新延遲Ikeda-Mackey-Glass系統的渾沌、渾沌同步、渾沌控制、參數估測,與應用GYC部分區域穩定理論實現新Ikeda-Lorenz系統之渾沌廣義同步及渾沌控制

國 立 交 通 大 學

機械工程學系

碩士論文

新延遲 Ikeda-Mackey-Glass 系統的渾沌、渾沌同步、

渾沌控制、參數估測,與應用 GYC 部分區域穩定理論

實現新 Ikeda-Lorenz 系統之渾沌廣義同步及渾沌控制

Chaos, Chaos Synchronization, Chaos Control,

Estimation of Parameters of a New

Ikeda-Mackey-Glass Time-delayed System, and

Generalized Synchronization and Chaos Control of a

New Ikeda-Lorenz System by GYC Partial Region

Stability Theory

研 究 生: 何俊諺

指導教授: 戈正銘 教授

中華民國九十七年六月

新延遲 Ikeda-Mackey-Glass 系統的渾沌、渾沌同步、

渾沌控制、參數估測,與應用 GYC 部分區域穩定理論

實現新 Ikeda-Lorenz 系統之渾沌廣義同步及渾沌控制

Chaos, Chaos Synchronization, Chaos Control,

Estimation of Parameters of a New

Ikeda-Mackey-Glass Time-delayed System, and

Generalized Synchronization and Chaos Control of a

New Ikeda-Lorenz System by GYC Partial Region

Stability Theory

研究生: 何俊諺 Student: Chun-Yen Ho 指導教授: 戈正銘 Advisor: Zheng-Ming Ge

國 立 交 通 大 學

機 械 工 程 研 究 所

碩士論文

A Thesis

Submitted to Department of Mechanical Engineering

College of Engineering National Chiao Tung University

in Partial Fulfillment of the Requirement

For the Degree of Master of Science

in Mechanical Engineering June 2008

Hsinchu, Taiwan, Republic of China

  i   

新延遲 Ikeda-Mackey-Glass 系統的渾沌、

渾沌同步、渾沌控制、參數估測,與應用 GYC 部

分區域穩定理論實現新 Ikeda-Lorenz 系統之渾

沌廣義同步及渾沌控制

學生: 何俊諺 指導教授: 戈正銘

摘要

本篇論文以相圖、分歧圖等之數值方法來研究新延遲 Ikeda-Mackey-Glass 系統及 新 Ikeda-Lorenz 系統的渾沌行為。 發現新延遲 Ikeda-Mackey-Glass 系統不需加 入控制器,只需調整延遲項, 即可讓系統達到渾沌廣義同步、渾沌反同步及渾沌廣 義延遲同步。 接著利用此新延遲 Ikeda-Mackey-Glass 系統之渾沌訊號, 使新延遲 Ikeda-Mackey-Glass 系統實現渾沌化控制。 此外本篇論文以相圖、李亞普諾夫指 數等數值方法來研究新 Ikeda-Lorenz 系統之渾沌行為。 並應用 GYC 部分區域 穩定理論, 可以設計出較簡單的控制器使誤差較小。 以新 Ikeda-Lorenz 系統得 出渾沌廣義同步及渾沌控制, 以驗證此方法之有效。最後, 利用渾沌同步使新延 遲 Ikeda-Mackey-Glass 系統達到參數估測之效果。根據最小平方法則, 得出系統 參數之微分方程式。模擬 12 個微分方程式, 當系統達渾沌同步時, 使所估測的 兩個參數達目標值,模擬的結果非常成功。

 

ii   

Chaos, Chaos Synchronization, Chaos

Control, Estimation of Parameters of a New

Ikeda-Mackey-Glass Time-delayed System, and

Generalized Synchronization and Chaos Control

of a New Ikeda-Lorenz System by GYC Partial

Region Stability Theory

Student: Chun-Yen Ho Advisor: Zheng-Ming Ge

Abstract

In this thesis, a new Ikeda-Mackey-Glass (IMG) time-delayed system and a new Ikeda-Lorenz (IL) system are studied. Their chaotic behaviors are presented by phase portraits, bifurcation diagrams, and Lyapunov exponent. When one of delay times is zero, two identical IMG systems cannot be synchronized with slightly different initial conditions. It is found that when one of delay time is positive, different types of synchronization can be obtained with slightly different initial conditions, such as generalized synchronization, anti-synchronization, and generalized lag- synchronization. One chaotization method is presented by using different types of chaos signals as parameters, it can be obtained the chaotic behaviors of a new Ikeda-Mackey-Glass time-delayed system. A new strategy to achieve chaos

 

iii   

generalized synchronization and chaos control by GYC partial region stability theory is proposed. The control design method is simple and a less simulation error because they are in lower degree than that of traditional controllers. A new IL system is used to show the effectiveness of the scheme. Finally, estimation of parameters of a new IMG system through synchronization is studied. By a minimization problem, a system of differential equations governing the evolution of parameters is constructed. Two time delay IMG systems are synchronized and their corresponding two parameters converge to same values by solving twelve differential equations. The simulation results are very satisfactory.

iv

誌謝

本篇碩士論文得以完成， 最先要感謝的人， 就是我的指導教授戈正 銘老師，戈老師在我的碩士生涯中，扮演了很重要的角色。在研究上，戈 老師給我許多意見及指引我正確的研究方向，在研究過程中，也讓我學習 到如何發現問題、解決問題。戈老師也不厭其煩的修訂論文，得以讓論文 更加完整。還有戈老師的文學素養，讓我在跟老師交談中，感染到老師詩 詞的涵養，增進我對文學上更深的了解。最後，感謝老師兩年來的教導， 讓我對未來的人生有了新的啟發。 在兩年的碩士生涯裡，感謝博士班楊振雄、張晉銘、李仕宇學長，碩 士班李乾豪、吳宗訓、李式中、林森生學長及翁郁婷學姊，在我研究遇到 瓶頸時，給予我寶貴的意見及傳授經驗；也感謝我的同學李彥賢、許凱銘、 陳聰文…等，大家彼此互相扶持成長，共同度過這兩年研究的時光，留下 許多快樂的回憶；另外要感謝學弟陳志銘、徐瑜韓、張育銘，幫忙處理繁 瑣雜事，得以讓我們專心致力於研究。 最後，感謝我的家人，您們全力支持我攻讀碩士學位，使我無後顧之 憂地專致於研究。雖然您們在南，我在北，但是久久返家一次，一通關心 的電話，都讓我感到非常的溫暖。感謝您們的教養，得以讓我順利拿到碩 士學位。也謝謝女友盈琇，妳對我的關心、體諒及支持，讓我於疲累時能 繼續支撐下去，妳是我支撐下去的動力。最後，僅以此論文獻給你們。

v CONTENTS ABSTRACT………...i ACKNOWLEDGEMENT………..iv CONTENTS………..v LIST OF FIGURES………...vii Chapter 1 Introduction……….………...1

