ 100, σ ¼ 0:05, ε ¼ 0:04, Q ¼ 10, Du ¼ 0:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

160 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Les ¼ 100, σ ¼ 0:05, ε ¼ 0:04, Q ¼ 10, Du ¼ 0:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5:

Figure 9. The bifurcations in the three-dimension solution space ð Þ <sup>Y</sup>1; <sup>Y</sup>2; <sup>Y</sup><sup>3</sup> for (a) <sup>R</sup> <sup>¼</sup> <sup>10</sup><sup>2</sup>

when Da ¼ 0:05, Les ¼ 100, σ ¼ 0:05, ε ¼ 0:04, Q ¼ 10, Du ¼ 0:2, Sr ¼ 0:3, Le ¼ 2, N ¼ 25 and Rn ¼ 5.

For low buoyancy ratio parameters, the flow structure is significantly influenced by the buoyancy within the whole enclosure. Increasing the buoyancy ratio causes the boundary layer thickness to Three different types of eddies are observed for the isoconcentration contours when N ¼ 25. Of these, two have a clockwise rotation and one is anticlockwise. It is seen that the small eddy at the right bottom edge is diminished as N decreases from 1 to �25. Here, the concentration boundary layer decreases due to increasing N values, hence the buoyancy ratio has a significant influence on the concentration gradient. As the buoyancy ratio N increases from 1 to 25 the isoconcentrations become very dense at the bottom of nanofluid layer.

Figure 12. The pattern of streamlines (top), isotherms (middle) and isoconcentration (bottom) for different values of the buoyancy ratio N.

The effect of the Darcy number on the nanofluid flow in the porous medium is shown in detail in Figure 13. The streamline patterns are similar to those depicted in Figure 12. However, as Da increases from 0.05 to 0.07, the rotation of the streamlines changes. Similarly, the isotherm patterns change with increasing Darcy numbers. The value of the center eddies increases with increasing Da. Increasing Da has the effect of increasing the effective fluid viscosity and reducing the thermal and solutal boundary layers.

number increases to 104

number Da.

Acknowledgements

eter increases the rate of mass transfer.

South Africa for financial support.

, the trajectories spiral many times before reaching a fixed point. The

, and the

nanofluid convection regime is complex for Rayleigh numbers higher than <sup>R</sup> <sup>¼</sup> 104

Additionally, a change in system parameters, such as an increase in the flow Lewis number, improves the rate of heat and mass transfer in the nanofluid saturated porous media. The Dufour parameter has the effect of increasing heat transfer, while increasing the Soret param-

Figure 13. The streamlines (top), isotherms (middle) and isoconcentration (bottom) for different values of the Darcy

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

http://dx.doi.org/10.5772/intechopen.71066

163

The authors are grateful to the University of KwaZulu-Natal and the Claude Leon Foundation,

flow pattern presents difficulties in interpreting correctly.

#### 7. Conclusion

We have investigated the onset of thermal instability in a horizontal porous layer of infinite extent in a cross-diffusive nanofluid flow. The focus of the study has been on stress free boundary conditions with zero nanoparticle flux at the wall. A multidomain spectral collocation method was used to solve the system of nonlinear evolution equations. As the Rayleigh Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral… http://dx.doi.org/10.5772/intechopen.71066 163

Figure 13. The streamlines (top), isotherms (middle) and isoconcentration (bottom) for different values of the Darcy number Da.

number increases to 104 , the trajectories spiral many times before reaching a fixed point. The nanofluid convection regime is complex for Rayleigh numbers higher than <sup>R</sup> <sup>¼</sup> 104 , and the flow pattern presents difficulties in interpreting correctly.

Additionally, a change in system parameters, such as an increase in the flow Lewis number, improves the rate of heat and mass transfer in the nanofluid saturated porous media. The Dufour parameter has the effect of increasing heat transfer, while increasing the Soret parameter increases the rate of mass transfer.

#### Acknowledgements

The effect of the Darcy number on the nanofluid flow in the porous medium is shown in detail in Figure 13. The streamline patterns are similar to those depicted in Figure 12. However, as Da increases from 0.05 to 0.07, the rotation of the streamlines changes. Similarly, the isotherm patterns change with increasing Darcy numbers. The value of the center eddies increases with increasing Da. Increasing Da has the effect of increasing the effective fluid viscosity and reducing

Figure 12. The pattern of streamlines (top), isotherms (middle) and isoconcentration (bottom) for different values of the

We have investigated the onset of thermal instability in a horizontal porous layer of infinite extent in a cross-diffusive nanofluid flow. The focus of the study has been on stress free boundary conditions with zero nanoparticle flux at the wall. A multidomain spectral collocation method was used to solve the system of nonlinear evolution equations. As the Rayleigh

the thermal and solutal boundary layers.

162 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

7. Conclusion

buoyancy ratio N.

The authors are grateful to the University of KwaZulu-Natal and the Claude Leon Foundation, South Africa for financial support.

#### A. Appendix

The terms ai ð Þ <sup>j</sup>;<sup>n</sup> <sup>r</sup> and Ri jr for j ¼ 1, 2, …, 7 in Eq. (38) are given by

ai ð Þ <sup>1</sup>;<sup>1</sup> <sup>r</sup> ¼ �γ3B, ai ð Þ <sup>1</sup>;<sup>2</sup> <sup>r</sup> ¼ �γ3, a<sup>i</sup> ð Þ <sup>1</sup>;<sup>4</sup> <sup>r</sup> ¼ �γ3N, ai ð Þ <sup>1</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup>3NARn Ra , ai ð Þ <sup>1</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>1</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>1</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai ð Þ <sup>2</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>R</sup> � <sup>Y</sup><sup>i</sup> 3,r, ai ð Þ <sup>2</sup>;<sup>2</sup> <sup>r</sup> ¼ �1, ai ð Þ <sup>2</sup>;<sup>3</sup> <sup>r</sup> ¼ �Y<sup>i</sup> 1,r, ai ð Þ <sup>2</sup>;<sup>4</sup> <sup>r</sup> ¼ �Du, ai ð Þ <sup>2</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>2</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>2</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>ai ð Þ <sup>3</sup>;<sup>1</sup> <sup>r</sup> ¼ � απ 2 ui 2,r, ai ð Þ <sup>3</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,r, ai ð Þ <sup>3</sup>;<sup>3</sup> <sup>r</sup> ¼ �G, ai ð Þ <sup>3</sup>;<sup>5</sup> <sup>r</sup> ¼ �GDua<sup>i</sup> ð Þ <sup>3</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> <sup>a</sup><sup>i</sup> ð Þ <sup>3</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>3</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai ð Þ <sup>4</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>R</sup> � <sup>Y</sup><sup>i</sup> 5,r, ai ð Þ <sup>4</sup>;<sup>2</sup> <sup>r</sup> ¼ �Sr, ai ð Þ <sup>5</sup>;<sup>3</sup> <sup>r</sup> ¼ �GSr ai ð Þ <sup>4</sup>;<sup>4</sup> <sup>r</sup> ¼ � <sup>1</sup> Les, ai ð Þ <sup>4</sup>;<sup>5</sup> <sup>r</sup> ¼ �Y<sup>i</sup> 1,r, ai ð Þ <sup>4</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>4</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>4</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai ð Þ <sup>5</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 4,r, ai ð Þ <sup>5</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,r, ai ð Þ <sup>5</sup>;<sup>5</sup> <sup>r</sup> ¼ � <sup>G</sup> Les, ai ð Þ <sup>5</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>5</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>5</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, a<sup>i</sup> ð Þ <sup>6</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup><sup>1</sup> � <sup>γ</sup>2Y<sup>i</sup> 7,r, ai ð Þ <sup>6</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> <sup>σ</sup> Le, a<sup>i</sup> ð Þ <sup>6</sup>;<sup>6</sup> <sup>r</sup> ¼ � <sup>σ</sup> Le, ai ð Þ <sup>6</sup>;<sup>7</sup> <sup>r</sup> ¼ �γ2Y<sup>i</sup> 1,r, a<sup>i</sup> ð Þ <sup>6</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>6</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> <sup>a</sup><sup>i</sup> ð Þ <sup>6</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai ð Þ <sup>7</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup> 2 Yi 6,r, ai ð Þ <sup>7</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> <sup>G</sup> Le, a<sup>i</sup> ð Þ <sup>7</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup> 2 Yi 1,r, ai ð Þ <sup>7</sup>;<sup>7</sup> <sup>r</sup> ¼ � <sup>G</sup> Le, ai ð Þ <sup>7</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>7</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> ai ð Þ <sup>7</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, Ri <sup>1</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup>, Ri <sup>2</sup><sup>r</sup> ¼ �Y<sup>i</sup> 1,rY<sup>i</sup> 3,rRi <sup>3</sup><sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,rY<sup>i</sup> 2,r, R<sup>i</sup> <sup>4</sup><sup>r</sup> ¼ �Y<sup>i</sup> 1,rY<sup>i</sup> 5,r, R<sup>i</sup> <sup>5</sup><sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,rY<sup>i</sup> 4,r, Ri <sup>6</sup><sup>r</sup> ¼ � <sup>σ</sup>NA ε Yi 1,rY<sup>i</sup> 7,r, Ri <sup>7</sup><sup>r</sup> <sup>¼</sup> <sup>σ</sup>NA 2ε Yi 1,rY<sup>i</sup> 6,r,

<sup>A</sup><sup>71</sup> <sup>¼</sup> diag ai

Author details

References

240-250

2012;48(5):863-874

8162-8178

425-436

Osman A.I. Noreldin<sup>1</sup>

matrix.

ð Þ 7;1 r

, A<sup>73</sup> <sup>¼</sup> diag ai

, Precious Sibanda1

\*Address all correspondence to: sibandap@ukzn.ac.za

ð Þ 7;3 r

1 School of Mathematical Sciences, University of KwaZulu-Natal, South Africa 2 Department of Mathematics, Amity University, Kolkata, West Bengal, India

Technical Report. IL, United States: Argonne National Lab; 1995

by dispersing ultra-fine particles. Netsu Bussei. 1993;4:227-233

Heat and Mass Transfer. 2008;51(11):2967-2979

, A<sup>76</sup> <sup>¼</sup> diag ai

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

where O is an ð Þ� N þ 1 ð Þ N þ 1 matrix of zeros and diag is an ð Þ� N þ 1 ð Þ N þ 1 diagonal

[1] Choi SU, Eastman JA. Enhancing Thermal Conductivity of Fluids with Nanoparticles,

[2] Buongiorno J. Convective transport in nanofluids. Journal of Heat Transfer. 2006;128(3):

[3] Masuda H, Ebata A, Teramae K. Alteration of thermal conductivity and viscosity of liquid

[4] Tzou D. Thermal instability of nanofluids in natural convection. International Journal of

[5] Narayana M, Sibanda P, Motsa S, Lakshmi-Narayana P. Linear and nonlinear stability analysis of binary maxwell fluid convection in a porous medium. Heat and Mass Transfer.

[6] Narayana M, Sibanda P, Siddheshwar PG, Jayalatha G. Linear and nonlinear stability analysis of binary viscoelastic fluid convection. Applied Mathematical Modelling. 2013;37(16):

[7] Yadav D, Bhargava R, Agrawal G. Numerical solution of a thermal instability problem in a rotating nanofluid layer. International Journal of Heat and Mass Transfer. 2013;63:313-322

[8] Kuznetsov A, Nield D. Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transport in Porous Media. 2010;83(2):

[9] Kuznetsov A, Nield D. Thermal instability in a porous medium layer saturated by a

nanofluid: Brinkman model. Transport in Porous Media. 2010;81(3):409-422

\* and Sabyasachi Mondal<sup>2</sup>

ð Þ 7;6 r

, A<sup>72</sup> <sup>¼</sup> <sup>A</sup><sup>74</sup> <sup>¼</sup> <sup>A</sup><sup>75</sup> <sup>¼</sup> <sup>O</sup>,

http://dx.doi.org/10.5772/intechopen.71066

165

where <sup>γ</sup><sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>R</sup> <sup>ε</sup> , <sup>γ</sup><sup>2</sup> <sup>¼</sup> NA<sup>σ</sup> <sup>ε</sup> and <sup>γ</sup><sup>3</sup> <sup>¼</sup> Pr Da.

#### B. Appendix

The matrices Aij in Eq. (42) are given by

Ann ¼ diag an,nÞr <sup>i</sup> � <sup>D</sup>, A<sup>12</sup> <sup>¼</sup> diag ai ð Þ 1;2 r , A<sup>16</sup> <sup>¼</sup> diag <sup>a</sup><sup>i</sup> ð Þ 1;6 r , A<sup>13</sup> ¼ A<sup>14</sup> ¼ A<sup>15</sup> ¼ A<sup>17</sup> ¼ O, <sup>A</sup><sup>21</sup> <sup>¼</sup> diag ai ð Þ 2;1 r , A<sup>23</sup> <sup>¼</sup> diag ai ð Þ 2;3 r , A<sup>24</sup> <sup>¼</sup> diag ai ð Þ 2;4 r , A<sup>25</sup> <sup>¼</sup> <sup>A</sup><sup>26</sup> <sup>¼</sup> <sup>A</sup><sup>27</sup> <sup>¼</sup> <sup>O</sup>, <sup>A</sup><sup>31</sup> <sup>¼</sup> diag ai ð Þ 3;1 r , A<sup>32</sup> <sup>¼</sup> diag ai ð Þ 3;2 r , A<sup>35</sup> <sup>¼</sup> diag ai ð Þ 3;5 r , A<sup>34</sup> <sup>¼</sup> <sup>A</sup><sup>36</sup> <sup>¼</sup> <sup>A</sup><sup>37</sup> <sup>¼</sup> <sup>O</sup>, <sup>A</sup><sup>41</sup> <sup>¼</sup> diag ai ð Þ 4;1 r , A<sup>42</sup> <sup>¼</sup> diag ai ð Þ 4;2 r , A<sup>45</sup> <sup>¼</sup> diag ai ð Þ 4;5 r , A<sup>43</sup> <sup>¼</sup> <sup>A</sup><sup>46</sup> <sup>¼</sup> <sup>A</sup><sup>47</sup> <sup>¼</sup> <sup>O</sup>, <sup>A</sup><sup>51</sup> <sup>¼</sup> diag ai ð Þ 5;1 r , A<sup>53</sup> <sup>¼</sup> diag ai ð Þ 5;3 r , A<sup>54</sup> <sup>¼</sup> diag ai ð Þ 5;4 r , A<sup>52</sup> <sup>¼</sup> <sup>A</sup><sup>56</sup> <sup>¼</sup> <sup>A</sup><sup>57</sup> <sup>¼</sup> <sup>O</sup>, <sup>A</sup><sup>61</sup> <sup>¼</sup> diag ai ð Þ 6;1 r , A<sup>62</sup> <sup>¼</sup> diag ai ð Þ 6;2 r , A<sup>67</sup> <sup>¼</sup> diag ai ð Þ 6;7 r , A<sup>63</sup> <sup>¼</sup> <sup>A</sup><sup>64</sup> <sup>¼</sup> <sup>A</sup><sup>65</sup> <sup>¼</sup> <sup>O</sup>,

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral… http://dx.doi.org/10.5772/intechopen.71066 165

$$A\_{71} = \mathbf{diag}\left(a\_{(7,1)r}^i\right), \quad A\_{73} = \mathbf{diag}\left(a\_{(7,3)r}^i\right), \quad A\_{76} = \mathbf{diag}\left(a\_{(7,6)r}^i\right), \quad A\_{72} = A\_{74} = A\_{75} = \mathbf{O}\_i$$

where O is an ð Þ� N þ 1 ð Þ N þ 1 matrix of zeros and diag is an ð Þ� N þ 1 ð Þ N þ 1 diagonal matrix.

