Recent Advances in Chaotic Systems and Synchronization: From Theory to Real World Applications

Recent Advances in Chaotic Systems and Synchronization: From Theory to Real World Applications is a major reference for scientists and engineers interested in applying new computational and mathematical tools for solving complex problems related to modeling, analyzing and synchronizing chaotic systems. Furthermore, it offers an array of new, real-world applications in the field. Written by eminent scientists in the field of control theory and nonlinear systems from 19 countries (Cameroon, China, Ethiopia, France, Greece, India, Italia, Iran, Japan, Mexico, and more), this book covers the latest advances in chaos theory, along with the efficiency of novel synchronization approaches.

Readers will find the fundamentals and algorithms related to the analysis and synchronization of chaotic systems, along with key applications, including electronic design, text and image encryption, and robot control and tracking.

• Explores and evaluates the latest real-world applications of chaos across various engineering and biomedical engineering fields
• Investigates advances in chaos synchronization techniques, including the continuous sliding-mode control approach, hybrid synchronization between chaotic and hyperchaotic systems, and neural network synchronization
• Presents recent advances in chaotic systems through an overview of new systems and new proprieties

• Language: English
• Publisher: Academic Press
• Release date: Nov 5, 2018
• ISBN: 9780128162668

Book preview

Cameroon

Preface

Olfa Boubaker; Sajad Jafari

Chaos is a science full of surprises and unpredictability. It teaches us to expect the unexpected. Chaos is also the property of complex systems whose behaviors are so unpredictable as to appear at random, owing to its great sensitivity to small changes in initial conditions. It is believed that chaos exists in realworld systems (all of which are nonlinear) like weather, brain waves, turbulence in fluids, erratic flows of epidemics, arrhythmic heartbeats in the moments before death, and so on.

The main objective of this book is to explore new developments related to chaos theory and its pivotal role in several fields. The selected contributions shed lights on a series of interesting issues related to a range of novel chaotic and hyperchaotic systems, with the aim of demonstrating several novel proprieties as well relevant real-world applications. Experimental results are presented from a series of selected disciplines. A set of advanced chaos synchronization schemes are also included which are supported by relevant applications and experiments.

The book is a timely and comprehensive referencing guide for graduate students, researchers, and practitioners in the area of chaos theory. It presents a clear and concise introduction to the field of chaos theory, suitable not only for researchers in nonlinear systems, control theory, telecommunications, mathematics, and physics, but also in chemistry, medicine, economy, and natural and social sciences. It covers a wide range of topics not usually found in similar books. The motivations of the respective subjects and a clear presentation ease understanding. The book contains worked examples, codes, and videos which make it ideal for an introductory course for students as well as for researchers starting to work in the field. It is also particularly suitable for engineers wishing to enter the field quickly and efficiently.

Written by eminent scientists in the field from 15 countries (Algeria, Cameroon, China, Ethiopia, Greece, India, Iran, México, Poland, Saudi Arabia, the Slovak Republic, Turkey, Tunisia, the United Kingdom, Vietnam), this book offers a concise introduction to the many facets of chaos theory. It covers the latest advances in concise methodologies and key concepts for modeling and analyzing chaos dynamics. A range of new chaotic systems are presented and analyzed over an array of novel proprieties, from circuit design to real-time experiments. New applications highlighting the potential role of chaos in real-world applications are extensively exposed. Last but not least, new trends in chaos synchronization are exposed, where proposed novel synchronization schema are verified for real-world applications, not only via simulation results, but also via some experimental tests.

In total, 17 chapters written by active researchers in the field are compiled in this book to provide an overall picture of the most challenging problems to be solved in chaos theory. The book covers the topic in three parts: Part I, organized in seven chapters, introduces new chaotic systems from design to analysis; Part II, structured in five chapters, describes realworld applications of chaos theory; and finally, Part III, presented in five chapters, introduces new trends in chaos synchronization.

The following researchers are particularly acknowledged for their considerable efforts:

•We are grateful to our collaborators in this artwork, Professor Guanrong Chen from City University of Hong Kong, China, Professor Jun Ma from Lanzhou University of Technology, China, Professor Nikolay Vladimirovich Kuznetsov from Saint Petersburg State University, Russia, and Professor Seyed Mohammad Reza Hashemi Golpayegani from Amirkabir University of Technology, Iran, for their helpful and professional efforts to provide precious comments and reviews.

•We particularity thank Emeritus Professor Mohamed Benrejeb from National Engineering School of Tunis, Tunisia, Professor Sundarapandian Vaidyanathan from Vel Tech University, India, and Professor Tomasz Kapitaniak from Lodz University of Technology, Poland, for the honor they give us by contributing to this artwork.

•Our particular thanks are given to our main collaborators in this work, Professor Viet-Thanh Pham from Hanoi University of Science and Technology, Vietnam, Professor Rachid Dhifaoui from National Institute of Applied Sciences and Technology, Tunisia, and Assistant Professor Christos Volos from Aristotle University of Thessaloniki, Greece.