Chapter 2 Chaos of a New Ikeda-Mackey-Glass System………...………...4

2.1 Preliminaries 2.2 Ikeda-Mackey-Glass system Chapter 3 Chaos Synchronization of the Two Identical Ikeda-Mackey-Glass Systems…...………...8

3.1 Preliminaries 3.2 Synchronization Scheme 3.2.1 Case1: If the Delay Time τ2₌₀ 3.2.2 Case2: If the Delay Time τ2₌₁ Chapter 4 Chaos Control of a New Ikeda-Mackey-Glass System by Chaos Signals as Parameters...………17

4.1 Preliminaries 4.2 Chaotization Scheme 4.3 Simulation Results Chapter 5 Chaos of A New Ikeda-Lorenz System………...37

8 18 4 4 17 17 9 9 9

vi

5.1 Preliminaries

5.2 Ikeda-Lorenz System

Chapter 6 Chaos Generalized Synchronization of a New Ikeda-Lorenz System by

GYC Partial Region Stability Theory...39

6.1 Preliminaries

6.2 Chaos Generalized Synchronization Strategy 6.3 Simulation Results

Chapter 7 Chaos Control of a New Ikeda-Lorenz System

by GYC Partial Region Stability Theory……….55

7.1 Preliminaries

7.2 Chaos Control Scheme 7.3 Simulation Results

Chapter 8 Estimation of Parameters of a New Ikeda-Mackey-Glass System

through Chaos Synchronization with Random Disturbance……..….67

8.1 Preliminaries

8.2 Chaos Synchronization Scheme

8.3 Parameters Estimation of a New Ikeda-Mackey-Glass System without Disturbance.

8.4 Parameters Estimation of a New Ikeda-Mackey-Glass System

with Random Disturbance.

Chapter 9 Conclusions………..………..….……..86 Appendix……….88 References…..……….…………96 37 37 39 39 40 55 55 56 67 67 69 71

vii

LIST OF FIGURES

Fig. 2.1. An IMG chaotic attractor when the delay timesτ₁=5,τ₂=1 Fig.2.2. The bifurcation diagram of the IMG system when the delay Fig.2.3. An IMG chaotic attractor when the delay timesτ₁=5,τ₂=0 Fig.2.4. The bifurcation diagram of the IMG system when the delay

timesτ₁=5,τ₂=0.

Fig.3.1. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0,y1(0)=-1 and y2(0)=0.5, whenτ2=0.

Fig.3.2. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0,y1(0)=-1 and y2(0)=0.5, whenτ2=0.

Fig.3.3. Error of two identical IMG systems with

x1(0)=100, x2(0)=10, y1(0)=101 and y2(0)=10.001, whenτ2 = . 1

Fig.3.4. Error of two identical IMG systems with

x1(0)=100, x2(0)=10, y1(0)=101 and y2(0)=10.001, whenτ2 = . 1

Fig.3.5. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

Fig.3.6. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

Fig.3.7. Error of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

Fig.3.8. Error of two identical IMG systems with

6 6 7 7 11 11 12 12 13 13 14 14

viii

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

Fig.3.9. Time response of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ₂ = , 1 μ₁ =1.2427sec.

Fig.3.10. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ₂ = ,1 μ₂ =1.08sec.

Fig.3.11. Error of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ2 = . 1

Fig.3.12. Error of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ2 = . 1

Fig.4.1. Phase portrait of an IMG system in period 2 whenα₁=25, β=24.8 ,K = 13.4 , ₁ α₂=4.7, b=1.2348,c=10, K =8, ₂

1

τ =5 and τ₂=1.

Fig.4.2. The time history of x1 of an IMG system in period 2

when α₁=25, β=24.8 ,K = 13.4 , ₁ α₂=4.7, b=1.2348, c=10, K =8, ₂ τ₁=5 and τ₂=1.

Fig.4.3. The time history of x2 of an IMG system in period 2

when α₁=25, β=24.8 ,K = 13.4 , ₁ α₂=4.7, b=1.2348, c=10, K =8, ₂ τ₁=5 and τ₂=1.

Fig.4.4. The bifurcation diagram of an IMG system when α₁=25,

15 15 16 16 21 21 22 22

ix

β=24.8, α₂=4.7, b=1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1. Fig4.5. An IMG chaotic attractor when α₁=25, β=24.8 ,K = 14.1 , ₁

2

α =4.7, b=1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1.

Fig4.6. An IMG chaotic attractor when parameter is a chaos signal for CASE I.

Fig4.7. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE I.

Fig.4.8. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE I.

Fig.4.9. An IMG chaotic attractor when parameter is a chaos signal for CASE II.

Fig.4.10. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE II.

Fig.4.11. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE II.

Fig.4.12. An IMG chaotic attractor when parameter is a chaos signal for CASE III.

Fig.4.13. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE III.

Fig.4.14. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE III.

Fig.4.15. An IMG chaotic attractor when parameter is a chaos signal for CASE IV.

23 23 24 24 25 25 26 26 27 27 28

x

Fig.4.16. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE IV.

Fig.4.17. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE IV.

Fig.4.18. An IMG chaotic attractor when parameter is a chaos signal for CASE V.

Fig.4.19. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE V.

Fig.4.20. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE V.

Fig.4.21. An IMG chaotic attractor when parameter is a chaos signal for CASE VI.

Fig.4.22. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE VI.

Fig.4.23. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE VI.

Fig.4.24. An IMG chaotic attractor when parameter is a chaos signal for CASE VII.

Fig.4.25. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE VII.

Fig.4.26. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE VII.

Fig.4.27. An IMG chaotic attractor when parameter is a chaos signal for CASE VIII.

28 29 29 30 30 31 31 32 32 33 33 34

xi

Fig.4.28. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE VIII.

Fig.4.29. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal for CASE VIII.

Fig.4.30. An IMG chaotic attractor when parameter is a chaos signal for CASE IX.

Fig.4.31. The time history of z1 of an IMG system in chaotic behavior

when parameter is a chaos signal CASE IX..

Fig.4.32. The time history of z2 of an IMG system in chaotic behavior

when parameter is a chaos signal CASE IX.

Fig.5.1. The chaotic attractor of a new Ikeda-Lorenz system with parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92,

a3=0.05, b3=1.8, c=4 and initial conditions x1(0)=1,x2(0)=2,x3(0)=3.