#### Author details

A. Appendix

ð Þ <sup>j</sup>;<sup>n</sup> <sup>r</sup> and Ri

ð Þ <sup>1</sup>;<sup>1</sup> <sup>r</sup> ¼ �γ3B, ai

ð Þ <sup>2</sup>;<sup>4</sup> <sup>r</sup> ¼ �Du, ai

ð Þ <sup>3</sup>;<sup>3</sup> <sup>r</sup> ¼ �G, ai

ð Þ <sup>4</sup>;<sup>2</sup> <sup>r</sup> ¼ �Sr, ai

2 Yi 1,r, ai

Le, a<sup>i</sup>

2 Yi 6,r, ai

ð Þ <sup>7</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> ai

ε Yi 1,rY<sup>i</sup> 7,r, Ri

<sup>2</sup><sup>r</sup> ¼ �Y<sup>i</sup>

<sup>ε</sup> and <sup>γ</sup><sup>3</sup> <sup>¼</sup> Pr

<sup>i</sup> � <sup>D</sup>, A<sup>12</sup> <sup>¼</sup> diag ai

, A<sup>23</sup> <sup>¼</sup> diag ai

, A<sup>32</sup> <sup>¼</sup> diag ai

, A<sup>42</sup> <sup>¼</sup> diag ai

, A<sup>53</sup> <sup>¼</sup> diag ai

, A<sup>62</sup> <sup>¼</sup> diag ai

ð Þ <sup>4</sup>;<sup>4</sup> <sup>r</sup> ¼ � <sup>1</sup>

ð Þ <sup>5</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup>

ð Þ <sup>6</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> <sup>σ</sup>

ð Þ <sup>7</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup>

ð Þ <sup>7</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> ai

<sup>1</sup><sup>r</sup> <sup>¼</sup> <sup>0</sup>, Ri

<sup>6</sup><sup>r</sup> ¼ � <sup>σ</sup>NA

<sup>ε</sup> , <sup>γ</sup><sup>2</sup> <sup>¼</sup> NA<sup>σ</sup>

The matrices Aij in Eq. (42) are given by

A<sup>13</sup> ¼ A<sup>14</sup> ¼ A<sup>15</sup> ¼ A<sup>17</sup> ¼ O,

ð Þ 2;1 r 

ð Þ 3;1 r 

ð Þ 4;1 r 

ð Þ 5;1 r 

ð Þ 6;1 r 

ð Þ <sup>1</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> ai

Les, ai

ð Þ <sup>1</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> ai

jr for j ¼ 1, 2, …, 7 in Eq. (38) are given by

ð Þ <sup>1</sup>;<sup>4</sup> <sup>r</sup> ¼ �γ3N, ai

3,r, ai

ð Þ <sup>3</sup>;<sup>1</sup> <sup>r</sup> ¼ � απ

ð Þ <sup>2</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>R</sup> � <sup>Y</sup><sup>i</sup>

ð Þ <sup>3</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> <sup>a</sup><sup>i</sup>

ð Þ <sup>4</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> ai

ð Þ <sup>5</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> ai

ð Þ <sup>6</sup>;<sup>7</sup> <sup>r</sup> ¼ �γ2Y<sup>i</sup>

2 Yi 1,r, ai

ð Þ <sup>7</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup>

<sup>7</sup><sup>r</sup> <sup>¼</sup> <sup>σ</sup>NA 2ε Yi 1,rY<sup>i</sup> 6,r,

> ð Þ 1;2 r

ð Þ 2;3 r 

ð Þ 3;2 r 

ð Þ 4;2 r 

ð Þ 5;3 r 

ð Þ 6;2 r 

ð Þ <sup>2</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>ai

ð Þ <sup>3</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai

ð Þ <sup>4</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai

ð Þ <sup>5</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai

1,r, a<sup>i</sup>

ð Þ <sup>1</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup>3NARn

ð Þ <sup>2</sup>;<sup>2</sup> <sup>r</sup> ¼ �1, ai

2 ui 2,r, ai

ð Þ <sup>4</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai

ð Þ <sup>6</sup>;<sup>4</sup> <sup>r</sup> <sup>¼</sup> <sup>a</sup><sup>i</sup>

Le,

1,rY<sup>i</sup> 5,r, R<sup>i</sup>

ð Þ 1;6 r ,

ð Þ 2;4 r 

ð Þ 3;5 r 

ð Þ 4;5 r 

ð Þ 5;4 r 

ð Þ 6;7 r 

ð Þ <sup>5</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, a<sup>i</sup>

ð Þ <sup>6</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> ai

ð Þ <sup>7</sup>;<sup>7</sup> <sup>r</sup> ¼ � <sup>G</sup>

<sup>4</sup><sup>r</sup> ¼ �Y<sup>i</sup>

, A<sup>16</sup> <sup>¼</sup> diag <sup>a</sup><sup>i</sup>

, A<sup>24</sup> <sup>¼</sup> diag ai

, A<sup>35</sup> <sup>¼</sup> diag ai

, A<sup>45</sup> <sup>¼</sup> diag ai

, A<sup>54</sup> <sup>¼</sup> diag ai

, A<sup>67</sup> <sup>¼</sup> diag ai

ð Þ <sup>3</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai

Ra ,

ð Þ <sup>2</sup>;<sup>3</sup> <sup>r</sup> ¼ �Y<sup>i</sup>

5,r,

ð Þ <sup>3</sup>;<sup>2</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,r,

ð Þ <sup>4</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>R</sup> � <sup>Y</sup><sup>i</sup>

ð Þ <sup>5</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 4,r,

ð Þ <sup>6</sup>;<sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>γ</sup><sup>1</sup> � <sup>γ</sup>2Y<sup>i</sup>

ð Þ <sup>6</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>,

<sup>5</sup><sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,rY<sup>i</sup> 4,r,

, A<sup>25</sup> ¼ A<sup>26</sup> ¼ A<sup>27</sup> ¼ O,

, A<sup>34</sup> ¼ A<sup>36</sup> ¼ A<sup>37</sup> ¼ O,

, A<sup>43</sup> ¼ A<sup>46</sup> ¼ A<sup>47</sup> ¼ O,

, A<sup>52</sup> ¼ A<sup>56</sup> ¼ A<sup>57</sup> ¼ O,

, A<sup>63</sup> ¼ A<sup>64</sup> ¼ A<sup>65</sup> ¼ O,

1,r,

7,r,

ð Þ <sup>1</sup>;<sup>2</sup> <sup>r</sup> ¼ �γ3, a<sup>i</sup>

ð Þ <sup>1</sup>;<sup>7</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ai

ð Þ <sup>2</sup>;<sup>6</sup> <sup>r</sup> <sup>¼</sup> ai

1,r, ai

Les, ai

Le, ai

Le, a<sup>i</sup>

Da.

ð Þ <sup>2</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> ai

164 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

ð Þ <sup>3</sup>;<sup>5</sup> <sup>r</sup> ¼ �GDua<sup>i</sup>

ð Þ <sup>5</sup>;<sup>3</sup> <sup>r</sup> ¼ �GSr

ð Þ <sup>4</sup>;<sup>5</sup> <sup>r</sup> ¼ �Y<sup>i</sup>

ð Þ <sup>5</sup>;<sup>5</sup> <sup>r</sup> ¼ � <sup>G</sup>

ð Þ <sup>7</sup>;<sup>3</sup> <sup>r</sup> <sup>¼</sup> <sup>G</sup>

ð Þ <sup>7</sup>;<sup>5</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>,

1,rY<sup>i</sup> 3,rRi <sup>3</sup><sup>r</sup> <sup>¼</sup> <sup>1</sup> 2 Yi 1,rY<sup>i</sup> 2,r, R<sup>i</sup>

ð Þ <sup>6</sup>;<sup>6</sup> <sup>r</sup> ¼ � <sup>σ</sup>

The terms ai

ai

ai

ai

ai

ai

ai

ai

ai

ai

ai

Ri

Ri

where <sup>γ</sup><sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>R</sup>

B. Appendix

Ann ¼ diag an,nÞr

<sup>A</sup><sup>21</sup> <sup>¼</sup> diag ai

<sup>A</sup><sup>31</sup> <sup>¼</sup> diag ai

<sup>A</sup><sup>41</sup> <sup>¼</sup> diag ai

<sup>A</sup><sup>51</sup> <sup>¼</sup> diag ai

<sup>A</sup><sup>61</sup> <sup>¼</sup> diag ai

Osman A.I. Noreldin<sup>1</sup> , Precious Sibanda1 \* and Sabyasachi Mondal<sup>2</sup>

\*Address all correspondence to: sibandap@ukzn.ac.za


#### References


[10] Nield D, Kuznetsov A. The onset of double-diffusive convection in a nanofluid layer. International Journal of Heat and Fluid Flow. 2011;32(4):771-776

[25] Rana G, Kango S, Chadha K. Magneto-thermal convection in Walters' (model b') elasticoviscous fluid saturated by a Darcy-Brinkman porous medium. Engineering Mechanics.

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral…

http://dx.doi.org/10.5772/intechopen.71066

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[26] Motsa S, Dlamini P, Khumalo M. A new multistage spectral relaxation method for solving chaotic initial value systems. Nonlinear Dynamics. 2013;72(1–2):265-283

2014;21:425-435


[25] Rana G, Kango S, Chadha K. Magneto-thermal convection in Walters' (model b') elasticoviscous fluid saturated by a Darcy-Brinkman porous medium. Engineering Mechanics. 2014;21:425-435

[10] Nield D, Kuznetsov A. The onset of double-diffusive convection in a nanofluid layer.

[11] Nield D, Kuznetsov A. Thermal instability in a porous medium layer saturated by a nanofluid: A revised model. International Journal of Heat and Mass Transfer. 2014;68:211-214

[12] Bhadauria B. Double diffusive convection in a porous medium with modulated temper-

[13] Bhadauria B, Siddheshwar P, Kumar J, Suthar OP. Weakly nonlinear stability analysis of temperature/gravity-modulated stationary rayleigh–bénard convection in a rotating

[14] Gresho P, Sani R. The effects of gravity modulation on the stability of a heated fluid layer.

[15] Kiran P, Bhadauria B. Chaotic convection in a porous medium under temperature mod-

[16] Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Oxford University Press;

[17] Thompson W. Cxliii. Thermal convection in a magnetic field. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1951;42(335):1417-1432 [18] Heris SZ, Salehi H, Noie S. The effect of magnetic field and nanofluid on thermal performance of two-phase closed thermosyphon (tpct). International Journal of Physical Sci-

[19] Ghasemi B, Aminossadati S, Raisi A. Magnetic field effect on natural convection in a nanofluid-filled square enclosure. International Journal of Thermal Sciences. 2011;50(9):

[20] Hamad M, Pop I, Ismail AM. Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate. Nonlinear Analysis: Real World Applications.

[21] Bergman MI, Fearn DR. Chimneys on the earth's inner-outer core boundary? Geophysi-

[22] Furmanski P, Banaszek J. Modelling of the mushy zone permeability for solidification of

[23] Gupta U, Ahuja J, Wanchoo R. Rayleigh-bénard convection of nanofluids with magnetic

[24] Imomnazarov KK. Modified darcy laws for conducting porous media. Mathematical and

binary alloys. Materials Science Forum. Trans Tech Publ. 2006;508:411-418

field and permeability effects. Procedia Engineering. 2015;127:325-332

ature on the boundaries. Transport in Porous Media. 2007;70(2):191-211

International Journal of Heat and Fluid Flow. 2011;32(4):771-776

166 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

porous medium. Transport in Porous Media. 2012;92(3):633-647

Journal of Fluid Mechanics. 1970;40(4):783-806

1961

1748-1756

ences. 2012;7(4):534-543

2011;12(3):1338-1346

cal Research Letters. 1994;21(6):477-480

Computer Modelling. 2004;40(1–2):5-10

ulation. Transport in Porous Media. 2015;107(3):745-763

[26] Motsa S, Dlamini P, Khumalo M. A new multistage spectral relaxation method for solving chaotic initial value systems. Nonlinear Dynamics. 2013;72(1–2):265-283

**Chapter 10**

Provisional chapter

**Small-Angle Scattering from Mass and Surface Fractals**

DOI: 10.5772/intechopen.70870

Small-Angle Scattering from Mass and Surface Fractals

The concepts of mass and surface fractals are introduced, and the corresponding smallangle scattering (SAS; X-rays, neutrons) intensities are computed. It is shown how to resolve the fractal structure of various complex systems from experimental scattering measurements, and how obtained data are related to specific features of the fractal models. We present and discuss various mass and surface fractal structures, including fractals generated from iterated function systems and cellular automata. In addition to the fractal dimension and the overall fractal size, the suggested analysis allows us to obtain the iteration number, the number of basic units which form the fractal and the scaling factor.

Keywords: small-angle scattering, form factor, structure factor, fractals, iterated

A great number of natural systems provide us with examples of nano- and micro- structures, which appear similar under a change of scale. These structures are called fractals [1], and can be observed in various disordered materials, rough surfaces, aggregates, metals, polymers, gels, colloids, thin films, etc. Quite often, the physical properties (mechanical, optical, statistical, thermodynamical, etc.) depend on their spatial configurations, and a great deal of activity has been performed in elucidating such correlations [2–4]. To this aim, in the last decade, important steps have been performed in three directions: instrumentation [5–7], computer programs [8, 9], and the development of new methods for sample preparation [10–14]. The first two have been proved to be very useful for the physical execution and data analysis, and the third one for

preparation of nano- and micro-materials with the pre-defined structure and functions.

These achievements have stimulated a great interest in the development of theoretical investigations for structural modeling of self-similar objects at nano- and micro-scale. In particular, models based on deterministic or exact self-similar fractals (i.e., fractals that are self-similar at every point, such as the Koch snowflake, Cantor set, or Mandelbrot cube) have been frequently

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Eugen Mircea Anitas

Eugen Mircea Anitas

Abstract

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70870

function system, cellular automata

#### **Small-Angle Scattering from Mass and Surface Fractals** Small-Angle Scattering from Mass and Surface Fractals

DOI: 10.5772/intechopen.70870

#### Eugen Mircea Anitas Eugen Mircea Anitas

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70870

#### Abstract

The concepts of mass and surface fractals are introduced, and the corresponding smallangle scattering (SAS; X-rays, neutrons) intensities are computed. It is shown how to resolve the fractal structure of various complex systems from experimental scattering measurements, and how obtained data are related to specific features of the fractal models. We present and discuss various mass and surface fractal structures, including fractals generated from iterated function systems and cellular automata. In addition to the fractal dimension and the overall fractal size, the suggested analysis allows us to obtain the iteration number, the number of basic units which form the fractal and the scaling factor.