•The authors of Chapter 1, Abir Lassoued and collaborators, of Chapter 2, Sifeu T. Kingni and collaborators, of Chapter 8, Branislav Sobota and Milan Guzan, and of Chapter 13, Hafiz Ahmed and collaborators, are particularly acknowledged for their efforts to provide experimental data and videos.

Finally, we would like to express our gratefulness to all authors of the book for their valuable contributions and to all reviewers for their helpful and professional efforts to provide precious comments and feedback. In the end, we dedicate this book to the memory of Professor Gennady A. Leonov, who helped and guided us in many stages of the preparation of this book.

Part I

New Chaotic Systems: Design and Analysis

Chapter 1

Experimental Observations and Circuit Realization of a Jerk Chaotic System With Piecewise Nonlinear Function

Abir Lassoued⁎; Olfa Boubaker⁎; Rachid Dhifaoui⁎; Sajad Jafari†    ⁎ National Institute of Applied Sciences and Technology, Tunis, Tunisia

† Biomedical Engineering Department, Amirkabir University of Technology, Tehran, Iran

Abstract

In this chapter, a novel chaotic system characterized by only one nonlinear term is introduced. This single term is composed by a piecewise nonlinear function taken as the only chaotic generator. On the other hand, the proposed chaotic system is based on the jerk equation, which has a simple algebraic structure highly adapted for circuit design. Despite the simplicity of the designed structure, the chaotic system exhibits highly complex dynamic behaviors compared to related ones based on jerk systems. The dynamic behaviors of the proposed system are investigated by theoretical analysis focusing on its elementary characteristics such as Lyapunov exponents, Kaplan-Yorke dimension, attractor forms, and equilibrium points. To enhance the applicability of the proposed system, an electronic circuit is designed by using the MultiSIM Software. Finally, experimental investigations show the efficiency of the designed circuit and prove the good qualitative agreement between theoretical analysis, numerical simulations, and experimental results.

Keywords

Jerk system; Chaotic behaviors; Nonlinear piecewise function; Circuit design; Experimental chaos

1 Introduction

Currently, new chaotic systems and circuits have received considerable interest in the research community due to their potential application in nontraditional areas such as secure communication [1, 2], robotics [3], and encryption applications [4]. In order to satisfy the real needs imposed by these technologies, a great number of chaotic circuits have been discovered these last few years [5–7]. In fact, many researchers have attempted to build strange attractors with as simple as possible nonlinear algebraic structures. These systems are strongly recommended for electrical implementations. However, for chaotic circuits, building chaotic attractors is still an open research direction, and further results are expected.

, where G(x) is a classical nonlinear function [8]. Remarkably, a variety of conventional nonlinear functions were used to build chaos as well as particularly simple cases. Among these ones, we can cite the sine function [9], the hyperbolic tangent function [10], the polynomials with integer orders [11], and the polynomials with fractional order terms [12]. Usually, the well-known piecewise linear function is still the most used elementary function in jerk systems [13, 14]. In this framework, piecewise nonlinear function could be used as a chaotic generator, and to the best of our knowledge, it has not been harnessed until now. Nevertheless, particular applications (namely, secure communication and encryption) require not only simple algebraic models but also very complex dynamic behaviors. This compromise between a simple mathematical model and a complex dynamic behavior is very difficult to satisfy and remains infrequently exploited.

From the circuitry point of view, many research works have tried to design electronic circuits generating chaotic behaviors [15, 16]. On the other hand, peculiar nonlinear terms are quite complicated for implementation, such as the Chua circuit, which desires inductance elements [17, 18]. Indeed, inductance components are not recommended for chaos applications because they introduce uncontrollable parameters due to their inherent impedance. Thus, the common analog electronic elements are known in electrical engineering to be solely operational amplifiers, resistors, capacitors, diodes, and transistors. On the other hand, the previous compromise between a simple algebraic structure and complex dynamic behaviors should be satisfied in order to design high performance chaotic circuits.

The main objective of this chapter is to achieve the compromise to build simple jerk model with richer chaotic dynamics than those proposed by related works. Expecting that the piecewise nonlinear function gives us more complex chaotic proprieties than the piecewise linear one, the proposed system is characterized by only one nonlinear term based on the absolute function. The chaotic system presents interesting dynamical behaviors, it can exhibit regular and strange attractors. The corresponding oscillator circuit of the jerk system is designed using MultiSIM software. Experimental investigations also prove the efficiency of the designed circuit.

The remainder of this chapter is arranged as follows. In Section 2, the new jerk system with a piecewise nonlinear function is proposed, and its basic properties are described. In Section 3, the chaotic system is analyzed by focusing on its elementary characteristics such as the Lyapunov exponents, the regular and strange attractor exhibited, and the equilibrium points. In Section 3, the oscillator circuit of the jerk system is designed using simple electronic components. In Section 4, the experimental results of the implemented circuit are presented and compared to simulation results with the MultiSIM software.

