量子胞神經網路奈米系統之超渾沌、渾沌控制、渾沌化與同步研究

઻ ϱ ϸ ఻ ́ ጮ

機 械 工 程 學 系

博 士 論 文

量子胞神經網路奈米系統之超渾沌、渾沌控制、渾沌

化與同步研究

Hyperchaos, Chaos Control, Chaotization and

Synchronization for a Quantum Cellular Neural

Networks Nano System

研 究 生：楊振雄

指導教授：戈正銘

_榮譽教授

量子胞神經網路奈米系統之超渾沌、渾沌控制、渾沌化與

同步研究

Hyperchaos, Chaos Control, Chaotization and Synchronization for a

Quantum Cellular Neural Networks Nano System

研 究 生：楊振雄 Student：Cheng-Hsiung Yang 指導教授：戈正銘 榮譽教授 Advisor：Zheng-Ming Ge

國 立 交 通 大 學

機 械 工 程 學 系

博 士 論 文

A Dissertation

Submitted to Department of Mechanical Engineering College of Engineering

National Chiao Tung University In Partial Fulfillment of the Requirements

For the Degree of Doctor of Philosophy in

Mechanical Engineering June 2008

Hsinchu, Taiwan, Republic of China

量子胞神經網路奈米系統之超渾沌、渾沌控制、渾沌化與

同步研究

學生：楊振雄 指導教授：戈正銘 榮譽教授 機械工程學系 博士班 國立交通大學

摘

要

渾沌現象普遍存在於宇宙之間，大至星際，小至原子。本文研究耦合之量子 點胞。將它們用於量子點胞自動機，以構成神經胞網路。即使只有兩個量子點胞， 亦可因胞間極化與基底之些許差異而產生渾沌運動。研究此種奈米尺寸之渾沌運 動對於未來超小型非線性胞網路之製成及用標準的非線性胞網路硬體有效地進 行量子計算都是非常重要的。作為前瞻性的研究，本文將對此系統作全面之研 究，研究重點為： 1. 系統渾沌行為之研究。用相圖、功率頻譜圖、參數圖及李亞普諾夫指數分析超 渾沌之行為。 2. 系統之渾沌控制。利用外加常值項、GYC 控制理論、實用適應控制、可變結構 控制、脈衝控制及最佳控制將渾沌運動控制到週期運動或平衡態，同樣地可將 規則運動渾沌化。 3.系統之渾沌同步。利用非線性控制、線性耦合、GYC 控制理論、可變參數線性 耦合實用適應控制、實用適應控制、可變結構控制、最佳控制及脈衝控制方法 研究渾沌同步。

Hyperchaos, Chaos Control, Chaotization and Synchronization

for a Quantum Cellular Neural Networks Nano System

Student：Cheng-Hsiung Yang Advisor：Zheng-Ming Ge

Department of Mechanical Engineering National Chiao Tung University

Abstract

Chaos exists everywhere in the universe, from interstellar space to interatomic interval. In this thesis, coupled quantum-dot cells will be studied. They are used for Quantum-dots Cellular Automata (QCA), to build Cellular Neural Networks (CNN). It is shown that the connection of even only two quantum-dot cells can cause the onset of chaotic motion by small difference of polarizations and template between cells. This study could be very important for future ultra-small realization of CNNs and, on the other hand, for performing quantum computation efficiently via hardware by using standard CNN. As a foresighted study, a detailed study of this system will be developed. The main parts of our study are:

1. The study of chaotic system: By phase portraits, power spectra, parameter diagram and Lyapunov exponents, the various chaotic behaviors of this hyperchaos system are studied.

2. Chaos control and chaotization for the system: By addition of constant term, by GYC control, by pragmatical adaptive control, by variable structure control, by impulsive control and by optimal control, the chaotic motions of the system can be controlled to

periodic motions or to equilibrium states, as well as the regular motions of the system can be chaotized.

3. Chaos synchronization for the system: By nonlinear control, by linear coupling, by GYC control, by variable parameters linear couple pragmatical adaptive control, by pragmatical adaptive control, by variable structure control, by impulsive control and by optimal control, the synchronization of the chaos motions of the two systems are studied.

ᄪ!ᔀ!