Keywords: small-angle scattering, form factor, structure factor, fractals, iterated function system, cellular automata

#### 1. Introduction

A great number of natural systems provide us with examples of nano- and micro- structures, which appear similar under a change of scale. These structures are called fractals [1], and can be observed in various disordered materials, rough surfaces, aggregates, metals, polymers, gels, colloids, thin films, etc. Quite often, the physical properties (mechanical, optical, statistical, thermodynamical, etc.) depend on their spatial configurations, and a great deal of activity has been performed in elucidating such correlations [2–4]. To this aim, in the last decade, important steps have been performed in three directions: instrumentation [5–7], computer programs [8, 9], and the development of new methods for sample preparation [10–14]. The first two have been proved to be very useful for the physical execution and data analysis, and the third one for preparation of nano- and micro-materials with the pre-defined structure and functions.

These achievements have stimulated a great interest in the development of theoretical investigations for structural modeling of self-similar objects at nano- and micro-scale. In particular, models based on deterministic or exact self-similar fractals (i.e., fractals that are self-similar at every point, such as the Koch snowflake, Cantor set, or Mandelbrot cube) have been frequently

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

used, since this type of fractals allows an analytical representation of various geometrical parameters (radius of gyration) or of the scattering intensity spectrum. Although for most fractals generated by natural processes, this is only an approximation, in the case of deterministic nano- and micro-materials obtained recently such as 2D Sierpinski gaskets [15] and Cantor sets [16], or 3D Menger sponge [17] and octahedral structures [18], this approximation becomes exact.

This chapter focuses on the interpretation of SAXS/SANS data from deterministic mass and surface fractals. First, a brief theoretical background on the basics of SAS theory, and on description of mass and surface fractals, is presented. Here, we also include the theory of some well-known methods of generating fractals, such as iterated function system of cellular automata. Novel data analysis methods for extracting additional structural information are presented

In this section, some important concepts for the analysis of SAXS/SANS are reviewed, and analytical and numerical procedures for calculating the scattering intensity from some basic geometrical shapes are described. As we will see in the next section, these geometrical shapes will form the "scattering units" of the fractals. Then, the basic notions of fractal theory including fractal dimension, mass, and surface fractals are presented and defined in a rigorous manner, and two general methods for generating fractal structures are presented. These concepts are then applied in calculating the SAXS/SANS patterns from several theoretical models

In a SAS experiment, a beam of X-rays or neutrons is emitted from a source and strikes the sample. A small fraction is scattered by the sample and is recorded by the detector. In Figure 1, the incident beam has a wave vector ki, and the scattered beam with the wave vector k<sup>s</sup> makes

To describe the scattering from assemblies of objects with scattering length bj, we write the

where v is the total volume irradiated by the beam. In the following, we consider scattering occurring in a particulate system where particles of density ρ<sup>m</sup> are dispersed in a uniform solid matrix of density ρp. Then, the excess scattering SLD is defined by Δρ = ρ<sup>m</sup> � ρp. We also consider that the objects are fractals that are randomly distributed and with uncorrelated positions and orientations. Thus, the scattering intensities of each object are added, and the intensity from the entire sample can be obtained from a single object averaged over all

!

!� �:

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 171

<sup>j</sup> is the object positions. Then, the

r, (1)

the angle 2θ with the direction of the incoming or transmitted beam.

total scattering amplitude is defined by the Fourier transform of ρ r

Að Þ� q

ð v ρð Þr e �iq:r d3

scattering length density SLD as ρ(r) = ∑jbj(r � rj) [21], where r

and illustrated by applications to various models of mass and surface fractals.

2. Theoretical background

of mass and surface fractals.

2.1. Small-angle scattering

orientations, according to [29]:

2.1.1. General remarks

Small-angle scattering of X-rays (SAXS) and/or neutrons (SANS) are well established techniques for probing the nano/micro scale structure in disordered materials [19–21]. While in the case of X-rays, the scattering is mostly determined by the interaction of the incident radiation with electrons, in the case of neutrons, the scattering is determined by their interaction with the atomic nuclei and with the magnetic moments in magnetic materials. Since the wavelengths of X-rays (0.5–2 Å) are of the same order of magnitude as those of thermal neutrons (1–10 Å), often, the data analysis and interpretation procedures for SANS can be interchanged with SAXS, and the developed theoretical models can be applied, generally, to both techniques [22]. However, using neutrons is very important in studying magnetic properties of materials as well as in emphasizing or concealing certain features of the investigated sample [23]. In the later case, the possibility of wide variations in the neutron scattering lengths (which can be negative sometimes) is exploited, and this is a unique feature of SANS, which makes it a preferred method over SAXS in structural analysis of biological materials [24, 25].

As compared with other methods of structural investigations, SAXS/SANS have the advantage that they are noninvasive, the physical quantities of interest (specific surface, radius of gyration, volume, or the fractal dimension) are averaged over a macroscopic volume and they can be extracted with almost no approximation [26]. In particular, for self-similar objects (either exact or statistical), the most important advantage is that SAXS/SANS can distinguish between mass [27] and surface fractals [28]. Experimentally, the difference is revealed through the value of the scattering exponent τ in the region where the scattering intensity I(q) decays as a powerlaw, i.e., I(q)∝ q<sup>τ</sup> , where q = (4π/λ) sin θ is the scattering vector, λ is the wavelength of the incident radiation, and 2θ is the scattering angle. For mass fractals τ = Dm, where D<sup>m</sup> is the mass fractal dimension with 0 < D<sup>m</sup> < 3. For surface fractals τ = 6 Ds, where D<sup>s</sup> is the surface fractal dimension with 2 < D<sup>s</sup> < 3. Thus, in practice if the absolute value of the measured scattering exponent is smaller than 3, the sample is a mass fractal with fractal dimension τ (in the measured q-range), and if the exponent is between 3 and 4, the sample is a surface fractal with fractal dimension 6 τ.

Besides the fractal dimension (either mass or surface one), traditionally from SAXS/SANS patterns, we can also obtain the overall size of the fractal, as well as the size of the basic structural units composing the fractal. The last years have brought a breakthrough in the theoretical analysis of SAXS/SANS experimental data, allowing for the extraction of additional structural information and detailed modeling of fractals using a deterministic approach [29–40]. This progress was stimulated by recent advances in nanotechnology, which allows preparation of both mass and surface deterministic fractals at sub-micrometer scale [15–18, 41, 42], as well as by instrumentation which allows novel structural features to be recorded in experimental data [43].

This chapter focuses on the interpretation of SAXS/SANS data from deterministic mass and surface fractals. First, a brief theoretical background on the basics of SAS theory, and on description of mass and surface fractals, is presented. Here, we also include the theory of some well-known methods of generating fractals, such as iterated function system of cellular automata. Novel data analysis methods for extracting additional structural information are presented and illustrated by applications to various models of mass and surface fractals.

#### 2. Theoretical background

used, since this type of fractals allows an analytical representation of various geometrical parameters (radius of gyration) or of the scattering intensity spectrum. Although for most fractals generated by natural processes, this is only an approximation, in the case of deterministic nano- and micro-materials obtained recently such as 2D Sierpinski gaskets [15] and Cantor sets [16], or 3D Menger sponge [17] and octahedral structures [18], this approximation

170 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Small-angle scattering of X-rays (SAXS) and/or neutrons (SANS) are well established techniques for probing the nano/micro scale structure in disordered materials [19–21]. While in the case of X-rays, the scattering is mostly determined by the interaction of the incident radiation with electrons, in the case of neutrons, the scattering is determined by their interaction with the atomic nuclei and with the magnetic moments in magnetic materials. Since the wavelengths of X-rays (0.5–2 Å) are of the same order of magnitude as those of thermal neutrons (1–10 Å), often, the data analysis and interpretation procedures for SANS can be interchanged with SAXS, and the developed theoretical models can be applied, generally, to both techniques [22]. However, using neutrons is very important in studying magnetic properties of materials as well as in emphasizing or concealing certain features of the investigated sample [23]. In the later case, the possibility of wide variations in the neutron scattering lengths (which can be negative sometimes) is exploited, and this is a unique feature of SANS, which makes it a preferred method over SAXS in structural analysis of biological materials [24, 25]. As compared with other methods of structural investigations, SAXS/SANS have the advantage that they are noninvasive, the physical quantities of interest (specific surface, radius of gyration, volume, or the fractal dimension) are averaged over a macroscopic volume and they can be extracted with almost no approximation [26]. In particular, for self-similar objects (either exact or statistical), the most important advantage is that SAXS/SANS can distinguish between mass [27] and surface fractals [28]. Experimentally, the difference is revealed through the value of the scattering exponent τ in the region where the scattering intensity I(q) decays as a power-

, where q = (4π/λ) sin θ is the scattering vector, λ is the wavelength of the

incident radiation, and 2θ is the scattering angle. For mass fractals τ = Dm, where D<sup>m</sup> is the mass fractal dimension with 0 < D<sup>m</sup> < 3. For surface fractals τ = 6 Ds, where D<sup>s</sup> is the surface fractal dimension with 2 < D<sup>s</sup> < 3. Thus, in practice if the absolute value of the measured scattering exponent is smaller than 3, the sample is a mass fractal with fractal dimension τ (in the measured q-range), and if the exponent is between 3 and 4, the sample is a surface fractal

Besides the fractal dimension (either mass or surface one), traditionally from SAXS/SANS patterns, we can also obtain the overall size of the fractal, as well as the size of the basic structural units composing the fractal. The last years have brought a breakthrough in the theoretical analysis of SAXS/SANS experimental data, allowing for the extraction of additional structural information and detailed modeling of fractals using a deterministic approach [29–40]. This progress was stimulated by recent advances in nanotechnology, which allows preparation of both mass and surface deterministic fractals at sub-micrometer scale [15–18, 41, 42], as well as by instrumentation which allows novel structural features to be recorded in experimental data [43].

becomes exact.

law, i.e., I(q)∝ q<sup>τ</sup>

with fractal dimension 6 τ.

In this section, some important concepts for the analysis of SAXS/SANS are reviewed, and analytical and numerical procedures for calculating the scattering intensity from some basic geometrical shapes are described. As we will see in the next section, these geometrical shapes will form the "scattering units" of the fractals. Then, the basic notions of fractal theory including fractal dimension, mass, and surface fractals are presented and defined in a rigorous manner, and two general methods for generating fractal structures are presented. These concepts are then applied in calculating the SAXS/SANS patterns from several theoretical models of mass and surface fractals.

#### 2.1. Small-angle scattering

#### 2.1.1. General remarks

In a SAS experiment, a beam of X-rays or neutrons is emitted from a source and strikes the sample. A small fraction is scattered by the sample and is recorded by the detector. In Figure 1, the incident beam has a wave vector ki, and the scattered beam with the wave vector k<sup>s</sup> makes the angle 2θ with the direction of the incoming or transmitted beam.

To describe the scattering from assemblies of objects with scattering length bj, we write the scattering length density SLD as ρ(r) = ∑jbj(r � rj) [21], where r ! <sup>j</sup> is the object positions. Then, the total scattering amplitude is defined by the Fourier transform of ρ r !� �:

$$A(\boldsymbol{\eta}) \equiv \int\_{\boldsymbol{v}} \rho(\boldsymbol{r}) e^{-i\boldsymbol{q}\cdot\boldsymbol{r}} \mathbf{d}^3 \boldsymbol{r},\tag{1}$$

where v is the total volume irradiated by the beam. In the following, we consider scattering occurring in a particulate system where particles of density ρ<sup>m</sup> are dispersed in a uniform solid matrix of density ρp. Then, the excess scattering SLD is defined by Δρ = ρ<sup>m</sup> � ρp. We also consider that the objects are fractals that are randomly distributed and with uncorrelated positions and orientations. Thus, the scattering intensities of each object are added, and the intensity from the entire sample can be obtained from a single object averaged over all orientations, according to [29]:

$$I(\mathfrak{q}) = n \left| \Delta \mathfrak{p} \right|^2 V^2 \left< \left| F(\mathfrak{q}) \right|^2 \right>\tag{2}$$

At low values of the scattering vector (q≲2π/D), the above expression can be further exploited.

and then using only the first two terms of this approximation into Eq. (5), the Guinier equation

is the radius of gyration of the object. In practice, a plot of logI(q) vs. q<sup>2</sup> is used to obtain the

However, as we shall see in the next sections, inside a fractal, the scattering units have defined positions and correlations, and the interference among rays scattered by different units may no longer be ignored, so that the scattering amplitudes of individual units have to be added. By considering that the fractal is composed of N balls of size R, its form factor becomes [29]:

I qð Þ¼ <sup>I</sup>ð Þ<sup>0</sup> S qð Þj j <sup>F</sup>0ð Þ qR <sup>2</sup>

A physical sample almost always consists of fractals that have different sizes, which is called polydispersity. An exception to this rule is protein solutions, in which all have the same size and shape. Thus, the corresponding scattering intensity from polydisperse fractals can be regarded as the sum of each individual form factor weighted with the corresponding volume V and contrast Δρ. We consider here a continuous distribution DN(l) of fractals with different sizes l, defined in such a way that DN(l)dl gives the probability of finding a fractal with

<sup>g</sup>=3, and then an overall size of the object. For a ball of radius R, it is known that

5=3 p [21]. If a scattering experiment is performed on an absolute scale, molecular

<sup>e</sup>�<sup>i</sup>q. <sup>r</sup><sup>j</sup> is the Fourier component of the density of units centers, <sup>F</sup><sup>0</sup> is the form

!

<sup>6</sup> <sup>þ</sup> <sup>q</sup><sup>4</sup>r<sup>4</sup>

g <sup>3</sup> <sup>þ</sup> <sup>⋯</sup> !

<sup>120</sup> � <sup>⋯</sup>, (6)

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 173

, (7)

dr, (8)

<sup>j</sup> are their positions. By introducing

=N, (10)

Fð Þ¼ q ρqF0ð Þ qR =N, (9)

D E=<sup>N</sup> is the structure factor and it describes the corre-

sin ð Þ qr

<sup>R</sup><sup>g</sup> <sup>¼</sup> <sup>1</sup> 2 ð<sup>D</sup> 0 γð Þr r 4 dr= ð<sup>D</sup> 0 γð Þr r 2

qr <sup>≃</sup> <sup>1</sup> � <sup>q</sup><sup>2</sup>r<sup>2</sup>

I qð Þ¼ <sup>I</sup>ð Þ<sup>0</sup> <sup>1</sup> � <sup>q</sup><sup>2</sup>R<sup>2</sup>

By considering first the MacLaurin series

<sup>0</sup> <sup>γ</sup>ð Þ<sup>r</sup> <sup>r</sup>2dr, and

is obtained:

slope R<sup>2</sup>

R ¼ R<sup>g</sup>

where <sup>ρ</sup><sup>q</sup> <sup>¼</sup> <sup>P</sup>

where I(0) = n|Δρ|2

where <sup>I</sup>ð Þ¼ <sup>0</sup> <sup>4</sup><sup>π</sup> <sup>Ð</sup> <sup>D</sup>

ffiffiffiffiffiffiffi

weight can also be obtained.

j

V2

factor of each scattering unit composing the fractal, and r

Eq. (9) into Eq. (2), the scattering intensity can be written as [29]:

, and S qð Þ� ρqρ-<sup>q</sup>

lations between the scattering units inside the fractal.

where <sup>n</sup> is the concentration of objects, <sup>V</sup> is the volume of each object, and F q!� � is the normalized scattering amplitude given by:

$$F(\mathbf{q}) = \frac{1}{V} \int\_{V} e^{-i\mathbf{q}\cdot\mathbf{r}} \mathbf{d}^3 r. \tag{3}$$

The symbol 〈⋯〉 stands for ensemble averaging over all orientations, and for an arbitrary function f, it is calculated according to:

$$
\left< f\left(q\_x, q\_y, q\_z\right) \right> = \frac{1}{4\pi} \int\_0^\pi \mathbf{d}\theta \sin\theta \int\_0^{2\pi} \mathbf{d}\phi f(q, \theta, \phi),\tag{4}
$$

where, in spherical coordinates qx = q cos ϕ sin θ, qy = q sin ϕ sin θ, qz = q cos θ.

