**2. Pendulum motion and dynamics behind a swing**

### **2.1 Pendulum motion**

Motion of nonlinear driven pendulum with friction are widely discussed through numerous literature in Physics and applied Mathematics, (e.g. here, [1–3]). The nonlinear analysis of driven nonlinear pendulum provides a basis for understanding the complexity of various nonlinear dynamical systems. Regular and chaotic motions are observed in such pendulums depending on the numerical values assigned to the parameters associated in their equations of motion. A swing dynamics is very similar to that of nonlinear driven pendulum, [4–7]. In the present text regular and chaotic motion of a pendulum and that of the child's swing is discussed mathematically. Numerical results are presented in various forms of graphics. The equation of motion of a driven pendulum having angular displacement, *θ*, from vertical with linear damping expressed as

$$\frac{d^2\theta}{dt^2} + k\frac{d\theta}{dt} + \omega\_0^2 \sin\theta = F\cos\left(\alpha t\right) \tag{1}$$

smaller than a certain threshold *f <sup>C</sup>* and chaotic when the force *f* is greater than this threshold. An interesting case would also be when the driving force becomes com-

Keeping fixed values for *k* ¼ 0*:*125, *ω*<sup>0</sup> ¼ 1, *ω* ¼ 2*=*3 and then varying forcing amplitude *F*, following regular (periodic) and chaotic motion of the pendulum is

Case (a): for a value *F* ¼ 0*:*2, a time-series plot, phase plot, surface of section and

Case (b): for a value *F* ¼ 0*:*8, corresponding figures of case (a) are obtained as a

The plots shown in **Figure 3** indicate at a value *F* ¼ 0*:*8 the pendulum oscillation is chaotic and this leads to unpredictability. Pendulum may be whirling irregularly

As an application of the foregoing analysis, in the following section, we extend the formalism to discuss the problem of swing oscillation where the length of the

Oscillations of a swing pumped by a child is very familiar to us. Every time the

<sup>2</sup> *dθ*

*dt* <sup>þ</sup> *mgl*sin *<sup>θ</sup>* <sup>¼</sup> <sup>0</sup> (3)

swing passes through its lowest point the child pumps it over and again. The dynamics of weightless rod with a point mass sliding along the length mimics like a pendulum swing whose length varies periodically with time. The motion of the

*Time-series plot, phase plot, surface of section and Poincaré map for periodic motion for F* ¼ 0*:*2*.*

time-series plot, phase plot, surface of section and Poincaré map, are drawn as

parable to the weight.

shown in **Figure 3**.

pendulum varies periodically.

**2.2 Problem of Swing oscillation**

Poincaré map, are drawn as shown in **Figure 2**.

*Chaotic Dynamics and Complexity in Real and Physical Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.96573*

or overturn or show very irregular oscillations.

swing governed by the dynamical system written as [5]:

*d dt ml*<sup>2</sup> *<sup>d</sup><sup>θ</sup> dt* <sup>þ</sup> *<sup>γ</sup><sup>l</sup>*

observed:

**Figure 2.**

**35**

where *F* and *ω* are respectively the amplitude and frequency of the driving force and *k* is the damping coefficient and *ω*<sup>0</sup> ¼ ffiffi *g L* q is the natural frequency for free small-amplitude oscillations. Here *g* refers to acceleration due to gravity and *L* the length of the pendulum. Often it is convenient to express frequency in units of *ω*<sup>0</sup> by setting *ω*<sup>0</sup> ! 1 and rescaling the time unit accordingly. The periodic force *F* cos*ωt* is active and influence the motion of the pendulum.