Fig.5.2. Lyapunov exponents of a new Ikeda-Lorenz system with parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92,

a3=0.05, b3=1.8, c=4 and initial conditions x1(0)=1,x2(0)=2,x3(0)=3. Fig. 6.1. Phase portrait of error dynamics for CASE I.

Fig. 6.2. Time histories of errors for CASE I.

Fig.6.3. Time histories of x1, x2, x3, y1, y2, y3 for CASE I.

Fig.6.4. Phase portrait of error dynamics for CASE II. Fig.6.5. Time histories of errors for CASE II.

Fig.6.6. Time histories of x_i− +y_i 80 and -sin2t⋅cost for CASE II.

Fig.6.7. Phase portrait of error dynamics for CASE III. Fig.6.8. Time histories of errors for CASE III.

38 38 49 49 50 50 51 51 52 52 34 35 35 36 36

xii

Fig.6.9. Phase portrait of error dymanics for CASE IV. Fig.6.10. Time histories of errors for CASE IV.

Fig.6.11. Time histories of x− y+100 and − for CASE IV. z

Fig.7.1. Phase portrait of error dynamics for CASE I.

Fig.7.2 Time histories of errors for CASE I.

Fig.7.3. Phase portrait of error dynamics for CASE II.

Fig.7.4. Time histories of errors for CASE II.

Fig.7.5. Time histories of errors for CASE II.

Fig.7.6. Phase portrait of error dynamics for CASE III.

Fig.7.7. Time histories of errors for CASE III.

Fig.7.8. Time histories of errors for CASE III.

Fig.8.1. Error of master system state x1 and response system y1 without disturbance.

Fig.8.2. Convergence of the estimated parameter βˆ₁ to its actual value without disturbance in this system.

Fig.8.3. Error of master system state x2 and response system y2 without disturbance.

Fig.8.4. Convergence of the estimated parameter βˆ₂ to its actual value without disturbance in this system.

Fig.8.5. Error of master system state x1 and response system y1 with disturbance for CASE I.

Fig.8.6. Convergence of the estimated parameter βˆ₁ to its actual value with disturbance in the system for CASE I.

Fig.8.7. Error of master system state x2 and response system y2

53 53 54 63 63 64 64 65 65 78 66 66 78 79 79 80 80 81

xiii

with disturbance for CASE I.

Fig.8.8. Convergence of the estimated parameter βˆ₂ to its actual value with disturbance in the system for CASE I.

Fig.8.9. Error of master system state x1 and response system y1 with disturbance for CASE II.

Fig.8.10. Convergence of the estimated parameter βˆ₁ to its actual value with disturbance in the system for CASE II.

Fig.8.11. Error of master system state x2 and response system y2 with disturbance for CASE II.

Fig.8.12. Convergence of the estimated parameter βˆ₂ to its actual value with disturbance in the system for CASE II.

Fig.8.13. Error of master system state x1 and response system y1 with disturbance for CASE III.

Fig.8.14. Convergence of the estimated parameter βˆ₁ to its actual value with disturbance in the system for CASE III.

Fig.8.15. Error of master system state x2 and response system y2 with disturbance for CASE III.

Fig.8.16. Convergence of the estimated parameter βˆ₂ to its actual value with disturbance in the system for CASE III.

Fig.A1. Partial regions Ω and Ω1

81 83 83 84 84 85 85 95 82 82

1

Chapter 1

Introduction

Chaos, as an interesting nonlinear phenomenon, has been intensively investigated. It is well known that chaotic system has sensitive dependence on initial conditions. A chaotic system is a nonlinear deterministic system that displays complex dynamical behaviors [1].

Due to finite signal transmission times, switching speeds, and memory effects, time-delayed systems exist in everywhere, such as nature, technology, and society[2]. Mackey-Glass time-delayed system has been introduced as a model of blood generation for patients with leukemia. Nowadays this model is very popular in chaos theory[3]. The Ikeda time-delayed system has been introduced to describe the dynamics of an optical bistable resonator, plays an important role in electronics and physiological studies and is well-known for delay-induced chaotic behavior[4-6]. In 1963, Lorenz proposed a simple model for the unpredictable behavior of the weather. He used fluid convection theory to model the motion of a two-dimensional cell of fluid cooled from above and warmed from below [7].A new Ikeda-Mackey-Glass (IMG) time-delayed system and a new Ikeda-Lorenz system are studied in this thesis. There are different types of synchronization for interacting chaotic systems, such as complete synchronization [8,9], generalized synchronization [10], phase synchronization [11,12], lag synchronization[9,13,14], anticipating synchronization [15,16] and so on.

To achieve synchronization, different schemes, such as the Pecora and Carroll (PC) method [8], unidirectional coupling [9], bidirectional coupling [15], adaptive control [17,18] and impulsive control [19-21] are proposed.

2

In this thesis, it is found that when one of delay time of a new time-delayed system is positive, different types of synchronization can be obtained with slightly different initial conditions, such as generalized synchronization, anti-synchronization, and generalized lag- synchronization.

The theory of chaos control has developed since 1990[22-24] and today is at the forefront of research in the field of nonlinear dynamics. Techniques have been experimentally implemented in mechanical [25], chemical [26], electronic[27], laser[28], communication[29], and biological[30] systems.

There are many chaos control have been proposed such as different geometric method[31], feedback and non-feedback control[32-35], inverse optimal control[36], adaptive control[37,38], and backstepping control[39].

In this thesis, one chaos control method is presented by using different types of chaos signal as parameter, it can be obtained the chaotic behaviors of a New Ikeda-Mackey-Glass time-delayed system.

By using the GYC partial region stability theory[40,41], generalized chaos synchronization and chaos control can be obtained. A new Ikeda-Lorenz system is used as a simulation example.

Parameters estimation of chaotic system is an important issue[42-44]. Because chaotic system is very sensitive to initial conditions, parameters cannot be exactly known a priori. Parameter estimation through chaos synchronization is being further investigated.[45-47] .

In this thesis, estimation of parameters of a new Ikeda-Mackey-Glass system through synchronization is studied. By a minimization problem, a system of differential equations governing the evolution of parameters is constructed. Two time delay Ikeda-Mackey- Glass systems are synchronized and their corresponding two parameters converge to same values by solving twelve differential equations. The

3

simulation results are very satisfactory.

This thesis is organized as follows. In Chapter 2, the dynamic equation of a new Ikeda-Mackey-Glass(IMG) system is given. The phase portraits, bifurcation diagram of a new IMG system are presented. It is verified that the IMG system presents chaotic behaviors by numerical simulation.