本論文得以順利完成，首先要由衷地感謝我的指導老師 戈正銘榮譽教授在學術研 究上的細心及辛苦的指導。在兩年碩士班與四年博士班的學生生涯中，學生深切地感受 到老師對學術研究的熱情與願景，潛移默化間也成為激勵學生本身不斷追求進步的動 力；在待人處事方面，恩師諸多啟迪，使我面對任何事情都可以更圓融態度來處理。這 種無形智慧的累積，已踏入社會的我，自信能排除萬難，解決所有難題。浩瀚師恩永銘 於心，謹誌於此，以表由衷之感謝與敬意。 我也要感謝幾位口試委員張家歐教授、張江南教授、邵錦昌教授與陳献庚教授。諸 位老師給我許多精闢的建議，使本論文更臻完善。『聽君一席話，勝讀十年書』，正代表 我對每位口試委員的萬分謝意。 在交大這幾年間，我要感謝奈米與生科系統控制及動態研究室所有學長、同學及學 弟妹的幫助。特別感謝鄭普建博士、張晉銘、李仕宇、洪瑞祥、蔡岳穎、林宥緯、張安 瑞、易昌賢、歐展義、徐茂原、李乾豪、李小端、蔡秋玉、鄭惠先、林如茵、謝芬琪、 孔韻梅、……，感謝你們在我博士班的日子裡，給我很多幫助，使研究得以順利，也讓 生活更有樂趣。亦感謝在交通大學求學期間所有教導我的老師，使我在學識的領域上有 所成長與突破。 論語里仁篇，子曰：「父母在，不遠遊，遊必有方」。感謝爸 楊明瑞、先母 楊林桃、 後母 洪麗珠、岳父 彭校長源榮、岳母 蘇喜玉及在天國的爺爺 楊丁財，一直以來在經 濟與精神上無盡的付出與支持，使我擁有良好的學習環境，安心地完成學業，一切盡在 不言中。我衷心地將此學位的榮耀獻給最親愛的爸媽、岳父、岳母與爺爺。感謝伯父、 伯母們、姊姊意如、姊夫陳木富、妹妹意美、陳怡璇、陳思晴、陳佳伶、陳玠彥及陳卉 敏在作研究的過程中所給予支持與鼓勵。在此也將這份喜悅與溫文如雅、嫻淑的愛妻韻 文、兒子柏崧一起分享；在我毅然決然轉變人生跑道時，韻文總是尊重我的理想，支持 我的決定，並且在這段期間無怨無悔地包容與體諒及對兒子照顧與教導，使我能在無任 何後顧之憂的環境下順利取得博士學位，謝謝你。附帶一提的是，在研究這條路上，兒 子的笑聲總能舒緩我時而煩躁的心情，家庭的溫暖與安慰給了我最大的動力。 「用感恩的心送走過去，用虔誠的心迎接未來。」謹以此文獻給所有愛我、關心我 以及所有我所愛與關心的人們。 ! ဘડྂ 謹誌 戊子年于新竹交通大學

Contents

Abstract i

誌 謝 iv

Contents v

List of Figures

ix

List of Tables

xiii

Chapter 1 Introduction

1

Chapter 2 Bounded and Unbounded Chaos

5

2-1 Preliminary 5

2-2 Bounded Chaos 5

2-3 Unbounded Chaos 7

Chapter 3 Chaos Control and Chaotization

13

3-1 Preliminary 13

3-2 Chaos Control of the Quantum-CNN Systems 13 3-2-1 Model of Two Quantum-CNN Oscillator Systems 14

3-2-2 Controlling Quantum-CNN attractor to equilibriumO(0, 0, 0, 0) 14 3-3 Chaos Control of Quantum-CNN Oscillators Chaotic System by Variable Structure

Control 23

3-4 Chaos control of Quantum-CNN Oscillators Chaotic System by Impulse Control 25 3-5 Chaotization of Quantum-CNN Chaotic System by Optimum Control 26

Chapter 4 Generalized and Symplectic Synchronizations

28

4-1 Preliminary 28

4-2 The Generalized Synchronization of a Quantum-CNN Chaotic Oscillator with

Different Order Systems 29

4-2-2 Numerical Results of the Generalized Chaos Synchronization of the Quantum-CNN Oscillator with Different Order Systems 31 4-3 The Generalized Synchronization of a Quantum-CNN Chaotic Oscillator with a