The Eq. (1) can easily be replaced by equation

$$\frac{d^2\theta}{dt^2} + 2\beta \frac{d\theta}{dt} + a\_0^2 \sin\theta = f a\_0^2 \cos\left(\alpha t\right) \tag{2}$$

Here *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*β*, *<sup>β</sup>* <sup>¼</sup> *<sup>γ</sup> <sup>m</sup>*, i.e., ratio of damping coefficient per unit mass *m*, *ω*<sup>2</sup> <sup>0</sup> <sup>¼</sup> *<sup>g</sup> <sup>L</sup>* and *<sup>f</sup>* <sup>¼</sup> *<sup>F</sup> ω*2 0 . One obtains a bifurcation diagram for Eq. (2), shown in **Figure 1**. Bifurcation scenario indicates a period doubling phenomena followed by chaos. This implies the pendulum oscillations may be regular or chaotic depending on the magnitudes of external forcing.

System (1) or (2) are very common structure with very few degrees of freedom. The simple forced pendulum is periodic in *θ* when the driving force applied is

#### **Figure 1.**

*Bifurcation scenario of damped and driven pendulum for values β* ¼ 0*:*75, *ω* ¼ 2*π*, *ω*<sup>0</sup> ¼ 3*π and* 1*:*06 ≤*f* ≤1*:*09*.*

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

smaller than a certain threshold *f <sup>C</sup>* and chaotic when the force *f* is greater than this threshold. An interesting case would also be when the driving force becomes comparable to the weight.

Keeping fixed values for *k* ¼ 0*:*125, *ω*<sup>0</sup> ¼ 1, *ω* ¼ 2*=*3 and then varying forcing amplitude *F*, following regular (periodic) and chaotic motion of the pendulum is observed:

Case (a): for a value *F* ¼ 0*:*2, a time-series plot, phase plot, surface of section and Poincaré map, are drawn as shown in **Figure 2**.

Case (b): for a value *F* ¼ 0*:*8, corresponding figures of case (a) are obtained as a time-series plot, phase plot, surface of section and Poincaré map, are drawn as shown in **Figure 3**.

The plots shown in **Figure 3** indicate at a value *F* ¼ 0*:*8 the pendulum oscillation is chaotic and this leads to unpredictability. Pendulum may be whirling irregularly or overturn or show very irregular oscillations.

As an application of the foregoing analysis, in the following section, we extend the formalism to discuss the problem of swing oscillation where the length of the pendulum varies periodically.

### **2.2 Problem of Swing oscillation**

**2. Pendulum motion and dynamics behind a swing**

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

ment, *θ*, from vertical with linear damping expressed as

*F* cos*ωt* is active and influence the motion of the pendulum. The Eq. (1) can easily be replaced by equation

*dt* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

tion scenario indicates a period doubling phenomena followed by chaos. This implies the pendulum oscillations may be regular or chaotic depending on the

The simple forced pendulum is periodic in *θ* when the driving force applied is

*Bifurcation scenario of damped and driven pendulum for values β* ¼ 0*:*75, *ω* ¼ 2*π*, *ω*<sup>0</sup> ¼ 3*π and*

*d*2 *θ dt*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>β</sup> <sup>d</sup><sup>θ</sup>*

Here *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*β*, *<sup>β</sup>* <sup>¼</sup> *<sup>γ</sup>*

magnitudes of external forcing.

*<sup>f</sup>* <sup>¼</sup> *<sup>F</sup> ω*2 0

**Figure 1.**

**34**

1*:*06 ≤*f* ≤1*:*09*.*

*dθ dt* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*d*2 *θ dt*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*

and *k* is the damping coefficient and *ω*<sup>0</sup> ¼

Motion of nonlinear driven pendulum with friction are widely discussed through numerous literature in Physics and applied Mathematics, (e.g. here, [1–3]). The nonlinear analysis of driven nonlinear pendulum provides a basis for understanding the complexity of various nonlinear dynamical systems. Regular and chaotic motions are observed in such pendulums depending on the numerical values assigned to the parameters associated in their equations of motion. A swing

dynamics is very similar to that of nonlinear driven pendulum, [4–7]. In the present text regular and chaotic motion of a pendulum and that of the child's swing is discussed mathematically. Numerical results are presented in various forms of graphics. The equation of motion of a driven pendulum having angular displace-