In Chapter 3, synchronization scheme is given. It is found that no synchronization of the two identical IMG systems can be obtained with slightly different initial conditions when one of delay timeτ₂is zero and without any control scheme or coupling terms. It is also found that generalized synchronization, anti-synchronization and generalized lag-synchronization of the two identical IMG systems with slightly different conditions when two of delay time are positive and without any control scheme or coupling terms. Only by adjusting delay timeτ₂, chaos synchronization of the two identical IMG systems can be obtained.

In Chapter 4, one chaos control method is presented by using different types of chaos signal as parameter, it can be obtained the chaotic behaviors of a new Ikeda-Mackey-Glass time-delayed system.

In Chapter 5, a new Ikeda-Lorenz(IL) system is studied. The phase portraits, Lyapunov exponent of a new IL system are presented. It is verified that the IL system presents the chaotic behaviors by numerical simulation.

In Chapter 6, a new strategy to achieve chaos generalized synchronization by GYC partial region stability theory is proposed. Simulation results show that for the new IL system chaos generalized synchronization can be achieved by GYC partial region stability theory.

In Chapter 7, simulation results show that for the new IL system chaos control can be achieved by GYC partial region stability theory.

4

Chapter 2

Chaos of a New Ikeda-Mackey-Glass System

2.1 Preliminaries

In this chapter, the chaotic behaviors in IMG system with different parameters are studied numerically by phase portraits, Poincare maps and bifurcation diagrams.

2.2 A New Ikeda-Mackey-Glass System

A new IMG system is described by the following differential equations:

1 1 1 1 1 1 2 2 2 1 2 2 2 2 1 2 2 1 ( ) ( ) sin ( )+ ( ) ( ) ( ) ( ) ( ) 1 { ( )}c x t x t x t K x t x t x t x t b K x t x t α β τ τ τ α τ τ = − − − − − = − + + − + −   (2.1)

where the Ikeda model x₁ is the phase lag of the electric field across the resonator;

1

α is the relaxation coefficient for the driving x₁ dynamical variable; β is the

laser intensity injected into the driving system.τ₁, τ₂are the delay time in the new IMG system, and the dynamical variable x₂ in the Mackey-Glass model is the concentration of the mature cells in blood at time t and the delay time is the time between the initiation of cellular production in the bone marrow and release of mature cells into the blood[10]. α₂ is the relaxation coefficient for the driven x₂

dynamical variable, b is the feedback rate for the driven system, and K₁, K₂ is the coupling rate between the driver system x₁ and the response system x₂.

This system has a chaotic attractor shown in Fig.2.1. Fig.2.2 shows the bifurcation diagram, where α₁=25, β =24.8 ,K₁= 14.1 , α₂=4.7, b =1.2348,c=10, K₂=8, τ₁

5

=5 and τ₂=1.

If the delay time τ₂ is zero, also it is found that there is also a chaotic behavior for IMG system. Fig.2.3 show the chaotic attractor of this system. Fig.2.4 shows the bifurcation diagram, where α₁=25, β =24.8 ,k₁= 14.1 , α₂=4.7, b =1.2348, c=10,

2

6

Fig. 2.1. An IMG chaotic attractor when the delay timesτ₁=5,τ₂=1.

Fig.2.2. The bifurcation diagram of the IMG system when the delay timesτ₁=5,τ₂ =1. 1 x 2 x 2 x 1 K

7

Fig.2.3. An IMG chaotic attractor when the delay timesτ₁=5,τ₂=0.

Fig.2.4. The bifurcation diagram of the IMG system when the delay timesτ₁=5,τ₂=0.

1 x 2 x 2 x 1 K

8

Chapter 3

Chaos Synchronization of the Two Identical

Ikeda-Mackey-Glass Systems

3.1

Preliminaries

In this Chapter, synchronization of the two identical new Ikeda-Mackey-Glass (IMG) systems without any control is studied. Two identical IMG system cannot be synchronized with slightly different conditions if one of delay time is zero and different types of synchronization can be obtained with slightly different initial conditions, such as generalized synchronization, anti-synchronization, and generalized lag-synchronization when two of delay time are positive. It is shown that we only adjust delay time τ₂, chaos synchronization of the two identical IMG systems can be obtained.

3.2

Synchronization Scheme

Consider the time-delayed system:

x(t)= f(x(t),x(t−τ)) (3.1)

where x∈ represents the state of the system, and R

dt dx t

x( )= .

To synchronize system (3.1), the form of the other system is

y(t)= f(y(t),y(t−τ))+u (3.2) where u is the controlling term.

9

In this Chapter, we find that these two Ikeda-Mackey-Glass system can be synchronized without any controller, only by changing the delay timeτ₁andτ₂.

Consider synchronization between two Ikeda-Mackey-Glass systems:

1 1 1 1 1 1 2 2 2 1 2 2 2 2 1 2 2 1 ( ) ( ) sin ( )+ ( ) ( ) ( ) ( ) ( ) 1 { ( )} = − − − − ⎧ ⎪ − ⎨ _{= − + + −} ⎪ _{+ −} ⎩   _c x t x t x t K x t x t x t x t b K x t x t α β τ τ τ α τ τ (3.3) 1 1 1 1 1 1 2 2 1 2 1 2 2 2 2 1 2 2 2 1 ( ) ( ) sin ( )+ ( ) ( ) ( ) ( ) ( ) 1 { ( )} = − − − − + ⎧ ⎪ − ⎨ _{= − + + − +} ⎪ _{+ −} ⎩   _c y t y t y t K y t u y t y t y t b K y t u y t α β τ τ τ α τ τ (3.4)

where the controlling term u₁=u₂ = . 0

3.2.1 Case1: If the delay time

τ2

=0

In this Section it is shown that if the delay timeτ₂is zero, no synchronization can be obtained. Simulation results are shown in Fig.3.1 and Fig.3.2.

3.2.2 Case2: If the delay time

τ2

=1

In this Section it is shown that if the delay time τ₂ is not zero, different types of synchronization can be obtained.

Fig.3.3 and Fig.3.4 show that the generalized synchronization of the two identical IMG systems can be obtained. Define

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = = − = − = ) ( ) ( 2 2 1 1 2 2 2 1 1 1 t R e t R e y x e y x e (3.5) ) ( ), ( ₂ 1 t R t

10

Fig.3.5 and Fig.3.6 show that the time responses of two identical IMG systems. It is verified that the anti-synchronization can be obtained by Fig.3.7 and Fig.3.8, where error ⎩ ⎨ ⎧ + = + = 2 2 2 1 1 1 y x e y x e (3.6)

Fig.3.9 and Fig.3.10 show that the time responses of two identical IMG systems. It is verified that the generalized lag-synchronization can be obtained by Fig.3.11 and Fig.3.12, where error

1( ) 1( 1) 1( ) ( ) e t =x t−μ −y t +F t (3.7) ' 2( ) 2( ) 2( 2) ( ) e t =x t −y t−μ +F t (3.8) 1 1.2427

μ = sec,μ₂ =1.08sec, F t( )and '

( )

11

Fig.3.1.Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0,y1(0)=-1 and y2(0)=0.5, whenτ2=0.