Double Duffing Chaotic System 40

4-3-1 Generalized Synchronization Scheme 40 4-3-2 Numerical Results of Generalized Chaos Synchronization of the Quantum-CNN Oscillator with a Double Duffing Chaotic Systems 41 4-4 Symplectic Synchronization of Different Chaotic Systems 51

4-4-1 Symplectic Synchronization Scheme 51

4-4-2 Numerical Results for the Symplectic Chaos Synchronization of Quantum-CNN

Oscillator and Rössler System 52

4-5 Chaos Synchronization of Quantum-CNN Oscillators Chaotic System by Variable

Structure Control 62

Chapter 5 Linear Coupling Synchronization

67

5-1 Preliminaries 67

5-2 Synchronization of the Complex Chaotic Systems in Series Expansion Form 67 5-2-1 Synchronization Schemes of Complex Chaotic Systems in Series Expansion

Form 67

5-2-2 Numerical Results of the Synchronization of Two Quantum-CNN Oscillator Systems by Unidirectional and by Mutual Linear Coupling 69 5-3 Chaos Synchronization of Complex Chaotic Systems in Series Form by Optimal

control 83 5-3-1 Linearly Coupled Chaos Synchronization Scheme by Optimum Control 83

5-3-2 Numerical Results of the Synchronization of Two Quantum-CNN Oscillator Systems by Unidirectional and by Mutual Linear Coupling 85 5-4 Chaos Synchronization of Quantum-CNN Chaotic System by Impulse Control 92

Chapter 6 Pragmatical Synchronization and Control

95

6-1 Preliminary 95

6-2 Pragmatical Generalized Synchronization of Chaotic Systems with Uncertain

Parameters by Adaptive Control 96

6-2-1 Pragmatical Generalized Synchronization Scheme by Adaptive Control 97 6-2-2 Numerical Results of Pragmatical Generalized Chaos Synchronization of Two Quantum-CNN Oscillators by Adaptive Control 98 6-3 Pragmatical Adaptive Control for Different Chaotic Systems 106 6-3-1 Pragmatical Adaptive Control Scheme 106 6-3-2 Numerical Results of the Chaos Control 107 6-4 Synchronization of Chaotic System with Uncertain Variable/Chaotic Parameters by Linear Coupling and Pragmatical Adaptive Tracking 115

6-4-1 Theoretical Analyses 115

6-4-2 Numerical Examples 119

Chapter 7 Chaos Control, Chaotization and Synchronization by GYC

Partial Region Stability Theory

143

7-1 Preliminary 143

7-2 Chaos Control and Chaotization of Chaotic System to Different Systems 143 7-2-1 Chaos Control and Chaotization Scheme 143 7-2-2 Numerical Results of the Chaos Control 144 7-3 The Chaos Generalized Synchronization of a Quantum-CNN Chaotic Oscillator with a

Double Duffing Chaotic System 149

7-3-1 Chaos Generalized Synchronization Strategy 149

7-3-2 Numerical Simulations 150

Chapter 8 Conclusions

159

Appendix B GYC Partial Region Stability Theory

166

References

172

List of Figures

Fig. 1. Schema of the dissertation. 4

Fig. 2-1. Phase portraits of Quantum-CNN with a1=0.11, a2=0.13, ω1=0.11 and ω2=0.08.

6 Fig. 2-2. Power spectrum of x1 for Quantum-CNN with a1=0.11, a2=0.13, ω1=0.11 and

ω2=0.08. 6

Fig. 2-3. Lyapunov exponents of Quantum-CNN with a1=0.11, a2=0.13 and ω1=0.11. 6

Fig. 2-4. Lyapunov exponents diagram with a1=3×10−5, a2=2.3×10−5 and ω1=21×10−5. 8

Fig. 2-5. Power spectrum for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=10×10−5. 8