where *F* and *ω* are respectively the amplitude and frequency of the driving force

small-amplitude oscillations. Here *g* refers to acceleration due to gravity and *L* the length of the pendulum. Often it is convenient to express frequency in units of *ω*<sup>0</sup> by setting *ω*<sup>0</sup> ! 1 and rescaling the time unit accordingly. The periodic force

ffiffi *g L* q

<sup>0</sup> sin *<sup>θ</sup>* <sup>¼</sup> *<sup>f</sup>ω*<sup>2</sup>

. One obtains a bifurcation diagram for Eq. (2), shown in **Figure 1**. Bifurca-

System (1) or (2) are very common structure with very few degrees of freedom.

*<sup>m</sup>*, i.e., ratio of damping coefficient per unit mass *m*, *ω*<sup>2</sup>

<sup>0</sup> sin *θ* ¼ *F* cosð Þ *ωt* (1)

is the natural frequency for free

<sup>0</sup> cosð Þ *ωt* (2)

<sup>0</sup> <sup>¼</sup> *<sup>g</sup> <sup>L</sup>* and

**2.1 Pendulum motion**

Oscillations of a swing pumped by a child is very familiar to us. Every time the swing passes through its lowest point the child pumps it over and again. The dynamics of weightless rod with a point mass sliding along the length mimics like a pendulum swing whose length varies periodically with time. The motion of the swing governed by the dynamical system written as [5]:

$$\frac{d}{dt}\left[ml^2\frac{d\theta}{dt}\right] + \gamma l^2\frac{d\theta}{dt} + mgl\sin\theta = 0\tag{3}$$

**Figure 2.** *Time-series plot, phase plot, surface of section and Poincaré map for periodic motion for F* ¼ 0*:*2*.*

**Figure 3.** *Time-series plot, phase plot, surface of section and Poincaré map for periodic motion for F* ¼ 0*:*8*.*

where *m* is the mass, *l* is the length, *θ* is the angle made by the swing from the vertical position and *g* is the acceleration due to gravity. As the length of the swing varies periodically with time, one assumes

$$l = l\_0 + a\phi(\Omega t) \tag{4}$$

\_

*J* ¼

€*θ* þ

behavior.

**Figure 4.**

**37**

@

form of the equation of motion stands as

2*kλ* cosð Þ *λτ* <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* <sup>þ</sup> *βω* � �\_

**2.3 Regular and Chaotic motion of the swing**

**Figures 4** and **5** showing the case of regular motion.

*θ* ¼ *u* � *f*ð Þ *θ*, *u*

*DOI: http://dx.doi.org/10.5772/intechopen.96573*

*<sup>u</sup>*\_ ¼ � <sup>2</sup>*k<sup>λ</sup>* cosð Þ *λτ*

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* <sup>þ</sup> *βω* � �*<sup>u</sup>* � *<sup>ω</sup>*<sup>2</sup>

0 1

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* cos *<sup>θ</sup>* � <sup>2</sup>*k<sup>λ</sup>* cosð Þ *λτ* <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* <sup>þ</sup> *βω* � � <sup>0</sup>

When an external periodic force *F* cosð Þ *ϑτ* is applied to pump the swing, final

*<sup>θ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

The swing, Eq. (8), oscillates in regular motion for significant contribution of friction, (i.e. when the frictional coefficient *β* has sufficiently higher value) and it is in chaotic motion in case of small friction and higher values of driving force.