Another useful form of the scattering intensity, as a function of the correlation function, is the following [21]:

$$I(q) = 4\pi \int\_0^D \chi(r) \frac{\sin qr}{qr} r^2 \mathrm{d}r,\tag{5}$$

where γ(r) � 〈ρ(r) ∗ ρ(�r)〉 is the correlation function of the object, with γ(r) = 0 for r > D and D is the largest dimension in the object. The symbol ∗ denotes a convolution, and thus, the correlation function can be seen as an averaged self-convolution of density distribution.

Figure 1. Schematic representation of a small-angle scattering experiment. 2θ is the scattering angle and λ is the wavelength of incident beam. The sample shown is a two-phase system of polydisperse scatterers with the same shape and random orientations, embedded in a matrix or solution.

At low values of the scattering vector (q≲2π/D), the above expression can be further exploited. By considering first the MacLaurin series

$$\frac{\sin\left(qr\right)}{qr} \simeq 1 - \frac{q^2r^2}{6} + \frac{q^4r^4}{120} - \dotsb,\tag{6}$$

and then using only the first two terms of this approximation into Eq. (5), the Guinier equation is obtained:

$$I(q) = I(0)\left(1 - \frac{q^2 R\_{\rm g}^2}{3} + \cdots\right),\tag{7}$$

where <sup>I</sup>ð Þ¼ <sup>0</sup> <sup>4</sup><sup>π</sup> <sup>Ð</sup> <sup>D</sup> <sup>0</sup> <sup>γ</sup>ð Þ<sup>r</sup> <sup>r</sup>2dr, and

I qð Þ¼ <sup>n</sup> <sup>Δ</sup><sup>ρ</sup> � � � � 2

Fð Þ¼ q

¼ 1 4π ðπ 0

where, in spherical coordinates qx = q cos ϕ sin θ, qy = q sin ϕ sin θ, qz = q cos θ.

I qð Þ¼ 4π

normalized scattering amplitude given by:

function f, it is calculated according to:

following [21]:

f qx; qy; qz D E � �

172 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

where <sup>n</sup> is the concentration of objects, <sup>V</sup> is the volume of each object, and F q!� �

1 V ð V e �iq:r d3

The symbol 〈⋯〉 stands for ensemble averaging over all orientations, and for an arbitrary

Another useful form of the scattering intensity, as a function of the correlation function, is the

where γ(r) � 〈ρ(r) ∗ ρ(�r)〉 is the correlation function of the object, with γ(r) = 0 for r > D and D is the largest dimension in the object. The symbol ∗ denotes a convolution, and thus, the correlation function can be seen as an averaged self-convolution of density distribution.

Figure 1. Schematic representation of a small-angle scattering experiment. 2θ is the scattering angle and λ is the wavelength of incident beam. The sample shown is a two-phase system of polydisperse scatterers with the same shape

and random orientations, embedded in a matrix or solution.

ð<sup>D</sup> 0 γð Þr

dθ sin θ

sin qr qr r 2

ð<sup>2</sup><sup>π</sup> 0

<sup>V</sup><sup>2</sup> j j <sup>F</sup>ð Þ <sup>q</sup> <sup>2</sup> D E

, (2)

r: (3)

dϕf q; θ; ϕ � �, (4)

dr, (5)

is the

$$R\_{\rm g} = \frac{1}{2} \int\_0^D \gamma(r) r^4 \mathrm{d}r / \int\_0^D \gamma(r) r^2 \mathrm{d}r,\tag{8}$$

is the radius of gyration of the object. In practice, a plot of logI(q) vs. q<sup>2</sup> is used to obtain the slope R<sup>2</sup> <sup>g</sup>=3, and then an overall size of the object. For a ball of radius R, it is known that R ¼ R<sup>g</sup> ffiffiffiffiffiffiffi 5=3 p [21]. If a scattering experiment is performed on an absolute scale, molecular weight can also be obtained.

However, as we shall see in the next sections, inside a fractal, the scattering units have defined positions and correlations, and the interference among rays scattered by different units may no longer be ignored, so that the scattering amplitudes of individual units have to be added. By considering that the fractal is composed of N balls of size R, its form factor becomes [29]:

$$F(\mathfrak{q}) = \rho\_{\mathfrak{q}} F\_0(\mathfrak{q} \mathbb{R}) / \mathrm{N},\tag{9}$$

where <sup>ρ</sup><sup>q</sup> <sup>¼</sup> <sup>P</sup> j <sup>e</sup>�<sup>i</sup>q. <sup>r</sup><sup>j</sup> is the Fourier component of the density of units centers, <sup>F</sup><sup>0</sup> is the form factor of each scattering unit composing the fractal, and r ! <sup>j</sup> are their positions. By introducing Eq. (9) into Eq. (2), the scattering intensity can be written as [29]:

$$I(q) = I(0)S(q)|F\_0(q\mathbb{R})|^2/\text{N},\tag{10}$$

where I(0) = n|Δρ|2 V2 , and S qð Þ� ρqρ-<sup>q</sup> D E=<sup>N</sup> is the structure factor and it describes the correlations between the scattering units inside the fractal.

A physical sample almost always consists of fractals that have different sizes, which is called polydispersity. An exception to this rule is protein solutions, in which all have the same size and shape. Thus, the corresponding scattering intensity from polydisperse fractals can be regarded as the sum of each individual form factor weighted with the corresponding volume V and contrast Δρ. We consider here a continuous distribution DN(l) of fractals with different sizes l, defined in such a way that DN(l)dl gives the probability of finding a fractal with dimension l lying in the range (l, l + dl). Although any kind of broad distribution can be used, we take here, as an application, a log-normal distribution of fractal sizes, such as:

$$D\_N(l) = \frac{1}{\sigma l (2\pi)^{1/2}} e^{-\frac{\left(\log\left(l/\mu\_0\right) + s^2/2\right)^2}{2\sigma^2}},\tag{11}$$

F0ð Þ¼ q

ð1

u¼�1

Thus, the total scattering intensity (see also Eq. (2)) becomes:

I qð Þ� ½ � <sup>F</sup>0ð Þ qR <sup>2</sup> <sup>¼</sup> <sup>32</sup>

ð∞ r¼0 e �iqrur

F0ð Þ� q

tude of the ball of radius R becomes:

1 V ð<sup>2</sup><sup>π</sup> ϕ¼0

1 V ð<sup>2</sup><sup>π</sup> ϕ¼0 ðπ θ¼0 ð∞ r¼0 e �<sup>i</sup>q. <sup>r</sup> r

Note that since balls represent here the basic units of the fractal, we have chosen the notation F0(q) instead of F(q), in spirit of Eq. (9). We can choose the polar axis to coincide with the direction of q, and therefore q. r ¼ qr cos θ. By denoting u = cos θ, in the new variable, Eq. (16) becomes:

By performing an integration by parts of the last expression, the normalized scattering ampli-

<sup>F</sup>0ð Þ¼ qR 3 sin ð Þ ð Þ� qR ð Þ qR cos ð Þ qR

Figure 2a shows the scattering intensity of a ball of radius R = 10 nm. The scattering is represented on a double logarithmic scale and shows the presence of two main regions. At low values of the

which one can obtain the radius of gyration, as described in the previous section. At higher values (i.e., q≳π/R), there is a power-law decay of the type I(q)∝q�<sup>4</sup> and with many minima. This is called the Porod region and generally it gives information about the specific surface of the investigated object. The main feature here is that by increasing the relative variance σr, the scattering curve

Figure 2. (a) SAS intensity from a ball of radius R = 10 nm from Eq. (19) (lowest curve). The higher the relative variance, the smoother the curve. Polydisperse SAS intensity, according to Eq. (12) at various values of relative variances. (b) SAS

intensities from a ball of radius R = 10 nm (left curve), and R = 1 nm (right curve).

scattering vector (q≲π/R), we have the Guinier region, which is a plateau with I(q)∝q

<sup>2</sup> sin <sup>θ</sup>drdθd<sup>ϕ</sup> <sup>¼</sup> <sup>1</sup>

V ðR r¼0 4πr

ð Þ sin ð Þ� qR ð Þ qR cos ð Þ qR <sup>2</sup>

<sup>2</sup> sin θdrdθdϕ: (16)

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870

> <sup>2</sup> sin ð Þ qr qr

ð Þ qR <sup>3</sup> : (18)

ð Þ qR <sup>6</sup> : (19)

dr: (17)

175

0

, and from

where <sup>σ</sup> <sup>¼</sup> log 1 <sup>þ</sup> <sup>σ</sup><sup>2</sup> r � � � � <sup>1</sup>=<sup>2</sup> , μ<sup>0</sup> = 〈l〉<sup>D</sup> is the mean length, σ<sup>r</sup> ¼ l <sup>2</sup> � � � <sup>μ</sup><sup>2</sup> 0 � �<sup>1</sup>=<sup>2</sup> =μ<sup>0</sup> is the relative variance, and h i <sup>⋯</sup> <sup>D</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> ⋯DNð Þl dl. Since for a polydisperse fractal dispersion, the volume of each fractal has a continuous variation with its size, the polydisperse scattering intensity becomes:

$$I(q) = n \left| \Delta \rho \right|^2 \int\_0^\infty \left< \left| F(q) \right|^2 \right> V^2(l) D\_N(l) \, \mathrm{d}l,\tag{12}$$

where the form factor F(q) is given by Eq. (9). The effect of polydispersity is to smooth the scattering curves [20, 21] (see also Figure 2a).

#### 2.1.2. Debye-Pantos formula

In the next section, we shall make use of chaos game representation (CGR) and cellular automata (CA) to generate positions of the N scattering units/points. Thus, we can start with the Debye formula [44]

$$I^D(q) = \text{NI}\_s(q) + 2F\_s(q)^2 \sum\_{i=1}^{N-1} \sum\_{j=i+1}^N \frac{\sin qr\_{ij}}{qr\_{ij}} \,\text{,}\tag{13}$$

where Is(q) is the intensity scattered by each fractal unit, and rij is the distance between units i and j. When the number of units exceeds few thousands, the computation of the term sin(qrij)/(qrij) is time consuming, and thus it is handled via a pair-distance histogram g(r), with a bin-width commensurate with the experimental resolution [45]. Thus, Eq. (13) becomes

$$I^D(q) = \text{NI}\_s(q) + 2F\_s^2(q) \sum\_{i=1}^{N\_{\text{bin}}} g(r\_i) \frac{\sin qr\_i}{qr\_i} \,\text{}\,\tag{14}$$

where g(ri) is the pair-distance histogram at pair distance ri. For determining fractal properties, we can neglect the form factor, and consider <sup>I</sup>sð Þ¼ <sup>q</sup> <sup>F</sup><sup>2</sup> <sup>s</sup> ð Þ¼ q 1. Thus, Eq. (14) gives the structure factor:

$$I^D(q) \equiv S^D(q) = N + 2\sum\_{i=1}^{N\_{\rm lim}} g(r\_i) \frac{\sin qr\_i}{qr\_i}.\tag{15}$$

#### 2.1.3. Scattering from a ball and from a triangle

As a first example, we derive the scattering intensity of a ball with unit density, radius R, and volume V = (4π/3)R<sup>3</sup> . To do so, we can rewrite the normalized scattering amplitude given by Eq. (3), in spherical coordinates, such as:

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 175

$$F\_0(\mathbf{q}) = \frac{1}{V} \int\_{\phi=0}^{2\pi} \int\_{\theta=0}^{\pi} \int\_{r=0}^{\infty} e^{-i\mathbf{q}\cdot\mathbf{r}} r^2 \sin\theta \mathbf{d}r \mathbf{d}\theta \mathbf{d}\phi. \tag{16}$$

Note that since balls represent here the basic units of the fractal, we have chosen the notation F0(q) instead of F(q), in spirit of Eq. (9). We can choose the polar axis to coincide with the direction of q, and therefore q. r ¼ qr cos θ. By denoting u = cos θ, in the new variable, Eq. (16) becomes:

$$F\_0(\boldsymbol{q}) \equiv \frac{1}{V} \int\_{\phi=0}^{2\pi} \int\_{u=-1}^{1} \int\_{r=0}^{\boldsymbol{\alpha}} e^{-i\boldsymbol{q}\boldsymbol{r}\boldsymbol{u}} r^2 \sin\theta \mathbf{d}r \mathbf{d}\theta d\phi = \frac{1}{V} \int\_{r=0}^{R} 4\pi r^2 \frac{\sin\left(qr\right)}{qr} \mathbf{d}r. \tag{17}$$

By performing an integration by parts of the last expression, the normalized scattering amplitude of the ball of radius R becomes:

$$F\_0(q\mathbb{R}) = \frac{\Im(\sin\left(q\mathbb{R}\right) - \left(q\mathbb{R}\right)\cos\left(q\mathbb{R}\right))}{\left(q\mathbb{R}\right)^3}.\tag{18}$$

Thus, the total scattering intensity (see also Eq. (2)) becomes:

dimension l lying in the range (l, l + dl). Although any kind of broad distribution can be used,

<sup>1</sup>=<sup>2</sup> e

, μ<sup>0</sup> = 〈l〉<sup>D</sup> is the mean length, σ<sup>r</sup> ¼ l

each fractal has a continuous variation with its size, the polydisperse scattering intensity becomes:

j j Fð Þ q <sup>2</sup> D E

where the form factor F(q) is given by Eq. (9). The effect of polydispersity is to smooth the

In the next section, we shall make use of chaos game representation (CGR) and cellular automata (CA) to generate positions of the N scattering units/points. Thus, we can start with

where Is(q) is the intensity scattered by each fractal unit, and rij is the distance between units i and j. When the number of units exceeds few thousands, the computation of the term sin(qrij)/(qrij) is time consuming, and thus it is handled via a pair-distance histogram g(r), with a bin-width

> <sup>s</sup> ð Þq N Xbins i¼1

where g(ri) is the pair-distance histogram at pair distance ri. For determining fractal properties,

As a first example, we derive the scattering intensity of a ball with unit density, radius R, and

N Xbins i¼1

2 N X�1 i¼1

X N

sin qrij qrij

sin qri qri

sin qri qri

j¼iþ1

g rð Þ<sup>i</sup>

g rð Þ<sup>i</sup>

. To do so, we can rewrite the normalized scattering amplitude given by

<sup>D</sup>ð Þ¼ <sup>q</sup> NIsð Þþ <sup>q</sup> <sup>2</sup>Fsð Þ<sup>q</sup>

commensurate with the experimental resolution [45]. Thus, Eq. (13) becomes

<sup>D</sup>ð Þ¼ <sup>q</sup> NIsð Þþ <sup>q</sup> <sup>2</sup>F<sup>2</sup>

<sup>D</sup>ð Þ� <sup>q</sup> SDð Þ¼ <sup>q</sup> <sup>N</sup> <sup>þ</sup> <sup>2</sup>

� log <sup>l</sup>=<sup>μ</sup> ð Þ<sup>0</sup> <sup>þ</sup>σ<sup>2</sup> ð Þ <sup>=</sup><sup>2</sup> <sup>2</sup>

<sup>0</sup> ⋯DNð Þl dl. Since for a polydisperse fractal dispersion, the volume of

V2

<sup>2</sup>σ<sup>2</sup> , (11)

ð Þl DNð Þl dl, (12)

, (13)

, (14)

: (15)

<sup>s</sup> ð Þ¼ q 1. Thus, Eq. (14) gives the struc-

=μ<sup>0</sup> is the relative

<sup>2</sup> � � � <sup>μ</sup><sup>2</sup> 0 � �<sup>1</sup>=<sup>2</sup>

we take here, as an application, a log-normal distribution of fractal sizes, such as:

σlð Þ 2π

DNðÞ¼ <sup>l</sup> <sup>1</sup>

I qð Þ¼ <sup>n</sup> <sup>Δ</sup><sup>ρ</sup> � � � � 2 ð∞ 0

174 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

where <sup>σ</sup> <sup>¼</sup> log 1 <sup>þ</sup> <sup>σ</sup><sup>2</sup>

variance, and h i <sup>⋯</sup> <sup>D</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

2.1.2. Debye-Pantos formula

the Debye formula [44]

ture factor:

volume V = (4π/3)R<sup>3</sup>

r � � � � <sup>1</sup>=<sup>2</sup>

scattering curves [20, 21] (see also Figure 2a).