Fig. 2-6. Time histories of x1, x2, x3 and x4 with parameters a1=3×10−5, a2=2.3×10−5, ω1=21× 10−5 and ω2=10×10−5. 9 Fig. 2-7. Phase portrait of x1, x3 and x4 for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=10× 10−5. 9 Fig. 2-8. Power spectrum for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=40×10−5. 10 Fig. 2-9. Time histories of x1, x2, x3 and x4 with parameters a1=3×10−5, a2=2.3×10−5, ω1=21× 10−5 and ω2=40×10−5. 10 Fig. 2-10. Phase portrait of x1, x3 for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=40×10−5. 11 Fig. 3-1. Phase portraits of the Quantum-CNN System. 19 Fig. 3-2. Power Spectrum Density diagram of the Quantum-CNN System. 20 Fig. 3-3. Time histories of the states of the Quantum-CNN System (3-3-2). 20 Fig. 3-4. Time histories of the states of the Quantum-CNN System (3-2-6). 21 Fig. 3-5. Time histories of the states of the Quantum-CNN System (10). 21 Fig. 3-6. Performances of states of the system with linear, sine, or state cross cosine function

feedback control. 22

Fig. 3-7. Phase portraits of Quantum-CNN by variable structure control. 24 Fig. 3-8. Time histories of states by impulse control. 25

Fig. 3-9. Lyapunov exponent of Quantum-CNN system. 27 Fig. 4-1. Time histories of the master states, of the slave states, and of the synchronization

errors for the Quantum-CNN system and the Lorenz system. 36 Fig. 4-2. Time histories of the master states, of the slave states, and of the sine function synchronization errors for the Quantum-CNN system and the Lorenz system, where

ei=xi–yi+Fsinωt, i=1, 2, 3. 37

Fig. 4-3. Time histories of the master states, of the slave states, and of the Chen system state synchronization errors for the Quantum-CNN system and the Lorenz system, where

ei=xi–yi+z1, i=1, 2, 3. 38

Fig. 4-4. Time histories of the master states, of the slave states, and of the Chen system states synchronization errors for the Quantum-CNN system and the Lorenz system, where

ei=xi–yi+zi, i=1, 2, 3. 39

Fig. 4-5. Time histories of the master states, of the slave states, and of the errors for Case I. 47 Fig. 4-6. Time histories of the master states, of the slave states, and of the errors for Case II. 48 Fig. 4-7. The master state, the slave state, the error, F2 and F4 time histories for Case III. 49

Fig. 4-8. The master state, the slave state, the error, F1,F2, F3 and F4 time histories for Case IV.

50 Fig. 4-9. Time histories of states, state errors, F1, F2, F3, F4, H1, H2, H3 and H4 for Case I. 59

Fig. 4-10. Time histories of states, state errors, F1, F2, F3, F4, H1, H2, H3 and H4 for Case II. 60

Fig. 4-11. Time histories of states, state errors, F1, F2, F3, F4, H1, H2, H3 and H4 for Case III. 61

Fig. 4-12. Time histories of states, state errors. 66 Fig. 5-1. Phase portraits of master system (5-2-10). 78 Fig. 5-2. Time histories of B11, B12, B21, B22, B33, B34, B43 and B44 for Case I. 79

Fig. 5-3. Time histories of states and state errors for Case I. 80 Fig. 5-4. Time histories of B11, B12, B21, B22, B33, B34, B43 and B44 for Case II. 81

Fig. 5-6. Time histories of states, state errors for uni-direction linear coupling. 90 Fig. 5-7. Time histories of states, state errors for mutaul linear coupling. 91 Fig. 5-8. Time histories of states, state errors for uni-direction linear couple. 93 Fig. 5-9. Time histories of states, state errors for mutual linear couple. 94 Fig. 6-1. Time histories of states, state errors, z1, z2, z3, z4, â1, â2, ωˆ₁ and ωˆ₂ for Case I with

a1=4.9, a2=4.9, ω1=3.03 and ω2=1.83. 104

Fig. 6-2. Time histories of states, state errors, z1, z2, z3, z4, â1, â2, ωˆ₁ and ωˆ₂ for Case II with

a1=4.9, a2=4.9, ω1=3.03 and ω2=1.83. 105

Fig. 6-3. Phase portraits of modified nonlinear damped Mathieu system. 112 Fig. 6-4. Phase portraits and Poincaré map of a double harmonic system. 113 Fig. 6-5.Time histories of state errors, ωn1, ωn2, s1, s2, s3, s4, s5 and s6 for Case I. 113