Fig.3.2.Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0,y1(0)=-1 and y2(0)=0.5, whenτ2=0.

t (sec) x2 : blue y2 : r ed x1 :blu e y1 :re d t (sec)

12

Fig.3.3. Error of two identical IMG systems with

x1(0)=100, x2(0)=10, y1(0)=101 and y2(0)=10.001, whenτ2 = . 1

Fig.3.4. Error of two identical IMG systems with

x1(0)=100, x2(0)=10, y1(0)=101 and y2(0)=10.001, whenτ2 = . 1

t (sec)

t (sec)

e1

13

Fig.3.5. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

Fig.3.6. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1 t (sec) t (sec) x1 :blu e y1 :re d x2 :blu e y2 :re d

14

Fig.3.7. Error of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

Fig.3.8. Error of two identical IMG systems with

x1(0)=1, x2(0)=0, y1(0)=-1 and y2(0)=0, whenτ2 = . 1

t (sec)

e1

t (sec)

15

Fig.3.9. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ2 = , 1 μ1 =1.2427sec.

Fig.3.10. Time responses of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ2 = ,1 μ2 =1.08sec.

t (sec) x1 :blu e, y1 :re d t (sec) x2 :blu e y2 :re d

16

Fig.3.11. Error of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ2 = . 1

Fig.3.12. Error of two identical IMG systems with

x1(0)=1, x2(0)=0.1, y1(0)=-1 and y2(0)=0.5, whenτ2 = . 1

e1

t (sec)

e2

17

Chapter 4

Chaotization of a New Ikeda-Mackey-Glass System

by Chaos Signals as Parameters

4.1 Preliminaries

In this Chapter, another chaotization method is presented by using different types of chaos signal as parameter. A new Ikeda-Mackey-Glass system is described in Chapter 2.2. By using different types of chaos signals as parameters, a regular motion new IMG system becomes chaotic system.

4.2 Chaotization Scheme

Differential equations of two general delay systems are described as follows:

x = f(x (t),x (t−τ),K ) (4.1) y = f(y (t),y (t−τ),G ) (4.2)

where x, y∈ n

R are the state vector, K =[k1 k2...kn], G =[g1 g2...gn] n

R

∈ are the

parameter vectors. The dynamics of system (4.1) is periodic motion, while the dynamics of system (4.2) is chaotic motion. Replacing one of the parameters of system (4.1) by chaotic states of system (4.2), system (4.1) becomes:

18

where z is the state vector, P is the same as K , except that one of parameters of

K is replaced by chaotic states of system (4.2). Simulations show that system (4.3) becomes a chaotic system. In other worlds, a delay system (4.3) becomes chaotic system by parameter replacement method.

4.3 Simulation Results

By using Eq.(4.1), the periodic motion of a new Ikeda-Mackey-Glass system is

described as follows: ) ( )} ( { 1 ) ( ) ( ) ( ) ( ) ( sin ) ( ) ( 2 1 2 1 2 1 2 2 2 2 2 2 1 1 1 1 1 1 τ τ τ α τ τ β α − + − + − + − = − + − − − = t x k t x t x b t x t x t x k t x t x t x c   (4.4)

A periodic motion is obtained as shown by phase portraits in Fig.4.1, time histories in Fig.4.2 and Fig.4.3, and bifurcation diagram in Fig.4.4, where α₁=25, β =24.8 , k = 13.4 , ₁ α2=4.7, b =1.2348,c=10, k =8, 2 τ1=5 and τ2=1.

By using Eq.(4.2), the chaotic motion of a new Ikeda-Mackey-Glass system is described as follows: ) ( )} ( { 1 ) ( ) ( ) ( ) ( ) ( sin ) ( ) ( 2 1 2 1 2 1 2 2 2 2 2 2 1 1 1 1 1 1 τ τ τ α τ τ β α − + − + − + − = − + − − − = t y g t y t y b t y t y t y g t y t y t y c   (4.5)

19

diagram in Fig.4.4, where α₁=25, β =24.8 , g = 14.1 , ₁ α2=4.7, b =1.2348,c=10,

2

g =8, τ1=5 and τ2=1.

Replacing k by ₁ p , Eq.(4.4) becomes: ₁

) ( )} ( { 1 ) ( ) ( ) ( ) ( ) ( sin ) ( ) ( 2 1 2 1 2 1 2 2 2 2 2 2 1 1 1 1 1 1 τ τ τ α τ τ β α − + − + − + − = − + − − − = t z k t z t z b t z t z t z p t z t z t z c   (4.6)

where k =8. A new Ikeda-Mackey-Glass system(4.6) is a chaotic system by ₂

parameter replacement method.

CASE I: p1=y1

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.6 and time histories in Fig.4.7 and Fig.4.8.

CASE II: p1=y2

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.9 and time histories in Fig.4.10 and Fig.4.11.

CASE III: p1=y 12

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.12 and time histories in Fig.4.13 and Fig.4.14.

CASE IV: p1=y 22

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.15 and time histories in Fig.4.16 and Fig.4.17.

CASE V:p₁ = y₁y₂

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.18 and time histories in Fig.4.19 and Fig.4.20.

20

CASE VI: p₁ = y₂y₂(t−τ), where τ =30 sec.

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.21 and time histories in Fig.4.22 and Fig.4.23.

CASE VII: p₁ =cosy₁cosy₂

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.24 and time histories in Fig.4.25 and Fig.4.26.

CASE VIII: p₁ =R+y₂, where R is the Rayleigh noise.

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.27 and time histories in Fig.4.28 and Fig.4.29

CASE IX: p₁ = Ry₂, where R is the Rayleigh noise.

The chaotic simulation results obtained is achieved in phase portrait in Fig.4.30 and time histories in Fig.4.31 and Fig.4.32.

21

Fig.4.1. Phase portrait of an IMG system in period 2 when α₁=25, β =24.8 ,K = ₁

13.4 , α₂=4.7, b =1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1.

Fig.4.2. The time history of x1 of an IMG system in period 2 when α1=25, β =24.8 , 1 K = 13.4 , α₂=4.7, b =1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1. 1 x 2 x 1 x

22

Fig.4.3. The time history of x2 of an IMG system in period 2 when α1=25, β =24.8 , 1

K = 13.4 , α₂=4.7, b =1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1. .