I

I

we can neglect the form factor, and consider <sup>I</sup>sð Þ¼ <sup>q</sup> <sup>F</sup><sup>2</sup>

I

2.1.3. Scattering from a ball and from a triangle

Eq. (3), in spherical coordinates, such as:

$$I(q) \equiv \left[ F\_0(q\mathbb{R}) \right]^2 = \frac{\mathfrak{Z}^2 \left( \sin \left( q\mathbb{R} \right) - \left( q\mathbb{R} \right) \cos \left( q\mathbb{R} \right) \right)^2}{\left( q\mathbb{R} \right)^6}. \tag{19}$$

Figure 2a shows the scattering intensity of a ball of radius R = 10 nm. The scattering is represented on a double logarithmic scale and shows the presence of two main regions. At low values of the scattering vector (q≲π/R), we have the Guinier region, which is a plateau with I(q)∝q 0 , and from which one can obtain the radius of gyration, as described in the previous section. At higher values (i.e., q≳π/R), there is a power-law decay of the type I(q)∝q�<sup>4</sup> and with many minima. This is called the Porod region and generally it gives information about the specific surface of the investigated object. The main feature here is that by increasing the relative variance σr, the scattering curve

Figure 2. (a) SAS intensity from a ball of radius R = 10 nm from Eq. (19) (lowest curve). The higher the relative variance, the smoother the curve. Polydisperse SAS intensity, according to Eq. (12) at various values of relative variances. (b) SAS intensities from a ball of radius R = 10 nm (left curve), and R = 1 nm (right curve).

becomes smoother, and the value of the scattering exponent is preserved. Figure 2b shows that by decreasing the size of the ball, the corresponding scattering curve has the same characteristics and the Porod region is shifted to the right with the corresponding factor (here by 10).

The second example is an equilateral triangle of edge size a. This is a slightly more complicated structure since it does not have a center of symmetry as a ball, and thus an orientational averaging is required. Thus, its height is <sup>h</sup> <sup>¼</sup> <sup>a</sup> ffiffiffi 3 <sup>p</sup> <sup>=</sup>2 and its surface area is:

$$Area(a) = a^2 \frac{\sqrt{3}}{4}.\tag{20}$$

prefer here triangles due to the fact that the both well-known techniques for generating fractal structures: iteration function systems and cellular automata, in their basic form, involve triangles

Figure 3. (a) SAS intensity from an equilateral triangle of edge size a = 1 nm (right curve), and a = 3 nm (left curve), respectively. (b) A comparison between the SAS intensity of a ball of radius R = 1 nm (left curve) and a triangle of edge size

We start first with a simple model consisting of three triangles of edge size a/2, with a = 1 nm,

where β<sup>s</sup> is the scaling factor, F0(q) is given by Eq. (22), and the translation vectors are given by:

The corresponding scattering intensity is shown in Figure 4b, and as expected, it consists of a Guinier region followed by a Porod one with scattering exponent �3. For comparison, the

The second example is a system of 6 triangles of edge sizes a/3 arranged in such a way that they form a hexagon as shown in Figure 5a in black. For this configuration, the translation

The corresponding scattering intensity is shown in Figure 5b and shows a superposition of maxima and minima. Excepting the distribution of minima in the Porod region, there is no significant difference between this scattering curve and the one corresponding to the system of

π <sup>3</sup> <sup>j</sup>; sin <sup>π</sup> 3 j

3 2 � �; sin <sup>π</sup>

<sup>s</sup>Area að ÞF<sup>0</sup> <sup>β</sup>s<sup>q</sup> � �<sup>e</sup>

<sup>3</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup>

3 2 � � � � (25)

n o: (26)

�iqaj , (24)

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 177

2

j¼0 β2

π <sup>3</sup> <sup>2</sup><sup>j</sup> <sup>þ</sup>

as shown in Figure 4a. For this configuration, we can write [40]:

<sup>a</sup><sup>j</sup> <sup>¼</sup> <sup>a</sup> ffiffiffi 3 p

Area að Þ <sup>=</sup><sup>3</sup> <sup>F</sup>ð Þ¼ <sup>q</sup> <sup>X</sup>

<sup>6</sup> cos

same figure shows the scattering intensity of a single of edge size a = 1 nm.

<sup>b</sup><sup>j</sup> <sup>¼</sup> <sup>2</sup><sup>a</sup> 3 ffiffiffi 3 p cos

in the construction process.

a = 1 nm (right curve).

vectors can be written as [38]:

three triangles shown in Figure 4b.

We choose a Cartesian coordinate system where one edge is parallel to the x-axis and the opposite vertex coincides with the origin. Thus, Eq. (3) becomes a surface integral given by:

$$F\_0(\boldsymbol{q}) = \frac{1}{Area(\boldsymbol{a})} \int\_0^a \mathbf{d}y \int\_{-\frac{w}{2h}}^{\frac{w}{2h}} \mathrm{d}x e^{-i\left(x\eta\_x + y\eta\_y\right)}\,,\tag{21}$$

which can be calculated and transformed into:

$$F\_0(\mathfrak{q}) = 2e^{-i\alpha} \frac{\beta e^{i\alpha} - \beta \cos \beta - i\alpha \sin \beta}{\beta \left(\beta^2 - \alpha^2\right)},\tag{22}$$

where α = hqx and β = hqy. As we shall see in next sections, the scattering amplitude of a system of triangles can be obtained by properly taking into account their scaling, rotation, and translations.

The averaging over all orientations is performed by allowing the triangle to rotate in a 2D space, and thus the average given by Eq. (4) for 3D case, becomes now:

$$
\left\langle f\left(q\_x, q\_y\right)\right\rangle = \frac{1}{2\pi} \int\_0^{2\pi} \mathbf{d}\phi f\left(q, \phi\right),\tag{23}
$$

with qx = q cos ϕ and qy = q sin ϕ.

Similarly to scattering from a ball, the intensity curve of a triangle also shows the Guinier region at low-q (q≲ 2π/a), and a Porod region at high-q (q≳2π/a) as shown in Figure 3a. However, for a triangle, the absolute value of the scattering exponent in the Porod region is equal to 3. This is in contrast to the value of 4 obtained for the ball in the previous example (see Figure 3b). The difference arises due to the fact that the triangle is a 2D object while the ball is a 3D one. In addition, due to the lack of symmetry, scattering from a triangle does not show pronounced minima as in the case of scattering from a ball.

#### 2.1.4. Scattering from systems of triangles

In the previous section, we have seen that regardless of the shape and Euclidean dimension, the SAS intensity from basic geometrical structures always reproduces a Guinier region followed by a Porod one. Without loosing from generality, we will restrict in the following to calculate SAS intensity from systems of triangles. In principle, any geometrical shape can be chosen but we

becomes smoother, and the value of the scattering exponent is preserved. Figure 2b shows that by decreasing the size of the ball, the corresponding scattering curve has the same characteristics and

The second example is an equilateral triangle of edge size a. This is a slightly more complicated structure since it does not have a center of symmetry as a ball, and thus an orientational

Area að Þ¼ <sup>a</sup><sup>2</sup>

ða 0 dy ð ya 2h �ya 2h

We choose a Cartesian coordinate system where one edge is parallel to the x-axis and the opposite vertex coincides with the origin. Thus, Eq. (3) becomes a surface integral given by:

3

ffiffiffi 3 p

�i<sup>α</sup> <sup>β</sup>e<sup>i</sup><sup>α</sup> � <sup>β</sup> cos <sup>β</sup> � <sup>i</sup><sup>α</sup> sin <sup>β</sup>

where α = hqx and β = hqy. As we shall see in next sections, the scattering amplitude of a system of triangles can be obtained by properly taking into account their scaling, rotation, and translations. The averaging over all orientations is performed by allowing the triangle to rotate in a 2D

> ¼ 1 2π ð<sup>2</sup><sup>π</sup> 0

Similarly to scattering from a ball, the intensity curve of a triangle also shows the Guinier region at low-q (q≲ 2π/a), and a Porod region at high-q (q≳2π/a) as shown in Figure 3a. However, for a triangle, the absolute value of the scattering exponent in the Porod region is equal to 3. This is in contrast to the value of 4 obtained for the ball in the previous example (see Figure 3b). The difference arises due to the fact that the triangle is a 2D object while the ball is a 3D one. In addition, due to the lack of symmetry, scattering from a triangle does not show

In the previous section, we have seen that regardless of the shape and Euclidean dimension, the SAS intensity from basic geometrical structures always reproduces a Guinier region followed by a Porod one. Without loosing from generality, we will restrict in the following to calculate SAS intensity from systems of triangles. In principle, any geometrical shape can be chosen but we

<sup>p</sup> <sup>=</sup>2 and its surface area is:

<sup>4</sup> : (20)

dxe�i xqxþyq ð Þ<sup>y</sup> , (21)

β β<sup>2</sup> � <sup>α</sup><sup>2</sup> � � , (22)

dϕf q; ϕ � �, (23)

the Porod region is shifted to the right with the corresponding factor (here by 10).

1 Area að Þ

averaging is required. Thus, its height is <sup>h</sup> <sup>¼</sup> <sup>a</sup> ffiffiffi

176 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

which can be calculated and transformed into:

with qx = q cos ϕ and qy = q sin ϕ.

2.1.4. Scattering from systems of triangles

F0ð Þ¼ q

F0ð Þ¼ q 2e

space, and thus the average given by Eq. (4) for 3D case, becomes now:

f qx; qy D E � �

pronounced minima as in the case of scattering from a ball.

Figure 3. (a) SAS intensity from an equilateral triangle of edge size a = 1 nm (right curve), and a = 3 nm (left curve), respectively. (b) A comparison between the SAS intensity of a ball of radius R = 1 nm (left curve) and a triangle of edge size a = 1 nm (right curve).

prefer here triangles due to the fact that the both well-known techniques for generating fractal structures: iteration function systems and cellular automata, in their basic form, involve triangles in the construction process.

We start first with a simple model consisting of three triangles of edge size a/2, with a = 1 nm, as shown in Figure 4a. For this configuration, we can write [40]:

$$Area(a/3)F(\mathfrak{q}) = \sum\_{j=0}^{2} \beta\_s^2 Area(a)F\_0(\mathfrak{P}\_s \mathfrak{q})e^{-i\mathfrak{q}a\_j} \tag{24}$$

where β<sup>s</sup> is the scaling factor, F0(q) is given by Eq. (22), and the translation vectors are given by:

$$a\_{j} = \frac{a\sqrt{3}}{6} \left\{ \cos\frac{\pi}{3} \left( 2j + \frac{3}{2} \right), \sin\frac{\pi}{3} \left( 2j + \frac{3}{2} \right) \right\} \tag{25}$$

The corresponding scattering intensity is shown in Figure 4b, and as expected, it consists of a Guinier region followed by a Porod one with scattering exponent �3. For comparison, the same figure shows the scattering intensity of a single of edge size a = 1 nm.

The second example is a system of 6 triangles of edge sizes a/3 arranged in such a way that they form a hexagon as shown in Figure 5a in black. For this configuration, the translation vectors can be written as [38]:

$$\mathbf{b}\_{j} = \frac{2a}{3\sqrt{3}} \left\{ \cos \frac{\pi}{3} j, \sin \frac{\pi}{3} j \right\}. \tag{26}$$

The corresponding scattering intensity is shown in Figure 5b and shows a superposition of maxima and minima. Excepting the distribution of minima in the Porod region, there is no significant difference between this scattering curve and the one corresponding to the system of three triangles shown in Figure 4b.

In particular, if the triangles composing the system are arranged in such a way that they form a Sierpinski gasket, the intermediate region will correspond to a mass fractal region [29–31]. When the triangles have a power-law distribution in their sizes, the intermediate region will correspond to a surface fractal one [37, 38]. The great advantage of the SAS technique consists in the possibility to differentiate these two types of fractal regimes, as discussed at the beginning of this chapter.

As it was already pointed out before, the main characteristic of fractals obtained from a SAS experiment is the fractal dimension. Mathematically, the α-dimensional Hausdorff measure is

where A is a subset of an n-dimensional Euclidean space, {Vi} is a covering of A with ai = diam (Vi) ≤ a, and the infimum is on all possible coverings. Then, the Hausdorff dimension D of the

and it represents the value of α for which the Hausdorff measure changes its value from zero to

However, in practice, this definition is quite inconvenient to be used, and here we shall use the "mass-radius" relation for calculating the Hausdorff dimension of the fractals [1]. In this approach, the total fractal measure (i.e., mass, surface area, volume) of the fractal within a ball

M rð Þ¼ A rð Þr

As an application, for a deterministic mass fractal of length L, scaling factor βs, and k structural

and using Eq. (29), one obtains a formula for calculating the fractal dimension of the mass

Thus, if a finite iteration of the fractal consists of N scattering balls of radius a, the fractal

kβ<sup>D</sup>

<sup>N</sup> <sup>∝</sup> <sup>L</sup> a � �<sup>D</sup>

<sup>D</sup> � inf <sup>α</sup>i; <sup>m</sup><sup>α</sup> f g ð Þ¼ <sup>A</sup> <sup>0</sup> <sup>¼</sup> sup <sup>α</sup>; <sup>m</sup><sup>α</sup> f g ð Þ¼ <sup>A</sup> <sup>∞</sup> , (28)

<sup>i</sup> , α > 0 (27)

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 179

D, (29)

M Lð Þ¼ kM <sup>β</sup>s<sup>L</sup> � �, (30)

<sup>s</sup> ¼ 1: (31)

, (32)

<sup>m</sup><sup>α</sup>ð Þ¼ <sup>A</sup> lim

a!0 inf f g Vi X i aα

2.2. Mass and surface fractals

of radius r centered on the fractal is given by:

units in the first iteration, we can write [1]:

dimension is given by the asymptotic:

in the limit of large number of iterations.

where logA(r)/ log r!0 for r !∞.

defined by [29]

set A is given by:

infinity.

fractal:

Figure 4. (a) A model of three triangles (gray) of edge size a/2, with a = 1 nm; (b) the corresponding SAS intensity (highly oscillating curve). The smoother curve is the SAS intensity corresponding to a single triangle of edge size a = 1 nm and whose center coincides with the center of white triangle in part (a).