When the frictional contribution is insignificant, swing oscillations are chaotic and unpredictable. **Figure 6** stands for such chaotic motion of the swing when *β* ¼ 0. **Figure 7** show chaotic oscillation when *β* is not zero but small. Surface of section

We may thus conclude that the swing oscillates smoothly when the frictions are

and Poincare map shown in this figure are interesting showing typical chaotic

higher but for no friction or insignificant friction, swing oscillations would be

*A time-series and phase plots and plots of surface of section and Poincaré map for regular motion of the swing*

*for F* ¼ 0*:*8, *β* ¼ 0*:*5, *k* ¼ 0*:*1, *λ* ¼ 0*:*05, *ω* ¼ 1, *ϑ* ¼ 2*=*3*:*

The Jacobian for the above system may be written as,

*Chaotic Dynamics and Complexity in Real and Physical Systems*

� *<sup>ω</sup>*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* sin *<sup>θ</sup>* � *<sup>g</sup>*ð Þ *<sup>θ</sup>*, *<sup>u</sup>*

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* sin *<sup>θ</sup>* <sup>¼</sup> *<sup>F</sup>* cosð Þ *ϑτ* (8)

1 A (7)

where *l*<sup>0</sup> is the mean length of the swing and is constant, *a* and Ω are, respectively, the amplitude and frequency of excitation. The function *ϕ*ð Þ Ω*t* should be a periodic function of time. Then, by introducing the following dimensionless parameters and variables

$$
\tau = \Omega \, t, \quad \varepsilon = \frac{a}{l\_0}, \quad \Omega\_0 = \sqrt{\frac{\mathcal{g}}{l\_0}}, \quad \mathcal{o} = \frac{\Omega\_0}{\Omega}, \quad \beta = \frac{\mathcal{\chi}}{m\Omega\_0}, \quad \varepsilon
$$

equation of motion of the swing in dimensionless form written as:

$$\ddot{\theta} + \left[\frac{2\epsilon\dot{\phi}(\tau)}{\mathbf{1} + \epsilon\phi(\tau)} + \beta\alpha\right]\dot{\theta} + \frac{\alpha^2}{\mathbf{1} + \epsilon\phi(\tau)}\sin\theta = \mathbf{0} \tag{5}$$

where f g*:* in Eq. (5) corresponds to differentiation with respect *τ*.

Since *ϕ τ*ð Þ is a periodic function, we may take *ϕ τ*ð Þ¼ *A* sin ð Þ *λτ* and thus the foregoing equation may be rewritten as:

$$\ddot{\theta} + \left[ \frac{2k\lambda\cos\left(\lambda\tau\right)}{1 + k\sin\left(\lambda\tau\right)} + \beta\alpha \right] \dot{\theta} + \frac{\alpha^2}{1 + k\sin\left(\lambda\tau\right)} \sin\theta = 0 \tag{6}$$

where *k* ¼ *εA*.

For stability of motion of the swing a linear stability analysis is applied. We may write Eq. (6) as the following two first order equations:

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

$$\begin{aligned} \dot{\theta} &= u \equiv f(\theta, u) \\ \dot{u} &= -\left[ \frac{2k\lambda \cos(\lambda \tau)}{1 + k \sin(\lambda \tau)} + \beta o \right] u - \frac{\alpha^2}{1 + k \sin(\lambda \tau)} \sin \theta \equiv \mathbf{g}(\theta, u) \end{aligned} \tag{7}$$

The Jacobian for the above system may be written as,

$$J = \begin{pmatrix} 0 & 1 \\ -\frac{\alpha^2}{\mathbf{1} + k\sin\left(\lambda\pi\right)}\cos\theta & -\left[\frac{2k\lambda\cos\left(\lambda\pi\right)}{\mathbf{1} + k\sin\left(\lambda\pi\right)} + \beta\alpha\right] \end{pmatrix}$$

When an external periodic force *F* cosð Þ *ϑτ* is applied to pump the swing, final form of the equation of motion stands as

$$\ddot{\theta} + \left[ \frac{2k\lambda\cos\left(\lambda\tau\right)}{1 + k\sin\left(\lambda\tau\right)} + \beta\alpha \right] \dot{\theta} + \frac{\alpha^2}{1 + k\sin\left(\lambda\tau\right)} \sin\theta = F\cos\left(\theta\tau\right) \tag{8}$$

#### **2.3 Regular and Chaotic motion of the swing**

The swing, Eq. (8), oscillates in regular motion for significant contribution of friction, (i.e. when the frictional coefficient *β* has sufficiently higher value) and it is in chaotic motion in case of small friction and higher values of driving force. **Figures 4** and **5** showing the case of regular motion.