Fig.4.4. The bifurcation diagram of an IMG system when α₁=25,β =24.8, α₂ =4.7, b =1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1. 2 x 1 k 2 x

23

Fig4.5. An IMG chaotic attractor when α₁=25, β =24.8 ,K = 14.1 , ₁ α₂=4.7, b =1.2348,c=10, K =8, ₂ τ₁=5 and τ₂=1.

Fig4.6. An IMG chaotic attractor when parameter is a chaos signal for CASE I. 1 x 2 x 1 z 2 z

24

Fig4.7. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE I.

Fig.4.8. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE I.

.

1

z

2

25

Fig.4.9. An IMG chaotic attractor when parameter is a chaos signal for CASE II.

Fig.4.10. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE II.

. 1 z 2 z 1 z

26

Fig.4.11. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE II.

Fig.4.12. An IMG chaotic attractor when parameter is a chaos signal for CASE III.

2 z 1 z 2 z

27

Fig.4.13. The time history of z1 of an IMG system in chaotic behavior when

parameter is a chaos signal for CASE III.

Fig.4.14. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE III.

1

z

2

28

Fig.4.15. An IMG chaotic attractor when parameter is a chaos signal for CASE IV.

Fig.4.16. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE IV.

1 z 2 z 1 z

29

Fig.4.17. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE IV.

Fig.4.18. An IMG chaotic attractor when parameter is a chaos signal for CASE V. 2 z 1 z 2 z

30

Fig.4.19. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE V.

Fig.4.20. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE V.

1

z

2

31

Fig.4.21. An IMG chaotic attractor when parameter is a chaos signal for CASE VI.

Fig.4.22. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VI.

. 1 z 2 z 1 z

32

Fig.4.23. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VI.

Fig.4.24. An IMG chaotic attractor when parameter is a chaos signal for CASE VII.

2 z 1 z 2 z

33

Fig.4.25. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VII.

Fig.4.26. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VII.

1

z

2

34

Fig.4.27. An IMG chaotic attractor when parameter is a chaos signal for CASE VIII.

Fig.4.28. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VIII.

1 z 2 z 1 z

35

Fig.4.29. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VIII.

Fig.4.30. An IMG chaotic attractor when parameter is a chaos signal for CASE IX.

2 z 1 z 2 z

36

Fig.4.31. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal CASE IX.

Fig.4.32. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal CASE IX.

1

z

2

37

Chapter 5

Chaos of a New Ikeda-Lorenz System

5.1

Preliminaries

In this Chapter, given the equations of a new Ikeda-Lorenz system and verified the chaotic behavior by phase portraits and Lyapunov exponent.

5.2

Ikeda-Lorenz System

The Ikeda-Lorenz system is described as follows:

1 1 1 1 1 2 1 2 2 1 2 1 1 1 3 2 3 3 1 3 1 1 2 3 sin ( ) sin sin = − − + − = − − + − − = − − + −    x a x b x x x x a x b x rx x x x x a x b x x x cx σ (5.1)

which is a combination of Ikeda system without time delay and Lorenz system. The parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8, c=4 are used. The chaotic attractor and Lyapunov exponents of the new Ikeda-Lorenz system are shown in Fig.5.1 and Fig.5.2.

38

Fig.5.2. Lyapunov exponents of new Ikeda-Lorenz system with parameters a1=0.1, b1=1, σ =16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8, c=4 and initial conditions

x1(0)=1,x2(0)=2,x3(0)=3.

Fig.5.1. The chaotic attractor of a new Ikeda-Lorenz system with parameters a1=0.1,

b1=1,σ=16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8, c=4 and initial conditions

x1(0)=1,x2(0)=2,x3(0)=3.

x1

x2

x3

39

Chapter 6

Chaos Generalized Synchronization of New

Ikeda-Lorenz Systems by GYC Partial Region

Stability Theory

6.1

Preliminaries

The GYC partial region stability theory is proposed. By using the new strategy, the Lyapunov function is a simple linear homogeneous function of error states and the controllers are more simple and have less simulation error because they are in lower order than that of traditional controllers. Simulation results show that the new Ikeda-Lorenz system can be achieved chaos generalized synchronization by GYC partial region stability theory.

6.2 Chaos Generalized Synchronization Strategy

Consider the following unidirectional coupled chaotic systems

u y t h y x t f x + = = ) , ( ) , (   (6.1) where x=[x1,x2,...,xn]T ∈Rn,

[

1, 2, ,

]

T _n n

y= y y " y ∈R denote two state vectors, f

and h are nonlinear vector functions, and u=[u₁,u₂,...u_n]T ∈Rn is a control input vector.

The generalized synchronization can be accomplished when t→ ∞ , the limit of the error vector e=[e₁,e₂,...,e_n]T approaches zero:

40 0 lim = ∞ → e t (6.2) where y x G e= ( )− (6.3)

By using GYC partial region stability theory, the positive definite Lyapunov

function is a homogeneous linear function of error states and the controllers can be

designed in lower order than that of traditional controllers.

6.3 Simulation Results

Two new Ikeda-Lorenz systems with the unidirectional coupling are presented as follows: 1 1 1 1 1 2 1 2 2 1 2 1 1 1 3 2 3 3 1 3 1 1 2 3 sin ( ) sin sin = − − + − = − − + − − = − − + −    x a x b x x x x a x b x rx x x x x a x b x x x cx σ (6.4) 1 1 1 1 1 2 1 1 2 2 1 2 1 1 1 3 2 2 3 3 1 3 1 1 2 3 3 sin ( ) sin sin = − − + − + = − − + − − + = − − + − +    y a y b y y y u y a y b y ry y y y u y a y b y y y cy u σ

CASE I. The generalized synchronization error function is

90 + − = _{i i} i x y e , (i=1,2,3) (6.5) Our goal is

41

90 + = x

y , i.e.lim =lim( − +90)=0

∞ → ∞ → e t x y t (6.6) The error dynamics becomes

1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 1 3 1 3 2 2 2 3 3 3 3 1 1 3 1 1 1 2 1 2 ( ) (sin sin ) [( ) ( )] ( ) (sin sin ) ( ) ( ) ( ) ( ) (sin sin ) ( = − = − − − − + − − − − = − = − − − − + − − − − − − = − = − − − − + −          e x y a x y b x y x y x y u e x y a x y b x y r x y x x y y x y u e x y a x y b x y x x y y σ 3 3 3 )−c x( −y )−u (6.7)

Let initial states be( (0),x₁ x₂(0),x₃(0))=(1, 2, 3), ( (0),y₁ y₂(0),y₃(0))=(3.5, 4,1) and system parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8 and

c=4, we find the error dynamics always exists in first quadrant as shown in Fig.6.1.