Figure 5. (a) Two models consisting only of equilateral triangles: hexagon (black color) and Star of David (black and gray colors); (b) the corresponding SAS intensities of the hexagon (continuous curve) and Star of David (dashed curve).

The last example is slightly more complicated and it consists of one hexagon of edge size a/3 (black) and six triangles of edge size a/3 (gray), with a = 1 nm arranged as in Figure 5a. This configuration is equivalent to a system of one triangle of edge size a and three triangles of edge size a/3. This is known in the literature, also as the Star of David. The translation vectors of the three triangles can be obtained in a similar way, as in the case of the first example. The corresponding scattering intensity is shown in Figure 5b (red). It can be seen that due to various sizes of the triangles involved in the construction, an intermediate regime emerges between the Guinier and Porod regions, approximately at 2 ≲q≲8 nm<sup>1</sup> . As we shall see later, for even more complex structures, this intermediate region will evolve into a fractal one.

In particular, if the triangles composing the system are arranged in such a way that they form a Sierpinski gasket, the intermediate region will correspond to a mass fractal region [29–31]. When the triangles have a power-law distribution in their sizes, the intermediate region will correspond to a surface fractal one [37, 38]. The great advantage of the SAS technique consists in the possibility to differentiate these two types of fractal regimes, as discussed at the beginning of this chapter.

#### 2.2. Mass and surface fractals

As it was already pointed out before, the main characteristic of fractals obtained from a SAS experiment is the fractal dimension. Mathematically, the α-dimensional Hausdorff measure is defined by [29]

$$m^{\alpha}(A) = \lim\_{a \to 0} \inf\_{\{V\_i\}} \sum\_i a\_i^{\alpha}, a > 0 \tag{27}$$

where A is a subset of an n-dimensional Euclidean space, {Vi} is a covering of A with ai = diam (Vi) ≤ a, and the infimum is on all possible coverings. Then, the Hausdorff dimension D of the set A is given by:

$$D \equiv \inf \{ \alpha\_i, m^a(A) = 0 \} = \sup \{ \alpha, m^a(A) = \infty \},\tag{28}$$

and it represents the value of α for which the Hausdorff measure changes its value from zero to infinity.

However, in practice, this definition is quite inconvenient to be used, and here we shall use the "mass-radius" relation for calculating the Hausdorff dimension of the fractals [1]. In this approach, the total fractal measure (i.e., mass, surface area, volume) of the fractal within a ball of radius r centered on the fractal is given by:

$$M(r) = A(r)r^D,\tag{29}$$

where logA(r)/ log r!0 for r !∞.

The last example is slightly more complicated and it consists of one hexagon of edge size a/3 (black) and six triangles of edge size a/3 (gray), with a = 1 nm arranged as in Figure 5a. This configuration is equivalent to a system of one triangle of edge size a and three triangles of edge size a/3. This is known in the literature, also as the Star of David. The translation vectors of the three triangles can be obtained in a similar way, as in the case of the first example. The corresponding scattering intensity is shown in Figure 5b (red). It can be seen that due to various sizes of the triangles involved in the construction, an intermediate regime emerges

Figure 5. (a) Two models consisting only of equilateral triangles: hexagon (black color) and Star of David (black and gray colors); (b) the corresponding SAS intensities of the hexagon (continuous curve) and Star of David (dashed curve).

Figure 4. (a) A model of three triangles (gray) of edge size a/2, with a = 1 nm; (b) the corresponding SAS intensity (highly oscillating curve). The smoother curve is the SAS intensity corresponding to a single triangle of edge size a = 1 nm and

for even more complex structures, this intermediate region will evolve into a fractal one.

. As we shall see later,

between the Guinier and Porod regions, approximately at 2 ≲q≲8 nm<sup>1</sup>


0. 2

0. 4

0. 6

178 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals


whose center coincides with the center of white triangle in part (a).

(a)


As an application, for a deterministic mass fractal of length L, scaling factor βs, and k structural units in the first iteration, we can write [1]:

$$M(L) = kM(\beta\_s L),\tag{30}$$

and using Eq. (29), one obtains a formula for calculating the fractal dimension of the mass fractal:

$$k\mathfrak{f}\_s^D = \mathbf{1}.\tag{31}$$

Thus, if a finite iteration of the fractal consists of N scattering balls of radius a, the fractal dimension is given by the asymptotic:

$$N\mathfrak{a}\left(\frac{L}{a}\right)^D,\tag{32}$$

in the limit of large number of iterations.

If the quantity to be measured is the mass M(r) embedded in a disk of radius r, Eq. (30) becomes M(r) ∝r D<sup>m</sup> , which leads to I(q)∝ q�Dm. The lower the value of mass fractal dimension Dm, the less compact is the structure. In a similar way, for a surface fractal of fractal dimension Ds, its surface obeys S(r) ∝r <sup>2</sup> � <sup>D</sup><sup>s</sup> and thus the scattering intensity decays as I(q) ∝q�(2<sup>d</sup> � <sup>D</sup>s) .

More generally, in a two-phase system where one phase is of dimension D<sup>m</sup> and the second phase is its complement set of dimension D<sup>p</sup> ("pores"), the "boundary" between the two phases also forms a set of dimension D<sup>s</sup> ("surface"). Thus, for a mass fractal, we have D<sup>s</sup> =D<sup>m</sup> < d and D<sup>p</sup> = d, while for a surface fractal we have D<sup>m</sup> =D<sup>p</sup> = d and d� 1 < D<sup>s</sup> < d [37, 38]. The possibility of differentiating between mass and surface fractals makes SAS a very convenient technique for measuring fractal dimensions of materials at nano- and micro-scales.

#### 2.3. Iterated function system

As a first method of generating fractals, we consider an iterated function system (IFS). By definition, an IFS is a complete metric space (X, d) together with a finite set wn : X ! X of contraction mappings and contractivity factors sn, n > 1, 2, ⋯, N. In general, a transformation f : X!X on a metric space (X, d) is a contraction mapping if there is a constant (contractivity factor) 0 ≤ s < 1 such that

$$\mathbf{d}(\mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{y})) \le \mathbf{s}. \ \mathbf{d}(\mathbf{x}, \mathbf{y}) \quad \forall \mathbf{x}, \mathbf{y} \in \mathbf{X}. \tag{33}$$

applied alternatively for as many times as needed. The grid is n-dimensional but for our

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 181

In this chapter, we shall present Rule 90 [49] together with the corresponding structure factor based on Pantos formula given by Eq. (15). For particular values of the number of steps, Rule 90 generates exact shapes of the Sierpinski gasket (SG). However, for most of the number of steps, it generates intermediate structures between two consecutive iterations of SG. Thus, for Rule 90, CA extends considerably the number of structures generated, and therefore new classes of materials consisting of a "mixture" of SG at various iterations can

More generally, SAS from CA could be used to check whether a generated structure is a mass or surface fractal (or none of them), i.e., whether there exists a power-law distribution of some entities (collections of cells). In addition, through the oscillations of the scattering curve in the fractal region, SAS from CA can shed some light on the randomness of the generated structures. This could be of particular interest since some rules like Rule 30 generates so-called pseudorandom structures. However, this is beyond the scope of this chapter. Here, we shall restrict ourselves to calculation and interpretation of SAS intensities from basic structures, such as those based on SG. This shall facilitate a quick comparison with the theoretical model based on SG

As an example of a deterministic mas fractal, we calculate the scattering from a two dimensional Sierpinski gasket (SG), generated by three different methods. In order to calculate the

• When the size of a triangle is scaled as a !βsa, then the form factor scales as F(q)! F(βsq);

Zero-th iteration of SG consists of a single triangle of edge size a (here, a = 1 nm). First iteration (m = 1) consists of four smaller triangles, each of the edge length a/2 as shown in Figure 3a. At second iteration (m = 2), the same operation is repeated for each of the triangles of edge length a/2. In the limit of large number of iterations m, the total number of triangles of edge size

> log N log ð Þ a=am

�iq. b .

<sup>N</sup> <sup>¼</sup> <sup>3</sup><sup>m</sup>: (35)

≈ 1:585: (36)

presented in the next section, and thus to support the validity of the obtained results.

scattering intensities of an ensemble of triangles, we use the following properties:

• When the triangle is translated by a vector b such as r!rþr, then F(q) !F(q)e

Therefore using Eq. (31), one obtains the fractal dimension of SG as:

<sup>D</sup> <sup>¼</sup> lim<sup>m</sup>!<sup>∞</sup>

At m-th iteration, the positions of the triangles forming the SG are given by:

3. Small-angle scattering from mass fractals

purposes, we will choose n = 1.

be investigated.

am = a/2<sup>m</sup> is:

By considering a hyperbolic IFS and if we denote by (H(X), h(d)), the space of nonempty compact subsets with the Hausdorff metric h(d), then the transformations W : H(X) !H(X) defined by W Bð Þ¼ <sup>∪</sup><sup>N</sup> <sup>n</sup>¼<sup>1</sup>wnð Þ <sup>B</sup> , <sup>∀</sup>B<sup>∈</sup> <sup>H</sup>ð Þ <sup>X</sup> is a contraction mapping on the complete metric space ð Þ Hð Þ X ; h dð Þ with the contractivity factor s [46], i.e.,

$$h\left(\mathcal{W}(B), \mathcal{W}(\mathbb{C})\right) \le \operatorname{s} \cdot h(B, \mathbb{C}) \forall B, \mathbb{C} \in \mathcal{H}(\mathbf{X}).\tag{34}$$

The unique fix point <sup>A</sup> <sup>∈</sup> <sup>H</sup>(X) obeys <sup>A</sup> <sup>¼</sup> <sup>∪</sup><sup>N</sup> <sup>n</sup>¼<sup>1</sup>wnð Þ <sup>A</sup> and is called the attractor of the IFS, which is a deterministic fractal [46].

We generate here the attractor by using random iteration algorithm, and thus we assign the probability pn > 0 to wn for n = 1, 2, ⋯ where P<sup>N</sup> <sup>n</sup>¼<sup>1</sup> pn <sup>¼</sup> 1. Then, a point <sup>x</sup>0<sup>∈</sup> <sup>X</sup> is chosen, and we build recursively the sequence xk∈{w1(xk � 1), w2(xk � 1), ⋯, wN(xk � 1)}, where the probability of the event xk = wn(xk � 1) is pn, and k = 1, 2, ⋯. This leads to the sequence {xk � <sup>1</sup> : k = 0, 1, ⋯}, which converges to the attractor of IFS.

#### 2.4. Cellular automata

Another important method to generate exact self-similar fractals is by using cellular automata (CA) [47–49]. They provide simple models for dynamical systems dealing with the emergence of collective phenomena such as chaos, turbulence, or fractals. Basically, a cellular automaton is a set of cells on a grid (rectangular, hexagonal, etc.) that evolves through a number of discrete steps according to a set of rules based on the states of neighboring cells. The rules are then

applied alternatively for as many times as needed. The grid is n-dimensional but for our purposes, we will choose n = 1.

In this chapter, we shall present Rule 90 [49] together with the corresponding structure factor based on Pantos formula given by Eq. (15). For particular values of the number of steps, Rule 90 generates exact shapes of the Sierpinski gasket (SG). However, for most of the number of steps, it generates intermediate structures between two consecutive iterations of SG. Thus, for Rule 90, CA extends considerably the number of structures generated, and therefore new classes of materials consisting of a "mixture" of SG at various iterations can be investigated.

More generally, SAS from CA could be used to check whether a generated structure is a mass or surface fractal (or none of them), i.e., whether there exists a power-law distribution of some entities (collections of cells). In addition, through the oscillations of the scattering curve in the fractal region, SAS from CA can shed some light on the randomness of the generated structures. This could be of particular interest since some rules like Rule 30 generates so-called pseudorandom structures. However, this is beyond the scope of this chapter. Here, we shall restrict ourselves to calculation and interpretation of SAS intensities from basic structures, such as those based on SG. This shall facilitate a quick comparison with the theoretical model based on SG presented in the next section, and thus to support the validity of the obtained results.

#### 3. Small-angle scattering from mass fractals

If the quantity to be measured is the mass M(r) embedded in a disk of radius r, Eq. (30) becomes

less compact is the structure. In a similar way, for a surface fractal of fractal dimension Ds, its

More generally, in a two-phase system where one phase is of dimension D<sup>m</sup> and the second phase is its complement set of dimension D<sup>p</sup> ("pores"), the "boundary" between the two phases also forms a set of dimension D<sup>s</sup> ("surface"). Thus, for a mass fractal, we have D<sup>s</sup> =D<sup>m</sup> < d and D<sup>p</sup> = d, while for a surface fractal we have D<sup>m</sup> =D<sup>p</sup> = d and d� 1 < D<sup>s</sup> < d [37, 38]. The possibility of differentiating between mass and surface fractals makes SAS a very convenient technique for

As a first method of generating fractals, we consider an iterated function system (IFS). By definition, an IFS is a complete metric space (X, d) together with a finite set wn : X ! X of contraction mappings and contractivity factors sn, n > 1, 2, ⋯, N. In general, a transformation f : X!X on a metric space (X, d) is a contraction mapping if there is a constant (contractivity

By considering a hyperbolic IFS and if we denote by (H(X), h(d)), the space of nonempty compact subsets with the Hausdorff metric h(d), then the transformations W : H(X) !H(X)

We generate here the attractor by using random iteration algorithm, and thus we assign the

build recursively the sequence xk∈{w1(xk � 1), w2(xk � 1), ⋯, wN(xk � 1)}, where the probability of the event xk = wn(xk � 1) is pn, and k = 1, 2, ⋯. This leads to the sequence {xk � <sup>1</sup> : k = 0, 1, ⋯}, which

Another important method to generate exact self-similar fractals is by using cellular automata (CA) [47–49]. They provide simple models for dynamical systems dealing with the emergence of collective phenomena such as chaos, turbulence, or fractals. Basically, a cellular automaton is a set of cells on a grid (rectangular, hexagonal, etc.) that evolves through a number of discrete steps according to a set of rules based on the states of neighboring cells. The rules are then

dfx ð Þ ð Þ;f yð Þ ≤ s. d xð Þ ; y ∀x, y∈ X. (33)

<sup>n</sup>¼<sup>1</sup>wnð Þ <sup>B</sup> , <sup>∀</sup>B<sup>∈</sup> <sup>H</sup>ð Þ <sup>X</sup> is a contraction mapping on the complete metric

hWB ð Þ ð Þ; W Cð Þ ≤ s � h Bð Þ ;C ∀B, C∈ Hð Þ X : (34)

<sup>n</sup>¼<sup>1</sup>wnð Þ <sup>A</sup> and is called the attractor of the IFS, which

<sup>n</sup>¼<sup>1</sup> pn <sup>¼</sup> 1. Then, a point <sup>x</sup>0<sup>∈</sup> <sup>X</sup> is chosen, and we

measuring fractal dimensions of materials at nano- and micro-scales.

180 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

space ð Þ Hð Þ X ; h dð Þ with the contractivity factor s [46], i.e.,

The unique fix point <sup>A</sup> <sup>∈</sup> <sup>H</sup>(X) obeys <sup>A</sup> <sup>¼</sup> <sup>∪</sup><sup>N</sup>

probability pn > 0 to wn for n = 1, 2, ⋯ where P<sup>N</sup>

, which leads to I(q)∝ q�Dm. The lower the value of mass fractal dimension Dm, the

<sup>2</sup> � <sup>D</sup><sup>s</sup> and thus the scattering intensity decays as I(q) ∝q�(2<sup>d</sup> � <sup>D</sup>s)

.

