Preface

The book is devoted to recent achievements in chaos theory. It discusses the mathematical treatment of chaos phenomena, chaotic dynamical systems, and self‐organization processes. It consists of five chapters describing dynamical chaotic phenomena in evolutionary processes, medicine, chemistry, and electromagnetism.

Chapter 1 presents a short historical overview of chaos theory, focusing on the role of prominent scientists worldwide in developing this theory. The chapter emphasizes the importance of chaos theory and explains that many phenomena in nature, science, and human life can be explained correctly only when taking into account their irregularity. Some chaotic processes in nature are the prototypes for the development of research methods in science. One such example is Particle Swarm Optimization (PSO), which is widely applied to solve optimization problems in different areas of scientific research.

Chapter 2 by Prof. Masahiro Nakagawa studies the effectiveness of a new chaos auto‐associated (memory retrieval) model based on the Chebyshev‐type activation function. The approach foresees specific initial data as the autocorrelation connection matrix between neurons, the dependence of memory retrieval properties on the initial Hamming distance between the input, and target vectors.

The objective characteristic of the process, namely its retrieval ability, is measured in terms of the memory capacity of the associated memory. This work aims to try to improve the existing chaos retrieval models to increase the effectiveness of retrieving the chaotic neuron state. As is shown in the chapter, the Chebyshev‐type activation function is proper for this goal.

The theoretical background of the chaos neuron state is supplemented by the autocorrelation connections between neurons, the relation between the number of embedded vectors and the number of neurons, the activation function, the control parameter, and the relation between the Hamming distance and number of neurons. This allows for more effective retrieval in terms of memory capacities.

The advantage of the proposed approach is confirmed by a series of computations related to the extraction of memory retrieval characteristics for the Chebyshev, sinusoidal, and signum activation functions; the time dependence of internal states is examined comprehensively. The presented numerical data confirm that the proposed model is more effective in terms of memory capacities related to the robustness of the included noise.

Chapter 3 by Prof. Minoru Saito focuses on a statistical-physical approach to studying neurite morphology, which is based on the Loewner equation connected with stochastic driving function. This enables investigation of the neurite outgrowth process, which is described by the Loewner equation dealing with a chaotic driving function.

Such a process is different from the known stochastic Loewner evolution. The practical importance of the approach proposed is that the discussed function is presented in some physical systems and the stochastic driving function under consideration can be used for modeling the processes having complex curves in nature.

determination of the electrical field at the surface of particles. This allows for excluding integrating the Green function derivatives, which are contained in a kernel of this

Since progress in chaos theory predetermines its application in the many areas of pure and applied sciences, this book will be useful for scientists and industrial engineers as

It is my great pleasure to thank all authors for improving their chapters throughout the review process. In addition, I express my sincere thanks to Ms. Ana Cink at IntechOpen for her professional support during the preparation of this book.

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics,

**Mykhaylo Andriychuk**

Lviv Polytechnic National University,

NASU, Lviv, Ukraine

Lviv, Ukraine

Practical application of the approach yields an ability to create media or materials with the desired inhomogeneous distribution of effective refractive index and magnetic permeability. Explicit analytical formulas are derived for these physical parameters and supported by computations. The numerical results demonstrate that the chaotic distribution of particles in the initial medium makes it possible to obtain more contrasting material parameters in contradiction to the regular distribution of

boundary integral equation.

well as post‐graduate students and beyond.

particles.

It is substantiated that neurite morphology is useful for describing the scaling and chaotic properties of the driving functions, which are solutions of the Loewner equation.

To obtain the numerical data, the chapter author adopted a zipper algorithm based on the specific map that makes it possible to get the set of solutions to the Loewner equation as trajectories in the complex plane. For obtaining such trajectories, a method of small increments is applied. The proposed method allows for studying neurite morphology, which is described by the Loewner equation. The practical medical problem, which is investigated by the proposed approach, is an analysis of the neurite morphology of human‐induced pluripotent stem cells.

Chapter 4 by Prof. Anuj K. Shah and Prof. Enrique Peacock‐Lopez studies chemical self‐replication, which is described by a system of differential equations of the first order in regular derivatives. The work is a generalization of the author's previous study of Rebek's and Joyce's self‐replicating systems, which model ideal self‐replication using a self‐complementary template mechanism. This is based on the chemical model based on laboratory self‐implication. The used differential model foresees two types of initial conditions that consider both stoichiometry and the principle of conservation of mass; this requires investigation of two types of dynamic manifolds that describe the appearance of different types of dynamic curves.