When the frictional contribution is insignificant, swing oscillations are chaotic and unpredictable. **Figure 6** stands for such chaotic motion of the swing when *β* ¼ 0.

**Figure 7** show chaotic oscillation when *β* is not zero but small. Surface of section and Poincare map shown in this figure are interesting showing typical chaotic behavior.

We may thus conclude that the swing oscillates smoothly when the frictions are higher but for no friction or insignificant friction, swing oscillations would be

#### **Figure 4.**

*A time-series and phase plots and plots of surface of section and Poincaré map for regular motion of the swing for F* ¼ 0*:*8, *β* ¼ 0*:*5, *k* ¼ 0*:*1, *λ* ¼ 0*:*05, *ω* ¼ 1, *ϑ* ¼ 2*=*3*:*

where *m* is the mass, *l* is the length, *θ* is the angle made by the swing from the vertical position and *g* is the acceleration due to gravity. As the length of the swing

*Time-series plot, phase plot, surface of section and Poincaré map for periodic motion for F* ¼ 0*:*8*.*

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where *l*<sup>0</sup> is the mean length of the swing and is constant, *a* and Ω are, respectively, the amplitude and frequency of excitation. The function *ϕ*ð Þ Ω*t* should be a periodic function of time. Then, by introducing the following dimensionless

> ffiffiffiffi *g l*0 r

> > \_ *<sup>θ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

Since *ϕ τ*ð Þ is a periodic function, we may take *ϕ τ*ð Þ¼ *A* sin ð Þ *λτ* and thus the

\_

For stability of motion of the swing a linear stability analysis is applied. We may

*<sup>θ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

, *<sup>ω</sup>* <sup>¼</sup> <sup>Ω</sup><sup>0</sup>

*l* ¼ *l*<sup>0</sup> þ *aϕ*ð Þ Ω *t* (4)

<sup>Ω</sup> , *<sup>β</sup>* <sup>¼</sup> *<sup>γ</sup>*

*m*Ω<sup>0</sup> ,

<sup>1</sup> <sup>þ</sup> *εϕ τ*ð Þ sin *<sup>θ</sup>* <sup>¼</sup> <sup>0</sup> (5)

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* sin *<sup>θ</sup>* <sup>¼</sup> 0 (6)

varies periodically with time, one assumes

*<sup>τ</sup>* <sup>¼</sup> <sup>Ω</sup> *<sup>t</sup>*, *<sup>ε</sup>* <sup>¼</sup> *<sup>a</sup>*

€*θ* þ

foregoing equation may be rewritten as:

€*θ* þ

where *k* ¼ *εA*.

**36**

*l*0

2*εϕ τ* \_ð Þ <sup>1</sup> <sup>þ</sup> *εϕ τ*ð Þ <sup>þ</sup> *βω* � �

2*kλ* cosð Þ *λτ* <sup>1</sup> <sup>þ</sup> *<sup>k</sup>* sin ð Þ *λτ* <sup>þ</sup> *βω* � �

write Eq. (6) as the following two first order equations:

, Ω<sup>0</sup> ¼

equation of motion of the swing in dimensionless form written as:

where f g*:* in Eq. (5) corresponds to differentiation with respect *τ*.

parameters and variables

**Figure 3.**

chaotic or unpredictable. In such a case whirling, overturn or any unpredictable

section technique, we introduce the idea of Lyapunov characteristic exponents (LCE), correlation dimension and topological entropy which provide further insight of a complex dynamical system. In the following section, we analyze the complexity