Fig. 6-6. Phase portraits of Quantum-CNN system. 114 Fig. 6-7.Time histories of state errors, a1, a2, ω1, ω2, s1, s2, s3, s4, s5 and s6 for Case II. 114

Fig. 6-8. Phase portrait for Lorenz with σ =10,γ =28,b=8 3. 134 Fig. 6-9. Phase portrait for Eq. (6-4-30) with A1(t)=σ(1+d1sinϖ₁t), A2(t)=γ(1+d2sinϖ₂t) and

A3(t)=b(1+d3sinϖ₃t). 134

Fig. 6-10. Time histories of states, state errors, A1,A2, A3, Â1, Â2, Â3 and estimated Lipschitz

constant Ĝ for Case I. 134

Fig. 6-11. Phase portraits for chaotic system (6-4-42). 135 Fig. 6-12. Phase portraits for chaotic system (6-4-43). 135 Fig. 6-13. Time histories of states, state errors, A1, A2, A3, A4, â1, â2, ŵ1, ŵ2 and estimated

Lipschitz constant Ĝ for Case II. 136

Fig. 6-14. Phase portraits for chaotic system (6-4-54). 137 Fig. 6-15. Time histories of states, state errors, A1, A2, A3, â1, â2, ŵ1 and estimated Lipschitz

constant Ĝ for Case III. 138

Fig. 6-17. Phase portrait for Lorenz system (6-4-69) with σ=8, γ=27 and b=3.2. 139 Fig. 6-18. Time histories of states, state errors, A1, A2, A3, Â1, Â2, Â3 and estimated Lipschitz

constant Ĝ for Case IV. 139

Fig. 6-19. Time histories of states, state errors, A1, A2, Â1, Â2 and estimated Lipschitz constant

Ĝ for Case V. 140

Fig. 6-20. Phase portraits for chaotic system (6-4-93). 141 Fig. 6-21. Phase portraits for chaotic system (6-4-95). 141 Fig. 6-22. Time histories of states, state errors, A1, A2, A3, A4, â1, â2, ŵ1, ŵ2 and estimated

Lipschitz constant Ĝ for Case VI. 142

Fig. 7-1. Phase portraits of modified nonlinear damped Mathieu system. 147 Fig. 7-2. Phase portraits and Poincaré map of a double harmonic system. 147 Fig. 7-3. Phase portraits of error dynamic system for Case I. 148 Fig. 7-4.Time histories of state errors for Case I. 148 Fig. 7-5. Phase portraits of Quantum-CNN system. 148 Fig. 7-6. Phase portraits of error dynamic system for Case I. 148 Fig. 7-7.Time histories of state errors for Case II. 148 Fig. 7-8. Time histories of the master states, of the slave states and of the errors for Case I. 155 Fig. 7-9. Time histories of the master states, of the slave states and of the errors for Case II.

156 Fig. 7-10. The master state, the slave state, the error, F2 and F4 time histories for Case III. 157

Fig. 7-11. Time histories of the master states, of the slave states and of the errors for Case II. 158

List of Tables

Table 2-1. Sensitivity to initial conditions. One unbounded state x2. Time passage 1.2×1010s. 11

Table 2-2. Sensitivity to initial conditions. Two unbounded states x2 and x4. Time passage 1.2×

Chapter 1

Introduction

In nature, most of dynamic systems are nonlinear and can be described by the nonlinear equations of motion. If the nonlinear term can be ignored, it is possible to be linearized and easily be solved by the already known methods. For many nonlineared systems, the linearization process is reasonable, whereas for some nonlinear systems linearization is unevailable. Hence the researches of nonlinear systems spread quickly today. For the nonlinear system, the study of the types of periodic solutions, the effects to the solutions caused by different parameters and initial conditions, the stability analysis of the solutions, consist of the major tasks. Besides, a substantial understanding of the complicated phenomena raised from nonlinearity is also what we are interested in. The central characteristics are that a process like randomization happens in the deterministic system and small differences in the initial conditions produce very great ones in the final phenomena. The irregular and unpredictable motions of many nonlinear systems have been labeled ‘‘chaotic’’. In the end of nineteen century Poncaré first pointed out some important concepts of chaos theory like homoclinic, bifurcation, etc. Lorenz researched the strange changes in the atmosphere that is the first example to study chaos in 1963. Chaotic phenomena is quite useful in many applications such as fluid mixing [15], human brain study [16], and heart beat regulation [17], etc. Since Ott, Grebogi, and Yorke proposed the OGY method [78], a method of controlling chaos, ‘‘controlling of chaos’’ is receiving increasing attention within the area of non-linear dynamics. Chaos has many applications in various systems while it is unfavorable in many other cases due to its irregular behavior. Therefore, both chaos control and chaotization are important depending on the specific applications. They are effective method for both chaos elimination and utilization and have been thoroughly studied in various fields of science.