M(r) ∝r

D<sup>m</sup>

surface obeys S(r) ∝r

2.3. Iterated function system

factor) 0 ≤ s < 1 such that

defined by W Bð Þ¼ <sup>∪</sup><sup>N</sup>

is a deterministic fractal [46].

converges to the attractor of IFS.

2.4. Cellular automata

As an example of a deterministic mas fractal, we calculate the scattering from a two dimensional Sierpinski gasket (SG), generated by three different methods. In order to calculate the scattering intensities of an ensemble of triangles, we use the following properties:


Zero-th iteration of SG consists of a single triangle of edge size a (here, a = 1 nm). First iteration (m = 1) consists of four smaller triangles, each of the edge length a/2 as shown in Figure 3a. At second iteration (m = 2), the same operation is repeated for each of the triangles of edge length a/2. In the limit of large number of iterations m, the total number of triangles of edge size am = a/2<sup>m</sup> is:

$$N = \mathfrak{Z}^{\mathfrak{m}}.\tag{35}$$

Therefore using Eq. (31), one obtains the fractal dimension of SG as:

$$D = \lim\_{m \to \infty} \frac{\log N}{\log \left( a / a\_m \right)} \approx 1.585. \tag{36}$$

At m-th iteration, the positions of the triangles forming the SG are given by:

$$P\_m\left(\stackrel{\rightarrow}{\vec{q}}\right) \equiv \prod\_{i=1}^m G\_i\left(\stackrel{\rightarrow}{\vec{q}}\right). \tag{37}$$

most of the times, SAS experiments give this type of behavior, when the curve is completely smeared out. Please note that if we neglect the contribution of the form factor F<sup>0</sup> in Eq. (40), then the Porod region is replaced by an asymptotic region, from which the number of scattering units inside the fractal can be obtained (see Figure 6). This case is presented and discussed in the

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CGR representation gives directly the positions of the scattering units in the fractal. Figure 6 left part, shows the SG generated from CGR for N = 30, 130, 230, and for 430 points, respectively. The figure clearly shows that by increasing the number of points, the obtained structure approaches better the structure of SG. From the same figure, we can see that for N = 430 points, the second iteration of SG can be quite clearly distinguished. Thus, a convenient way to calculate the scattering intensity is to use Pantos formula given by Eq. (15), since we neglect

Figure 6 right part shows the scattering structure factor of SG built from the CGR, for the four structures in the left part. Generally, all scattering curves are characterized by the presence of the three main regions specific to SG, obtained using the analytic representation (Figure 5) right part: Guinier region at low q, fractal region at intermediate q, and here an asymptotic region instead of a Porod one, since we neglect the form factor. The same figure shows that by increasing the number of particles, the length of the fractal region also increases. This is to be expected, since increasing the number of points leads to a better approximation of the deterministic SG. However, for SG generated using CGR, a transition region appears (at 40 ≲q≲200), since in this region, the pair distance distribution function does not follow a power-law decay distribution of the number of distances. Finally, in the asymptotic region, the curves are proportional to 1/N, and thus, the number of points can be recovered. The asymptotic values of the curves are marked by horizontal dashes lines in Figure 5 right part. Rule 90 (Figure 7 left part) is an elementary cellular automaton rule, and it produces SG for particular values of the number of steps. Although the system is generated on a rectangular

Figure 6. Left part: first four iterations of the Sierpinski gasket generated using a deterministic algorithm; right part: The corresponding SAS intensities of the Sierpinski gasket. The curves are shifted vertically by a factor 10m, for clarity. The

lowest curve corresponds to m = 0, and the highest one corresponds to m = 3.

following, for the SG generated using CGR and IFS, respectively.

the shape of the scattering units.

where Gi q !� � is known as the generative function [29, 31, 38], and is given by

$$G\_1\left(\overrightarrow{q}\right) = \frac{1}{3} \sum\_{j=0}^{2} e^{-i\overrightarrow{q}\cdot\overrightarrow{a}\_j} \,\tag{38}$$

where the translation vectors are given by Eq. (25), Gm q !� � <sup>¼</sup> <sup>G</sup><sup>1</sup> <sup>β</sup><sup>m</sup>�<sup>1</sup> <sup>s</sup> q ! � �, and <sup>β</sup><sup>s</sup> = 1/2. Figure 5 left part, shows first three iterations of SG.

By using the property given in Eq. (37), the fractal structure factor can be written as [29]:

$$S(q)/N = \left\langle \prod\_{i=1}^{m} \left| G\_i \left( \overrightarrow{q} \right) \right|^2 . \right\rangle \tag{39}$$

Thus, by introducing Eqs. (39) and (22) into Eq. (10), we obtain an analytical expression for the scattering intensity:

$$I(q)/I(0) = \left\langle \left| F\_0 \left( \beta\_s^m a \,\overline{q} \right) \right|^2 \right\rangle \mathcal{S}(q)/N. \tag{40}$$

By ignoring the form factor F<sup>0</sup> in Eq. (40), an analytical expression of the structure factor is obtained. This case is discussed in [40].

The corresponding scattering intensities are shown in Figure 5 right part, for a triangle (m = 0) and for the first three iterations of SG (m = 1, 2, 3). At low q-values (q≲2 nm�<sup>1</sup> ), all the scattering curves are characterized by a Guinier region. A main feature of scattering from deterministic mass fractals is that after the Guinier region, it follows a fractal regime in which the absolute value of the scattering exponent equals the fractal dimension of the fractal. The length of the fractal regime increases with increasing the iteration number since the distances between the scattering units of the fractals (here triangles) decrease. In Figure 5 right part, the fractal regime is clearly seen within the range 2≲ q≲ 14 nm�<sup>1</sup> for m = 3. It is characterized by a succession of maxima and minima superimposed on a power-law decay (also known as a generalized power-law decay [29]). The number of minima in the fractal regime is equal with the fractal iteration number and from their periodicity, we can extract the value of the scaling factor [29]. Beyond the fractal regime, one obtain as expected, the Porod region where the exponent of the power-law decay is �3 (or �4 for three-dimensional objects). In Figure 5 right part, at m = 3 the Porod regime begins near q≳ 14 nm�<sup>1</sup> .

By taking into account, the effect of polydispersity or the random distribution of the scattering units, the scattering curve is smoothed and thus the maxima and minima are smeared out. The "smoothness" of the curve increases with increasing the relative variance of the distribution function, and they can be completely smeared out when a threshold is reached. Experimentally, most of the times, SAS experiments give this type of behavior, when the curve is completely smeared out. Please note that if we neglect the contribution of the form factor F<sup>0</sup> in Eq. (40), then the Porod region is replaced by an asymptotic region, from which the number of scattering units inside the fractal can be obtained (see Figure 6). This case is presented and discussed in the following, for the SG generated using CGR and IFS, respectively.

Pm q !� �

G<sup>1</sup> q !� � ¼ 1 3 X 2

where the translation vectors are given by Eq. (25), Gm q

182 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

left part, shows first three iterations of SG.

obtained. This case is discussed in [40].

part, at m = 3 the Porod regime begins near q≳ 14 nm�<sup>1</sup>

where Gi q

!� �

scattering intensity:

� <sup>Y</sup><sup>m</sup> i¼1

is known as the generative function [29, 31, 38], and is given by

By using the property given in Eq. (37), the fractal structure factor can be written as [29]:

i¼1

Thus, by introducing Eqs. (39) and (22) into Eq. (10), we obtain an analytical expression for the

By ignoring the form factor F<sup>0</sup> in Eq. (40), an analytical expression of the structure factor is

The corresponding scattering intensities are shown in Figure 5 right part, for a triangle (m = 0)

ing curves are characterized by a Guinier region. A main feature of scattering from deterministic mass fractals is that after the Guinier region, it follows a fractal regime in which the absolute value of the scattering exponent equals the fractal dimension of the fractal. The length of the fractal regime increases with increasing the iteration number since the distances between the scattering units of the fractals (here triangles) decrease. In Figure 5 right part, the fractal regime is clearly seen within the range 2≲ q≲ 14 nm�<sup>1</sup> for m = 3. It is characterized by a succession of maxima and minima superimposed on a power-law decay (also known as a generalized power-law decay [29]). The number of minima in the fractal regime is equal with the fractal iteration number and from their periodicity, we can extract the value of the scaling factor [29]. Beyond the fractal regime, one obtain as expected, the Porod region where the exponent of the power-law decay is �3 (or �4 for three-dimensional objects). In Figure 5 right

By taking into account, the effect of polydispersity or the random distribution of the scattering units, the scattering curve is smoothed and thus the maxima and minima are smeared out. The "smoothness" of the curve increases with increasing the relative variance of the distribution function, and they can be completely smeared out when a threshold is reached. Experimentally,

<sup>s</sup> a q ! � � � �

<sup>2</sup> � �

� � �

.

Gi q ! � � � � �

\* +

S qð Þ=<sup>N</sup> <sup>¼</sup> <sup>Y</sup><sup>m</sup>

I qð Þ=Ið Þ¼ <sup>0</sup> <sup>F</sup><sup>0</sup> <sup>β</sup><sup>m</sup>

and for the first three iterations of SG (m = 1, 2, 3). At low q-values (q≲2 nm�<sup>1</sup>

�

j¼0 e �i q !� <sup>a</sup> ! j

!� �

� � � 2 :

<sup>¼</sup> <sup>G</sup><sup>1</sup> <sup>β</sup><sup>m</sup>�<sup>1</sup> <sup>s</sup> q ! � �

Gi q !� �

, (37)

, (38)

S qð Þ=N: (40)

, and β<sup>s</sup> = 1/2. Figure 5

), all the scatter-

(39)

CGR representation gives directly the positions of the scattering units in the fractal. Figure 6 left part, shows the SG generated from CGR for N = 30, 130, 230, and for 430 points, respectively. The figure clearly shows that by increasing the number of points, the obtained structure approaches better the structure of SG. From the same figure, we can see that for N = 430 points, the second iteration of SG can be quite clearly distinguished. Thus, a convenient way to calculate the scattering intensity is to use Pantos formula given by Eq. (15), since we neglect the shape of the scattering units.

Figure 6 right part shows the scattering structure factor of SG built from the CGR, for the four structures in the left part. Generally, all scattering curves are characterized by the presence of the three main regions specific to SG, obtained using the analytic representation (Figure 5) right part: Guinier region at low q, fractal region at intermediate q, and here an asymptotic region instead of a Porod one, since we neglect the form factor. The same figure shows that by increasing the number of particles, the length of the fractal region also increases. This is to be expected, since increasing the number of points leads to a better approximation of the deterministic SG. However, for SG generated using CGR, a transition region appears (at 40 ≲q≲200), since in this region, the pair distance distribution function does not follow a power-law decay distribution of the number of distances. Finally, in the asymptotic region, the curves are proportional to 1/N, and thus, the number of points can be recovered. The asymptotic values of the curves are marked by horizontal dashes lines in Figure 5 right part.

Rule 90 (Figure 7 left part) is an elementary cellular automaton rule, and it produces SG for particular values of the number of steps. Although the system is generated on a rectangular

Figure 6. Left part: first four iterations of the Sierpinski gasket generated using a deterministic algorithm; right part: The corresponding SAS intensities of the Sierpinski gasket. The curves are shifted vertically by a factor 10m, for clarity. The lowest curve corresponds to m = 0, and the highest one corresponds to m = 3.

Figure 7. Left part: Sierpinski gasket generated using CGR, for different values of the number of scattering units; right part: the corresponding SAS intensities of the Sierpinski gasket from CGR. The highest curve corresponds to N = 30 and the lowest one corresponds to N = 430.

orange triangles in Figure 8 left part). By repeating the same procedure for each of the new

Figure 8. Left part: Rule 90 for p = 31, 41, 52 and p = 63 number of steps, respectively; right part: the corresponding

where am is the side length at mth iteration. In Figure 8, each of the triangle is scaled down by a factor of 0.6 � am. The case when the triangles are not scaled down corresponds exactly to the

Since a surface fractal can be constructed as a sum of mass fractals, the normalized scattering amplitude is written as a sum of the scattering amplitudes of mass fractals. If we denote

! � �β<sup>4</sup>

! h i � � <sup>þ</sup> <sup>β</sup><sup>2</sup>

<sup>¼</sup> log 4

!� �, then a recurrence formula for the scattering amplitude of the surface

<sup>s</sup> q

<sup>s</sup>Am�<sup>2</sup> <sup>β</sup><sup>2</sup>

j¼0 e �i q!� <sup>c</sup> ! j

j¼0 e �i q!� <sup>b</sup> ! j

log 3 <sup>≈</sup> <sup>1</sup>:26, (41)

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 185

<sup>s</sup>Am�<sup>1</sup> β<sup>s</sup> q

, (43)

: (44)

! � � <sup>1</sup> <sup>þ</sup> <sup>6</sup>G<sup>2</sup> <sup>q</sup>

! h i � � :

(42)

log 3 � <sup>4</sup><sup>m</sup> log ð Þ a=am

scattering intensities. The leftmost curve corresponds to p = 63 and the rightmost one corresponds to p = 31.

well-known KS and its scattering properties have been extensively studied in [38].

triangles, one obtains the KS having the fractal dimension:

fractal at arbitrarily iteration can be written as [38]:

<sup>s</sup>Am�<sup>1</sup> β<sup>s</sup> q

Am q

Am q

!� � � Area að ÞFm <sup>q</sup>

!� � <sup>β</sup><sup>2</sup>

The translation vectors are defined as:

!� � <sup>¼</sup> <sup>6</sup>G<sup>3</sup> <sup>q</sup>

and respectively by:

<sup>D</sup><sup>s</sup> <sup>¼</sup> lim<sup>m</sup>!<sup>∞</sup>

! � � � <sup>6</sup>G<sup>2</sup> <sup>β</sup><sup>s</sup> <sup>q</sup>

Here, the scaling factor is β<sup>s</sup> = 3/10, and the generative functions are given by:

G<sup>2</sup> q !� � <sup>¼</sup> <sup>1</sup> 6 X 5

G<sup>3</sup> q !� � <sup>¼</sup> <sup>1</sup> 6 X 5

grid with cells of finite size, we consider here (as in the previous example) that the scattering units are points centered in the cells. Figure 7 left part, shows the structure generated by Rule 90 for p = 31, 41, 51, and for p = 63 steps, respectively. Note that p = 31 and p = 63 correspond to SG at iterations m = 3, and m = 4, respectively, while p = 41 and p = 52 correspond to some intermediate structures between the two consecutive iterations.

The corresponding structure factors are shown in Figure 7 right part. In the scattering curve, all the three main regions are present: Guinier, fractal, and asymptotic regions. However, as opposed to the Guinier region of the SG generated deterministically, or through CGR, here its length decreases with increasing the number of steps. This is a consequence of the construction algorithm used, namely the structure increases its size at every step. Decreasing the length of the Guinier region leads to an increase of the fractal one, since the smallest distances between scattering points remain the same. This is indicated in the scattering curve, by approximately equal positions (q≃360 nm<sup>1</sup> ) of the first minima, for each of the four steps. As expected, the absolute value of the scattering exponent in the fractal region coincides with the fractal dimension of the structure, and the asymptotic behavior at large q (shown by horizontal dotted lines in Figure 7 right part) tens to 1/N, where N is the total number of scattering points.