**3. Complexity in prey-predator system with Allee effect**

*Yn*þ<sup>1</sup> ¼ *Yn* þ *aYn*ð Þ *Xn* � *Yn*

is defined as the Allee effect constant and the term

Beside the application of bifurcation diagram, phase plot and Poincare surface of

In recent years many type of predator-prey problems, originated in Biological sciences, investigated which depend on various environmental and social conditions, [8–11]. Some problems solved by the application of Allee effect, which is an interesting phenomenon, to some predator-prey systems appear to be very interesting, [12–16]. The Allee effect on prey-predator system is a phenomenon in biology which characterizes certain correlation between population size or density and the mean individual fitness of a population or species. In the

following study we investigate the complexity in a predator – prey problem with

A model for the prey-predator problem with Allee effect can represented as

*Xn*þ<sup>1</sup> ¼ *Xn* þ *rXn*ð Þ 1 � *Xn* ð1 � exp ð Þ �*εXn* Þ � *aXnYn*

where *Xn* and *Yn* refers to the density of prey and predators. Further, *r* correspond to the growth rate parameter of the prey population and *a* the predation

• 1 � exp ð Þ �*εXn* stands for mate finding Allee effect on prey population, here *ε*

stands for the Allee effect on predator and here, *μ* is the Allee effect constant. Bigger *μ* means the stronger the Allee effect on predator population.

For assumed values of parameters *a* ¼ 2*:*0, *r* ¼ 2*:*4, fixed points of system (9)

The phenomena of bifurcation provide a qualitative change in the behavior of a

<sup>2</sup> ð Þ 1, 0 , *<sup>P</sup>*<sup>∗</sup>

<sup>1</sup> ð Þ 0, 0 , *<sup>P</sup>*<sup>∗</sup>

system during evolution. Such a change occurs when a particular parameter is varied while keeping other parameters constant. Bifurcation diagram shows the splitting of stable solutions within a certain range of values of the parameter. During the processes of bifurcation, one observes different cycles of evolution which leading to the chaotic situation. Phenomena like bistability, periodic windows within chaos etc. may also be observed for some systems. A bifurcation can be taken as a

*Yn μ* þ *Yn*

(9)

<sup>3</sup> ð Þ 0*:*545455, 0*:*545455 and

situation may happen.

the Allee effect.

prameter. Here,

• *Yn μ*þ*yn*

**39**

**3.1 Discrete prey-predator model**

are obtained, approximately, as *P*<sup>∗</sup>

**3.2 Bifurcation diagrams**

by using stability analysis, we find all are unstable.

of Prey-predator system using such tools.

*DOI: http://dx.doi.org/10.5772/intechopen.96573*

*Chaotic Dynamics and Complexity in Real and Physical Systems*

**Figure 5.** *Regular two periodic motion of the swing when F* ¼ 0*:*9, *β* ¼ 0*:*5, *k* ¼ 0*:*3, *λ* ¼ 1, *ω* ¼ 1, *ϑ* ¼ 2*=*3*.*

**Figure 6.** *Chaotic oscillation of the swing when F* ¼ 0*:*2, *β* ¼ 0, *k* ¼ 0*:*1, *λ* ¼ 0*:*05, *ω* ¼ 1, *ϑ* ¼ 2*=*3*.*

**Figure 7.** *Chaotic oscillation of the swing when F* ¼ 0*:*2, *β* ¼ 0*:*01, *k* ¼ 0*:*1, *λ* ¼ 0*:*05, *ω* ¼ 1, *ϑ* ¼ 2*=*3*.*

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

chaotic or unpredictable. In such a case whirling, overturn or any unpredictable situation may happen.

Beside the application of bifurcation diagram, phase plot and Poincare surface of section technique, we introduce the idea of Lyapunov characteristic exponents (LCE), correlation dimension and topological entropy which provide further insight of a complex dynamical system. In the following section, we analyze the complexity of Prey-predator system using such tools.