Chaotic systems exhibit sensitive dependence on initial conditions. Because of this property, chaotic systems are thought difficult to be synchronized or controlled. From the earlier works

[1-3], especially after Pecora and Carroll [3], the researchers have realized that synchronization of chaotic motions is possible. From then on, synchronization of chaos was of great interest in these years [4-14]. In particular, it was pointed out that chaos synchronization has the potential in secure communication [18, 19], chemical and biological systems [20, 22], etc. Many engineers and scientists were attracted to this discipline [23-36].

Synchronization means that the states of slave system approach eventually to the ones of master system. Two kinds of chaos synchronization are discussed the most often. (1) Duplication: the first kind introduced by Pecora and Carroll [3] consists of a master system and a slave system. The former one evolves chaotic orbits and the latter is identical to the master system except some partial states replaced by that of the master one. (2) Coupling: the second kind consists of two identical chaotic systems except coupling term. Coupled systems can be unidirectional or mutual. Under certain conditions (appropriate coupling functions and/or system parameters with enough evolution time) the slave system will behave the same orbit with the master system.

There are many control methods to synchronize chaotic systems such as observer-based design methods [37-44], adaptive control [45-54], sliding mode control (or variable structure control) [41, 43, 44, 55-58], impulsive control [59-65] and other control methods [66-72]. A another kind of more general synchronism called generalized synchronization (GS) is studied in [73-77], this means that there is a functional relation between state variables of master and slave systems as time evolves. This function needs not to be defined on the whole phase space but on the attractor only. Three methods were proposed to detect GS in [73-75] respectively while another method measuring the smooth degree of this function in [77].

As numerical examples, recently developed Quantum Cellular Neural Networks (Quantum-CNN) model, based on Schrödinger equation in which cells composed of interacting quantum dots are employed in CNN architecture. One of their peculiarities lies in the further degree of freedom possessed by each cell due to the quantum interaction between dots (quantum mechanical phase difference). This fact allows obtaining complex dynamics even in a network

with only two cells. Our aim is therefore to investigate their dynamical behavior by suitable variation of coupling parameters and initial conditions. The study could be very important for future ultra-small realization of CNNs and, on the other hand, for performing quantum computation efficiently via hardware by using standard CNN as in literature [23].

In Fig. 1 shows the organization of this dissertation with the following eight chapters:

1. Chapter 1 is an introduction to this dissertation including the backgrounds, motivation and objectives, and the organization of this work.

2. In Chapter 2, hyper chaotic Quantum-CNN oscillator system is studied.

3. In Chapter 3, chaos control and chaotization for the Quantum-CNN oscillator system are studied. By addition of constant term, by variable structure control, by impulsive control and by optimal control, the chaotic motions of the system can be controlled to periodic motions as well as to equilibrium states or the regular motions of the system can be chaotized.

4. In Chapter 4, chaos synchronization for the Quantum-CNN oscillator system is studied by nonlinear control, and by variable structure control.

5. In Chapter 5, chaos synchronization for the Quantum-CNN oscillator system is studied by linear coupling, by impulsive control and by optimal control.