2 Mathematical Model and Basic Properties

Let consider the piecewise nonlinear function G(x) = kx|x| which can be also defined by the following expression

   (1)

Thus, the jerk chaotic system with only one nonlinear term is expressed by these three differential equations:

   (2)

where (x, y, z) are the state variables and (a, b, c, k) are the system parameters.

System (2) can exhibit chaotic behavior only if the general condition of disipativity is satisfied. Hence, let consider V the volume element of the flow of system trajectories given by

As long as a > 0 and time goes to infinity, each volume containing the trajectory of system (1) shrinks to zero at an exponential rate. As a result, all system orbits are finally confined to a subset of zero volume, and the asymptotic motion settles onto an attractor in the three-dimensional phase space.

In fact, when the initial conditions are chosen as (1, 1, 1) and the system parameters (a, b, c, k) are equal to (1, 1, − 2.625, − 0.25), system (2) generates a double scrolls chaotic attractor. As shown in Fig. 1, the exhibited attractor is symmetric with respect to the origin point O (0, 0, 0).

Fig. 1 Phase portraits of system ( 2): (A) x-y-z; (B) y-x; (C) x-z; (D) y-z.

The time series of the state variables x, y, and z are described in Fig. 2. These signals represent the chaotification rates of each state variable. More precisely, these curves reflect the variation of the dynamic behaviors of each variable in the course of time.

Fig. 2 Time series of the state trajectories of system ( 2): (A) x; (B) y; (C) z.

On the other hand, system (2) is sensitive to initial conditions which is the most visible signature of chaotic behaviors. Indeed, two time series of each state variables of system (2) for neighboring initial conditions are described in Fig. 3. These two curves are started from (1, 1, 1) and (1.001, 1, 1), respectively. It is clear that the two curves are perfectly superimposed in the beginning, but they diverge suddenly after that.

Fig. 3 Sensitive dependence to initial conditions of the state trajectories of system ( 2): (A) x; (B) y; (C) z.

3 Dynamic Analysis

In this section, the dynamic analysis of system (2) is focused on the elementary characteristic of chaotic behaviors. All numerical simulations are realized using the two Matlab package MatCont and Matds for the visualization of the attractor forms and the Lyapunov spectrum, respectively.

3.1 Equilibrium and Stability

The equilibrium points of system (2) are obtained by resolving the following equations

When the system parameters (a, b, c, k) are equal to (1, 1, − 2.625, − 0.25), since the parameters c and k are negative constants, then system (2) allows only three equilibrium points: P. It is clear that the equilibrium points P2 and P3 are symmetrical with respect to the origin point P1. This reflects the symmetry of the obtained chaotic attractor.

In order to study the stability analysis of system (2), we aim to make explicit in Table 1 the Jacobian matrix and their corresponding eigenvalues for each equilibrium point P1, P2, and P3.

Table 1

For the equilibrium points, two types of unstable dissipative points (UDS) are defined according to Campos-Cantn et al. [19]. Thus, we introduce below the corresponding definition of each type.

Definition 1

(Campos-Cantn et al. [19]) An equilibrium point, whose eigenvalues are (λ1, λ2, λ3), is said to be an UDS Type I, if the sum of its eigenvalues is negative, λ1 is a real negative and the other ones are complex conjugates with positive real parts.

Definition 2

(Campos-Cantn et al. [19]) An equilibrium point, whose eigenvalues are (λ1, λ2, λ3), is said to be an UDS Type II, if the sum of its eigenvalues is negative, λ1 is a real positive and the other ones are complex conjugates with negative real parts.

In strange attractors, only an equilibrium point USD type I can allows the generation of wings. According to Table 1 and Definitions 1 and 2, the stability results of system (2) are defined such as:

•P1 is an equilibrium USD type II since λ1 is a positive real and (λ2, λ3) are complex conjugates with negative real parts.

•P2 and P3 are two symmetrical points. These equilibrium points have the same eigenvalues and are USD type I points. Indeed, the corresponding eigenvalues are λ1 < 0 and (λ2, λ3) complex conjugates ones with positive real parts.

3.2 Routes to Chaos

When the system parameters are varied, system (2) generates different periodic and chaotic attractors. According to several tests using the MaCont package, we have chosen the parameter c as the only bifurcation parameters in the jerk system. Indeed, as parameter c increases and the parameters (a, b, k) are fixed, system (2) undergoes particular routes as shown in the bifurcation diagram illustrated in Fig. 4.

Fig. 4 Bifurcation diagram for − c ∈ [1, 2.7] while ( a , b , k ) = (1, 1, − 0.25).

The obtained behaviors are defined under peculiar conditions as follows:

•If − 1 ≤ c ≤−1.1, then system (2) converges to a fixed point as shown in Fig. 5A.

Fig. 5 Different regular and strange attractors exhibited by system ( 2) when (A) c = 0.3; (B) c = −1.3; (C) c = −1.59; (D) c = −1.85; (E) c = −2.5.

•If − 1.1 < c < −1.8, then jerk system exhibits periodic orbit around the equilibrium point P2. Fig. 5B shows this regular attractor with c = −1.5. At c = −1.59, a period doubling bifurcation is detected as presented in Fig.