By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

0 ) ( 2 1 2 3 2 2 2 1 3 2 1+ + + + + > =e e e e e e V (6.8)

Although V contains quadratic terms 2 2 2

1 2 3

1

( )

2 e +e +e , the degree of terms of following three controllers remain unchanged as that of V = + + . e₁ e₂ e₃

Its time derivative is

) 1 )( ) ( ) ( ) sin (sin ) ( ( ) 1 )( ) ( ) ( ) ( ) sin (sin ) ( ( ) 1 )( )] ( ) [( ) sin (sin ) ( ( ) 1 ( ) 1 ( ) 1 ( 3 3 3 3 2 1 2 1 1 1 3 1 1 3 2 2 2 2 3 1 3 1 1 1 1 1 2 1 1 2 1 1 1 1 2 2 1 1 1 1 1 1 3 3 2 2 1 1 e u y x c y y x x y x b y x a e u y x y y x x y x r y x b y x a e u y x y x y x b y x a e e e e e e V + − − − − + − − − − + + − − − − − − + − − − − + + − − − − + − − − − = + + + + + = σ     (6.9)

42 Choose 1 1 1 1 1 1 1 2 2 1 1 1 2 2 1 1 2 1 1 1 1 1 3 1 3 2 2 2 3 3 1 1 3 1 1 1 2 1 2 3 3 3 ( ) (sin sin ) [( ) ( )] ( ) (sin sin ) ( ) ( ) ( ) ( ) (sin sin ) ( ) ( ) = − − − − + − − − + = − − − − + − − − − − + = − − − − + − − − + u a x y b x y x y x y e u a x y b x y r x y x x y y x y e u a x y b x y x x y y c x y e σ (6.10) We obtain 0 ) 1 ( ) 1 ( ) 1 ( _{1 2 2 3 3} 1 + − + − + < − = e e e e e e V (6.11)

which is a negative definite function in first quadrant. Three state errors versus time and time histories of states are shown in Fig.6.2 and Fig.6.3.

CASE II. The generalized synchronization error function is

80 cos sin2 + + − =x y t t ei i i , (i=1,2, 3) (6.12) Our goal is , 80 cos sin2 + + =x t t y , (6.13) i.e. lim =lim( − +sin2 cos +80)=0

∞ → ∞ → e t x y t t t ,(i=1, 2, 3) The error dynamics become

43 t t t u y x c y y x x y x b y x a e t t t u y x y y x x y x r y x b y x a e t t t u y x y x y x b y x a e 3 2 3 3 3 2 1 2 1 1 1 3 1 1 3 3 3 2 2 2 2 3 1 3 1 1 1 1 1 1 1 1 2 2 3 2 1 1 1 2 2 1 1 1 1 1 1 1 sin cos sin ) ( ) ( ) sin (sin ) ( sin cos sin ) ( ) ( ) ( ) sin (sin ) ( sin cos sin )] ( ) [( ) sin (sin ) ( − + − − − − + − − − − = − + − − − − − − + − − − − = − + − − − − + − − − − =    σ (6.14)

Let initial states be( (0),x₁ x₂(0),x₃(0))=(1, 2, 3), ( (0),y₁ y₂(0),y₃(0))=(3.5, 4,1) and system parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8 and

c=4, we find the error dynamics always exists in first quadrant as showen in Fig.6.4.

By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

0 ) ( 2 1 2 3 2 2 2 1 3 2 1 + + + + + > =e e e e e e V (6.15)

Its time derivative is

) )(1 sin cos 2 sin ) ( ) ( ) sin (sin ) ( ( ) )(1 sin cos 2 sin ) ( ) ( ) ( ) sin (sin ) ( ( ) 1 )( sin cos 2 sin )] ( ) [( ) sin (sin ) ( ( ) 1 ( ) 1 ( ) 1 ( 3 3 3 3 2 1 2 1 1 1 3 1 1 3 2 2 2 2 3 1 3 1 1 1 1 1 1 1 1 2 1 1 1 1 2 2 1 1 1 1 1 1 3 3 2 2 1 1 e t t-t u y x c y y x x y x b y x a e t t-t u y x y y x x y x r y x b y x a e t t-t u y x y x y x b y x a e e e e e e V + + − − − − + − − − − + + + − − − − − − + − − − − + + + − − − − + − − − − = + + + + + = σ     (6.16)

44 Choose sin cos 2 sin ) ( ) ( ) sin (sin ) ( sin cos 2 sin ) ( ) ( ) ( ) sin (sin ) ( sin cos 2 sin )] ( ) [( ) sin (sin ) ( 3 3 3 2 1 2 1 1 1 3 1 1 3 3 2 2 2 3 1 3 1 1 1 1 1 1 1 1 2 2 1 1 1 2 2 1 1 1 1 1 1 1 e t t-t y x c y y x x y x b y x a u e t t-t y x y y x x y x r y x b y x a u e t t-t y x y x y x b y x a u + + − − − + − − − − = + + − − − − − + − − − − = + + − − − + − − − − = σ (6.17) We obtain 0 ) 1 ( ) 1 ( ) 1 ( 1 2 2 3 3 1 + − + − + < − = e e e e e e V (6.18)

which is a negative definite function. Three state errors versus time and time histories of 80x_i− +y_i are shown in Fig.6.5 and Fig.6.6.

CASE III. The generalized synchronization error function is

2 1 100 30 = − + i i i e x y , (i=1, 2, 3) (6.19) Our goal is 100 30 1 2 + = i i x y , (i=1, 2, 3) (6.20) i.e. 100) 0,( 1,2,3) 30 1 ( lim lim = 2 − + = = ∞ → ∞ → ei _t xi yi i t

45

The error dynamics become

2 3 2 1 1 3 1 3 3 2 1 1 3 1 3 3 3 3 3 3 2 2 3 1 1 1 2 1 2 2 3 1 1 1 2 1 2 2 2 2 2 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 ) sin ( ) sin ( 15 1 15 1 ) sin ( ) sin ( 15 1 15 1 )) ( sin ( )) ( sin ( 15 1 15 1 u cy y y y b y a cx x x x b x a x y x x e u y y y ry y b y a x x x rx x b x a x y x x e u y y y b y a x x x b x a x y x x e − − + − − − − + − − = − = − − − + − − − − − + − − = − = − − + − − − − + − − = − =          σ σ (6.21)

Let initial states be( (0),x₁ x₂(0),x₃(0))=(1, 2, 3), ( (0),y₁ y₂(0),y₃(0))=(3.5, 4,1)and system parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8 and

c=4, we find the error dynamics always exists in first quadrant as shown in Fig.6.7.