6. In Chapter 6, a pragmatical adaptive control method is applied to the chaos control and chaos synchronization.

7. In Chapter 7, a GYC partial region stability theorem is applied to the chaos control and chaos synchronization.

Hyperchaos and Chaos Control, Chaotization and Synchronization

for a Quantum Cellular Neural Networks Nano System

Chapter 1 Instruction Control of Chaos Chaos Synchronization Chaos Analysis Chapter 2 Bounded and Uunbound

chaos

Chapter 3

Chaos Control and Chaotization 1. Control to fixed point

2. Impulse control

3. Variable structure control 4. Optimal control

Chapter 7

Chaos Control, Chaotization and Synchronization by GYC Partial

Region Stability Theory 1. Chaos synchronization 2. Chaos Control

Chapter 5

Linear coupling synchronization 1. Series form

2. Optimal control 3. Impulse control

Chapter 4

Generalized and Symplectic Synchronization 1. Different order systems

2. Hybrid synchronization 3. Symplectic synchronization 4. Variable structure control

Chapter 6

Pragmatical Synchronization and Control 1. Chaos synchronization 2. Chaos control 3. Lipschitz condition Chapter 8 Conclusions

Chapter 2

Bounded and Unbounded Chaos

2-1 Preliminary

In order for a system to exhibit hyperchaos, the minimum dimension of its state space is four. This is because one Lyapunov exponent is always zero and the sum of the exponents must be negative in order for an attractor to form. A hyperchaotic system is characterized by the presence of two or more PLEs in its Lyapunov spectrum, indicating that it is unstable in more than one direction. Besides the theoretical interest in the dynamics of such nonlinear systems, there has been a practical interest in chaos and hyperchaos as means for secure communication. Hyperchaos was first reported from computer simulations of hypothetical ordinary differential equations in [106].

2-2 Bounded Chaos

For a two-cell Quantum-CNN, the following differential equations are obtained [23]:

2 1 1 1 2 1 2 1 1 3 1 ₂ 2 1 2 3 2 3 4 3 4 2 3 1 2 ₂ 4 3 2 1 sin ( ) 2 cos 1 2 1 sin ( ) 2 cos 1 x a x x x x x x a x x x a x x x x x x a x x ω ω ⎧ = − − ⎪ ⎪ _{= − − +} ⎪ ₋ ⎪ ⎨ = − − ⎪ ⎪ ⎪ _{= − − +} ⎪ ₋ ⎩ (2-1-1)

where x1 and x3 are polarizations, x2 and x4 are quantum phase displacements, a1 and a2 are

proportional to the inter-dot energy inside each cell and ω1 and ω2 are parameters that weigh

effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional CNNs.

The evolution of a set of trajectories emanating from various initial conditions is presented in the phase space. When the solution becomes stable, the asymptotic behaviors of the phase trajectories are particularly interested and the transient behaviors in the system are neglected. The phase portraits of the Quantum-CNN system, equation (2-1-1), are plotted in Fig. 2-1.

Fig. 2-1. Phase portraits of Quantum-CNN with a1=0.11, a2=0.13, ω1=0.11 and ω2=0.08.

If the states are not periodic, their spectrum must be in terms of oscillations with a continuum of frequencies. Such a representation of the spectrum is called Fourier integral. The power spectrum analysis of the nonlinear dynamical system, equation (2-1-1) is shown in Fig. 2-2. Apparently, the spectrum of the periodic motion only consists of discrete frequencies. The noise-like spectrum is the characteristics of chaotic dynamical system.

Fig. 2-2. Power spectrum of x1 for Quantum-CNN with a1=0.11, a2=0.13, ω1=0.11 and ω2=0.08.

The Lyapunov exponents of the solutions of the nonlinear dynamical system, equation (2-1-1), is plotted in Fig. 2-3. Since there exist two positive Lyapunov exponents, hyperchaos is obtained.

Fig. 2-3. Lyapunov exponents of Quantum-CNN with a1=0.11, a2=0.13 and ω1=0.11.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 4 5 6 7 8 x1 x2 (a) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x3 x4 (b) 0 0.05 0.1 0.15 -6 -4 -2 0 2 4 6x 10 -3 ω2 L. E. Frequency (rad/sec) log 10 (Sx 1 ) 0 0.5 1 1.5 2 2.5 3 3.5 10-12 10-10 10-8 10-6 10-4 10-2 100 102

2-3 Unbounded Chaos

In [101], _{Λ ⊂ R}n _{is a compact set invariant under flow ( , )φ t x . Then the definition is:}

Definition (Chaotic Invariant Set) Λ is said to be chaotic if

1. The flow ( , )φ t x generated by vector field x= f x( ) has sensitive dependence on initial conditions on Λ.

2. ( , )φ t x is topologically transitive on Λ.