#### 4. Small-angle scattering from surface fractals

Recently, it has been shown that surface fractals can be built as a sum of mass fractals [37, 38]. As an example, we consider here a generalized version of the Koch snowflake (KS). The construction algorithm starts from an equilateral triangle with edge length a, which is the zero-th mas fractal iteration (black triangle in Figure 8 left part). At the second step, each side is divided into three segments, each of length a/3, and a new triangle pointing outward is added. The base of the new triangle coincides with the central segment. This is the first mass fractal iteration (three

Figure 8. Left part: Rule 90 for p = 31, 41, 52 and p = 63 number of steps, respectively; right part: the corresponding scattering intensities. The leftmost curve corresponds to p = 63 and the rightmost one corresponds to p = 31.

orange triangles in Figure 8 left part). By repeating the same procedure for each of the new triangles, one obtains the KS having the fractal dimension:

$$D\_s = \lim\_{m \to \infty} \frac{\log 3 \cdot 4^m}{\log \left(a/a\_m\right)} = \frac{\log 4}{\log 3} \approx 1.26,\tag{41}$$

where am is the side length at mth iteration. In Figure 8, each of the triangle is scaled down by a factor of 0.6 � am. The case when the triangles are not scaled down corresponds exactly to the well-known KS and its scattering properties have been extensively studied in [38].

Since a surface fractal can be constructed as a sum of mass fractals, the normalized scattering amplitude is written as a sum of the scattering amplitudes of mass fractals. If we denote Am q !� � � Area að ÞFm <sup>q</sup> !� �, then a recurrence formula for the scattering amplitude of the surface fractal at arbitrarily iteration can be written as [38]:

$$A\_m\left(\stackrel{\rightarrow}{\bar{q}}\right) = 6G\_3\left(\stackrel{\rightarrow}{\bar{q}}\right)\left[\beta\_s^2 A\_{m-1}\left(\beta\_s \stackrel{\rightarrow}{\bar{q}}\right) - 6G\_2\left(\beta\_s \stackrel{\rightarrow}{\bar{q}}\right)\beta\_s^4 A\_{m-2}\left(\beta\_s^2 \stackrel{\rightarrow}{\bar{q}}\right)\right] + \beta\_s^2 A\_{m-1}\left(\beta\_s \stackrel{\rightarrow}{\bar{q}}\right)\left[1 + 6G\_2\left(\stackrel{\rightarrow}{\bar{q}}\right)\right].\tag{42}$$

Here, the scaling factor is β<sup>s</sup> = 3/10, and the generative functions are given by:

$$G\_2\left(\overrightarrow{q}\right) = \frac{1}{6} \sum\_{j=0}^{5} e^{-i\overrightarrow{q}\cdot\overrightarrow{c}\_j} \, \tag{43}$$

and respectively by:

grid with cells of finite size, we consider here (as in the previous example) that the scattering units are points centered in the cells. Figure 7 left part, shows the structure generated by Rule 90 for p = 31, 41, 51, and for p = 63 steps, respectively. Note that p = 31 and p = 63 correspond to SG at iterations m = 3, and m = 4, respectively, while p = 41 and p = 52 correspond to some

Figure 7. Left part: Sierpinski gasket generated using CGR, for different values of the number of scattering units; right part: the corresponding SAS intensities of the Sierpinski gasket from CGR. The highest curve corresponds to N = 30 and

The corresponding structure factors are shown in Figure 7 right part. In the scattering curve, all the three main regions are present: Guinier, fractal, and asymptotic regions. However, as opposed to the Guinier region of the SG generated deterministically, or through CGR, here its length decreases with increasing the number of steps. This is a consequence of the construction algorithm used, namely the structure increases its size at every step. Decreasing the length of the Guinier region leads to an increase of the fractal one, since the smallest distances between scattering points remain the same. This is indicated in the scattering curve, by approximately

absolute value of the scattering exponent in the fractal region coincides with the fractal dimension of the structure, and the asymptotic behavior at large q (shown by horizontal dotted lines in Figure 7 right part) tens to 1/N, where N is the total number of scattering points.

Recently, it has been shown that surface fractals can be built as a sum of mass fractals [37, 38]. As an example, we consider here a generalized version of the Koch snowflake (KS). The construction algorithm starts from an equilateral triangle with edge length a, which is the zero-th mas fractal iteration (black triangle in Figure 8 left part). At the second step, each side is divided into three segments, each of length a/3, and a new triangle pointing outward is added. The base of the new triangle coincides with the central segment. This is the first mass fractal iteration (three

) of the first minima, for each of the four steps. As expected, the

intermediate structures between the two consecutive iterations.

184 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

4. Small-angle scattering from surface fractals

equal positions (q≃360 nm<sup>1</sup>

the lowest one corresponds to N = 430.

$$\mathcal{G}\_3\left(\overrightarrow{\vec{q}}\right) = \frac{1}{6} \sum\_{j=0}^5 e^{-i\overrightarrow{q}\cdot\overrightarrow{b}\_j}.\tag{44}$$

The translation vectors are defined as:

$$\overrightarrow{\boldsymbol{c}}\_{j} = \frac{2a}{9} \{ \cos \pi (j + 1/2)/3, \sin \pi (j + 1/2)/3 \},\tag{45}$$

the scattering units are much bigger than the sizes of the units, the approximation of the uncorrelated positions of the triangles works fairy well in the fractal region [38] (Figure 10).

Figure 10. Left part: a generalized Koch snowflake model. Right part: highly oscillating curve—the total scattering intensity, smoother curve—the scattering intensity from a system containing the same number of triangles but whose

Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 187

In this chapter, we have presented and discussed some general concepts in small-angle scattering (SAS; neutrons, X-ray, light) from deterministic mass and surface fractals. To do so, we have considered the Sierpinski gasket (SG) as a model for deterministic mass fractals, and a generalized version of Koch snowflake (KS) as a model for deterministic surface fractals. The model for SG has been introduced through three main algorithms: deterministic and random iteration function system (IFS), and through cellular automata (CA). KS has been constructed

The SAS intensities (fractal and structure factor) has been calculated from a system of 2D, monodisperse diluted (i.e., spatial correlation can be neglected) and randomly oriented fractals. They are characterized by the presence of three main regions: Guinier (at low q), fractal (at intermediate q), and Porod/asymptotic (at high q). We have shown that in the case of mass fractals, we can extract structural information about the fractal dimension (from the exponent of the SAS curve in the fractal region), scaling factor (from the periodicity of minima in the fractal region), iteration number (from the number of minima in the fractal region), and the total number of scattering units inside a fractal (from the value of the structure factor in the asymptotic region). In addition, mass fractals generated using IFS are able to reproduce fairly well the scattering curve of deterministic IFS under the proper conditions. Mass fractals generated from CA increase their size with increasing the number of steps. This growing process is also reflected by a decrease of the length of Guinier region. We have also shown that

5. Conclusions

positions are random.

in the framework of deterministic IFS.

and the vectors b ! <sup>j</sup> are given by Eq. (26). Thus, using Eq. (2), the non-normalized scattering amplitude is I qð Þ∝ Am q ! <sup>2</sup> , provided the amplitudes at <sup>m</sup> = 0 and <sup>m</sup> = 1 are known.

Figures 8 and 9 left parts show the construction of the generalized KS when the ratio between the sizes of the triangles and the distances between them are 0.6 and 0.4, respectively. Figures 8 and 9 right parts show the corresponding scattering intensities. The general feature is that when the ratio is 0.4, in the fractal region the overall agreement between the total scattering intensity and the intensity corresponding to uncorrelated triangles is slightly better. The curve also shows that the absolute value of the scattering exponent is now 6 � Ds, with D<sup>s</sup> = 1.26 given by Eq. (41), which is a "signature" of scattering from surface fractals. The reason for such behavior is that at a given iteration, the surface fractal consists of scattering units of different sizes following a powerlaw distribution (see Figure 8 left part), while mass fractals consist of scattering units of the same size (see Figure 5 left part). A mathematical derivation of the value of the scattering exponent in the case of scattering from surface fractals can be found in [37, 38].

The centers of triangles in Figure 8 left part coincide with the centers of triangles of the regular KS. From another hand, it has been recently shown that the scattering intensity corresponding to a system of triangles whose sizes follow a power-law distribution and whose positions are uncorrelated, approximates under certain conditions the scattering intensity of the same system but when the positional correlations are not considered [38]. Figure 8 right part, shows also the scattering intensity of a system of triangles whose positions are uncorrelated (red curve). We can observe that in the fractal region (20≲q≲400 nm�<sup>1</sup> ), the approximation is not quite satisfactory. The reason of this behavior is that here the scaling factor is βs, and thus the distances between the centers of the triangles is of the same order as their size. However, when the distances between

Figure 9. Left part: a generalized Koch snowflake model. Right part: highly oscillating curve—the corresponding scattering intensity, smoother curve—the scattering intensity from a system containing the same number of triangles but whose positions are randomly.

Figure 10. Left part: a generalized Koch snowflake model. Right part: highly oscillating curve—the total scattering intensity, smoother curve—the scattering intensity from a system containing the same number of triangles but whose positions are random.

the scattering units are much bigger than the sizes of the units, the approximation of the uncorrelated positions of the triangles works fairy well in the fractal region [38] (Figure 10).

#### 5. Conclusions

c ! <sup>j</sup> <sup>¼</sup> <sup>2</sup><sup>a</sup>

! 

<sup>2</sup>

 

186 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

the case of scattering from surface fractals can be found in [37, 38].

observe that in the fractal region (20≲q≲400 nm�<sup>1</sup>

and the vectors b

positions are randomly.

!

amplitude is I qð Þ∝ Am q

<sup>9</sup> f g cos <sup>π</sup>ð Þ <sup>j</sup> <sup>þ</sup> <sup>1</sup>=<sup>2</sup> <sup>=</sup>3; sin <sup>π</sup>ð Þ <sup>j</sup> <sup>þ</sup> <sup>1</sup>=<sup>2</sup> <sup>=</sup><sup>3</sup> , (45)

, provided the amplitudes at m = 0 and m = 1 are known.

), the approximation is not quite satisfactory.

<sup>j</sup> are given by Eq. (26). Thus, using Eq. (2), the non-normalized scattering

Figures 8 and 9 left parts show the construction of the generalized KS when the ratio between the sizes of the triangles and the distances between them are 0.6 and 0.4, respectively. Figures 8 and 9 right parts show the corresponding scattering intensities. The general feature is that when the ratio is 0.4, in the fractal region the overall agreement between the total scattering intensity and the intensity corresponding to uncorrelated triangles is slightly better. The curve also shows that the absolute value of the scattering exponent is now 6 � Ds, with D<sup>s</sup> = 1.26 given by Eq. (41), which is a "signature" of scattering from surface fractals. The reason for such behavior is that at a given iteration, the surface fractal consists of scattering units of different sizes following a powerlaw distribution (see Figure 8 left part), while mass fractals consist of scattering units of the same size (see Figure 5 left part). A mathematical derivation of the value of the scattering exponent in

The centers of triangles in Figure 8 left part coincide with the centers of triangles of the regular KS. From another hand, it has been recently shown that the scattering intensity corresponding to a system of triangles whose sizes follow a power-law distribution and whose positions are uncorrelated, approximates under certain conditions the scattering intensity of the same system but when the positional correlations are not considered [38]. Figure 8 right part, shows also the scattering intensity of a system of triangles whose positions are uncorrelated (red curve). We can

The reason of this behavior is that here the scaling factor is βs, and thus the distances between the centers of the triangles is of the same order as their size. However, when the distances between

Figure 9. Left part: a generalized Koch snowflake model. Right part: highly oscillating curve—the corresponding scattering intensity, smoother curve—the scattering intensity from a system containing the same number of triangles but whose

In this chapter, we have presented and discussed some general concepts in small-angle scattering (SAS; neutrons, X-ray, light) from deterministic mass and surface fractals. To do so, we have considered the Sierpinski gasket (SG) as a model for deterministic mass fractals, and a generalized version of Koch snowflake (KS) as a model for deterministic surface fractals. The model for SG has been introduced through three main algorithms: deterministic and random iteration function system (IFS), and through cellular automata (CA). KS has been constructed in the framework of deterministic IFS.

The SAS intensities (fractal and structure factor) has been calculated from a system of 2D, monodisperse diluted (i.e., spatial correlation can be neglected) and randomly oriented fractals. They are characterized by the presence of three main regions: Guinier (at low q), fractal (at intermediate q), and Porod/asymptotic (at high q). We have shown that in the case of mass fractals, we can extract structural information about the fractal dimension (from the exponent of the SAS curve in the fractal region), scaling factor (from the periodicity of minima in the fractal region), iteration number (from the number of minima in the fractal region), and the total number of scattering units inside a fractal (from the value of the structure factor in the asymptotic region). In addition, mass fractals generated using IFS are able to reproduce fairly well the scattering curve of deterministic IFS under the proper conditions. Mass fractals generated from CA increase their size with increasing the number of steps. This growing process is also reflected by a decrease of the length of Guinier region. We have also shown that a surface fractal can be considered as a superposition of mass fractals at various iterations and the range of structural information which can be extracted is similar to the case of scattering from mass fractals.

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Small-Angle Scattering from Mass and Surface Fractals http://dx.doi.org/10.5772/intechopen.70870 189

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The IFS and CA algorithms used to generate the mass fractals models can be easily extended to more general structures and can be used to address various questions. For example, in the case of CA, SAS could be used to determine the fractal dimension of an arbitrary structure generated using one dimensional rules, it can shed some light on the randomness of some structures, or it can reveal the existence of a power-law distribution of some entities (of arbitrarily shape) generated by a specific rule.

#### Author details

Eugen Mircea Anitas1,2\*

\*Address all correspondence to: anitas@jinr.ru

1 Joint Institute for Nuclear Research, Dubna, Moscow Region, Russian Federation

2 Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania

#### References


[10] Ariga K. Manipulation of Nanoscale Materials: An Introduction to Nanoarhitectonics. Great Britain: RSC Publishing; 2012. 473 p. DOI: 10.1039/9781849735124

a surface fractal can be considered as a superposition of mass fractals at various iterations and the range of structural information which can be extracted is similar to the case of scattering

The IFS and CA algorithms used to generate the mass fractals models can be easily extended to more general structures and can be used to address various questions. For example, in the case of CA, SAS could be used to determine the fractal dimension of an arbitrary structure generated using one dimensional rules, it can shed some light on the randomness of some structures, or it can reveal the existence of a power-law distribution of some entities (of arbitrarily shape)

1 Joint Institute for Nuclear Research, Dubna, Moscow Region, Russian Federation

2 Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele,

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## *Edited by Ricardo López-Ruiz*

Modeling and simulating biological and physical systems are nowadays active branches of science. The diversity and complexity of behaviors and patterns present in the natural world have their reciprocity in life systems. Bifurcations, solitons and fractals are some of these ubiquitous structures that can be indistinctively identified in many models with the most diverse applications, from microtubules with an essential role in the maintenance and the shaping of cells, to the nano/microscale structure in disordered systems determined with small-angle scattering techniques. This book collects several works in this direction, giving an overview of some models and theories, which are useful for the study and analysis of complex biological and physical systems. It can provide a good guidance for physicists with interest in biology, applied research scientists and postgraduate students.

Published in London, UK © 2018 IntechOpen © Cappan / iStock

Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Complexity in Biological

and Physical Systems

Bifurcations, Solitons and Fractals

*Edited by Ricardo López-Ruiz*