By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

0 ) ( 2 1 2 3 2 2 2 1 3 2 1+ + + + + > =e e e e e e V (6.22)

Its time derivative is

) 1 ]( ) sin ( ) sin ( 15 1 [ ) 1 ]( ) sin ( ) sin ( 15 1 [ ) 1 ]( )) ( sin ( )) ( sin ( 15 1 [ 3 2 3 2 1 1 3 1 3 3 2 1 1 3 1 3 3 2 2 2 3 1 1 1 2 1 2 2 3 1 1 1 2 1 2 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 e u cy y y y b y a cx x x x b x a x e u y y y ry y b y a x x x rx x b x a x e u y y y b y a x x x b x a x V + − − + − − − − + − − + + − − − + − − − − − + − − + + − − + − − − − + − − = σ σ  (6.23)

46 Choose ) sin ( ) sin ( 15 1 ) sin ( ) sin ( 15 1 )) ( sin ( )) ( sin ( 15 1 3 3 2 1 1 3 1 3 3 2 1 1 3 1 3 3 3 2 2 3 1 1 1 2 1 2 2 3 1 1 1 2 1 2 2 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 e cy y y y b y a cx x x x b x a x u e y y y ry y b y a x x x rx x b x a x u e y y y b y a x x x b x a x u + − + − − − − + − − = + − − + − − − − − + − − = + − + − − − − + − − = σ σ (6.24) We obtain 0 ) 1 ( ) 1 ( ) 1 ( _{1 2 2 3 3} 1 + − + − + < − = e e e e e e V (6.25)

which is a negative definite function in first quadrant. Three state errors versus time are shown in Fig.6.8.

CASE IV. The generalized synchronization error function is

z y x

e= − + +K (6.26)

where z is the chaotic state vector of Genesio system[48], K=[100 100 100]T.

The goal system for synchronization is Genesio system and initial states is (1, 1, 1), system parameters a4=6, b4=2.92, c4=1.2.

47 1 2 2 3 2 3 1 4 1 4 2 4 3 z z z z z z a z b z c z = = = − − −    (6.27) We have + + − = ∞ → ∞ → e t x y z t lim( lim K)=0 (6.28)

The error dynamics becomes

1 1 1 1 1 1 1 2 2 1 1 1 2 2 2 1 1 2 1 1 1 1 1 3 1 3 2 2 2 3 3 3 1 1 3 1 1 1 2 1 2 3 3 3 2 1 ( ) (sin sin ) [( ) ( )] + ( ) (sin sin ) ( ) ( ) ( ) + ( ) (sin sin ) ( ) ( ) + = − − − − + − − − − = − − − − + − − − − − − = − − − − + − − − − −    e a x y b x y x y x y u z e a x y b x y r x y x x y y x y u z e a x y b x y x x y y c x y u z σ 4 1− 4 2− 4 3 a z b z c z (6.29)

Let initial states be( (0),x₁ x₂(0),x₃(0))=(1, 2, 3), ( (0),y₁ y₂(0),y₃(0))=(3.5, 4,1)and system parameters a1=0.1, b1=1,σ =16, a2=0.2, b2=0.3, r=45.92, a3=0.05, b3=1.8 and

c=4, we find the error dynamics always exists in first quadrant as shown in Fig.6.9.

By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

0 ) ( 2 1 2 3 2 2 2 1 3 2 1 + + + + + > =e e e e e e V (6.30)

48

Its time derivative is

) )(1 ) ( ) ( ) sin (sin ) ( ( ) )(1 ) ( ) ( ) ( ) sin (sin ) ( ( ) 1 )( )] ( ) [( ) sin (sin ) ( ( 3 3 4 2 4 1 4 2 1 3 3 3 2 1 2 1 1 1 3 1 1 3 2 3 2 2 2 3 1 3 1 1 1 1 1 1 1 1 2 1 2 1 1 1 2 2 1 1 1 1 1 1 e z -c z -b z -a z u y x c y y x x y x b y x a e z u y x y y x x y x r y x b y x a e z u y x y x y x b y x a V + + − − − − + − − − − + + + − − − − − − + − − − − + + + − − − − + − − − − = σ  (6.31) Choose 1 1 1 1 1 1 1 2 2 1 1 2 1 2 2 1 1 2 1 1 1 1 1 3 1 3 2 2 3 2 3 3 1 1 3 1 1 1 2 1 2 3 3 2 1 4 1 4 ( ) (sin sin ) [( ) ( )] + ( ) (sin sin ) ( ) ( ) ( ) ( ) (sin sin ) ( ) ( ) u a x y b x y x y x y z e u a x y b x y r x y x x y y x y z e u a x y b x y x x y y c x y z a z b σ = − − − − + − − − + = − − − − + − − − − − + + = − − − − + − − − + − − z₂−c z_{4 3}+e₃ We obtain 0 ) 1 ( ) 1 ( ) 1 ( _{1 2 2 3 3} 1 + − + − + < − = e e e e e e V (6.33)

which is a negative definite function in first quadrant. Three state errors versus time and time histories of x_i− +y_i 100, (i=1,2,3) are shown in Fig.6.10 and Fig.6.11.

49

Fig. 6.1. Phase portrait of error dynamics for CASE I.

50

Fig.6.3. Time histories of x1, x2, x3, y1, y2, y3 for CASE I.

51

Fig.6.5. Time histories of errors for CASE II.

52

Fig.6.7. Phase portrait of error dynamics for CASE III.

53

Fig.6.9. Phase portrait of error dymanics for CASE IV.

54

55

Chapter 7

Chaos Control of a New Ikeda-Lorenz System by

GYC Partial Region Stability Theory

7.1 Preliminaries

By using the GYC partial region stability theory in Appendix, the Lyapunov

function is a simple linear homogeneous function of error states and the controllers are more simple and have less simulation error because they are in lower order than that of traditional controllers. Simulation results show that for a new Ikeda-Lorenz system can be achieved chaos control by GYC partial region stability theory.

7.2 Chaos Control Scheme

Consider the following chaotic systems

( , )

x= f t x (7.1)

where x=

[

x x₁, ₂,",x_n

]

T∈Rn is a the state vector, f R: ₊×Rn →Rn is a vector function.

The goal system which can be either chaotic or regular, is

( , )t = y g y (7.2) where

[

1, 2, ,

]

T _n n y y y R = ∈

y " is a state vector, g:R₊×Rn →Rn is a vector