The behaviors of the chemical system in the absence of one of the competing templates are described using the system of five ordinary differential equations in the regular derivatives. The difference in the initial condition that is the result of two different initial concentrations introduces a mass constraint on the dynamic behavior of the concentrations. The phase plots of concentrations describe the dynamic characteristics of solutions of the dynamic system of differential equations. The analysis of initial conditions allows for checking the different regimes of the chemical processes under consideration. The time attractor confirms the existence of different types of non‐limiting reagents. The important conclusion is that the difference in initial concentrations introduces mass constraints on the dynamic behavior of the concentrations. The numerous plots demonstrate the correctness of the study of chemical processes using the system of dynamical differential equations.

Chapter 5 by Prof. Mykhaylo Andriychuk and Mr. Borys Yevstyneiev examines the problem of electromagnetic wave scattering on a set of chaotic placed small‐size impedance particles of arbitrary shape with the application of an asymptotic approach. The particles are distributed in a homogeneous domain with constant material parameters. The solution to the scattering problem is derived under the condition that the characteristic size of particles tends to zero; in addition, the quantity of particles tends to infinity at a specific rate.

To obtain a solution an explicit relation to the vector of the electrical field is derived that excludes the necessity to solve the governing integral equation for the determination of the electrical field at the surface of particles. This allows for excluding integrating the Green function derivatives, which are contained in a kernel of this boundary integral equation.

Practical application of the approach yields an ability to create media or materials with the desired inhomogeneous distribution of effective refractive index and magnetic permeability. Explicit analytical formulas are derived for these physical parameters and supported by computations. The numerical results demonstrate that the chaotic distribution of particles in the initial medium makes it possible to obtain more contrasting material parameters in contradiction to the regular distribution of particles.

Since progress in chaos theory predetermines its application in the many areas of pure and applied sciences, this book will be useful for scientists and industrial engineers as well as post‐graduate students and beyond.

It is my great pleasure to thank all authors for improving their chapters throughout the review process. In addition, I express my sincere thanks to Ms. Ana Cink at IntechOpen for her professional support during the preparation of this book.

> **Mykhaylo Andriychuk** Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, Lviv, Ukraine Lviv Polytechnic National University, Lviv, Ukraine

**1**

**Chapter 1**

Nature and Life

*Mykhaylo Andriychuk*

**1. Introduction**

Introductory Chapter: Chaos in

This chapter provides a short overview related to chaos theory, its basic, the impact of the prominent world scientists to develop the chaos theory, and some examples of dynamical phenomena, which are characterized by chaotic nature. J.C. Maxwell was one of the authors who started discussion about a chaos theory referring to the "butterfly effect" between the 1860s and 1870s (see [1, 2]). H. Poincaré was an early adherent of chaos theory as well. He observed that there can exist the nonperiodic orbits, which are increasing not forever and not approaching to a fixed point, at study of the three-body problem in the 1880s [3, 4]. J. Hadamard published a highpowered work about the chaotic motion of free particle, which glides frictionlessly on some surface with negative constant curvature in 1898 [5]; and this phenomenon got the name "Hadamard's billiards." It was shown in this work that all trajectories are unstable, and trajectories of all particles deviate exponentially one from one another,

and this deviation is characterized by a Lyapunov exponent, which is positive.

in spite of that the theory cannot explain what they were seeing.

An ergodic theory initiated the development of the chaos science. Later, the progress of the chaos theory was connected with studying the nonlinear differential equations. Many prominent scientists, namely Birkhoff [6], Kolmogorov [7], Cartwright and Littlewood [8], and Smale [9] dealt with this problem. A little bit later, chaos studies were inspired in physics: the Birkhoff research on three-body problems, the Kolmogorov turbulence and astronomical problems, and the Cartwright and Littlewood studies related to radio engineering. In the area of planetary mechanics, fluid dynamics, and electromagnetics, the scientists had observed a chaotic planetary motion, a turbulence in fluid flow, and a nonperiodic oscillation in the radio circuits

In spite of an initial understanding in the first half of the twentieth century, chaos theory was fully formalized as specific theory in its second half; when the linear theory, the theory of prevailing system could not explain clearly the observed properties of some experiments such as the logistic map, for example. Such misunderstanding phenomena were referred to the inaccuracy of measurement and appearing "noise" was counted by scientists as an inherent component of the systems under study. B. V. Chirikov introduced a criterion for the genesis of the chaos classical phenomena in the Hamiltonian systems, known as the Chirikov criterion, in 1959. This criterion was applied by him to explain series experimental results related to plasma imprisonment in the open mirror catches [10]. This research can be considered as a pioneer physical chaos theory, which allows to explain this

## **Chapter 1**