The compact invariant set Λ is bounded [102]. Other definitions are, for instance: (a) Chaos is defined as the phenomenon of occurrence of bounded nonperiodic evolution in completely deterministic nonlinear dynamical system with high sensitive dependence on initial conditions [103]. (b) Chaos is recurrent motion in simple systems or low-dimensional behavior that has some random aspect as well as a certain order. Exponential divergence from adjacent starts while remaining in a bounded region of phase space is a signature of chaotic motion [104]. (c) Chaos is the aperiodic, long-term behavior of a bounded, deterministic system that exhibits sensitive dependence on initial conditions [105].

Unbounded chaos is found for a two-cell Quantum-CNN oscillator chaotic system [23]:

2 1 1 1 2 1 2 1 1 3 1 ₂ 2 1 2 3 2 3 4 3 4 2 3 1 2 ₂ 4 3 2 1 sin ( ) 2 cos 1 2 1 sin ( ) 2 cos 1 x a x x x x x x a x x x a x x x x x x a x x ω ω ⎧ = − − ⎪ ⎪ _{= − − +} ⎪ ₋ ⎪ ⎨ = − − ⎪ ⎪ ⎪ _{= − − +} ⎪ ₋ ⎩ (2-2-1)

where x1 and x3 are polarizations, x2 and x4 are quantum phase displacements, a1 and a2 are

proportional to the inter-dot energy inside each cell and ω1 and ω2 are parameters that weigh

effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional CNNs.

We choose ω2 as abscissa, a1=3×10−5, a2=2.3×10−5 and ω1=21×10−5. Lyapunov exponent

exponents are smooth curves on unbounded chaos states and they are uneven curves on bouned chaos. They are shown in Fig. 2-3 and Fig. 2-4.

Fig. 2-4. Lyapunov exponents diagram with a1=3×10−5, a2=2.3×10−5 and ω1=21×10−5.

Then choose a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=10×10−5, the power spectrum is

shown in Fig. 2-5.

Fig. 2-5. Power spectrum for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=10×10−5.

Time histories of x1, x2, x3 and x4 are shown in Fig. 2-6, while x2 is unbounded.

0 1 2 3 4 5 6 x 10-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5x 10 -3 ω2 L. E. 0 0.1 0.2 0.3 0.4 0.5 0.6 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 log 10 (Sx 1 ) Frequency (rad/sec)

Fig. 2-6. Time histories of x1, x2, x3 and x4 with parameters a1=3×10−5, a2=2.3×10−5, ω1=21×10−5

and ω2=10×10−5.

The phase portrait of bounded states x1, x3 and x4 is shown in Fig. 2-7.

Fig. 2-7. Phase portrait of x1, x3 and x4 for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=10×10−5.

Choose a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=40×10−5. The power spectrum is

shown in Fig. 2-8. 0 2 4 6 8 10 12 x 109 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Time (sec) x4 (d) 0 2 4 6 8 10 12 x 109 -1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92 -0.91 Time (sec) x3 (c) 0 2 4 6 8 10 12 x 109 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0x 10 6 Time (sec) x2 (b) 0 2 4 6 8 10 12 x 109 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Time (sec) x1 (a) 0.75 0.8 0.85 0.9 0.95 -1 -0.98 -0.96 -0.94 -0.92 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x3 x4 x1

Fig. 2-8. Power spectrum for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=40×10−5.

Time histories of x1, x2, x3 and x4 are shown in Fig. 2-9, while x2 and x4 are unbounded.

Fig. 2-9. Time histories of x1, x2, x3 and x4 with parameters a1=3×10−5, a2=2.3×10−5, ω1=21×10−5

and ω2=40×10−5. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 log 10 (S x1 ) Frequency (rad/sec) 0 2 4 6 8 10 12 x 109 -1 0 1 2 3 4 5 6 7 8 9x 10 6 Time (sec) x4 (d) 0 2 4 6 8 10 12 x 109 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0x 10 6 Time (sec) x2 (b) 0 2 4 6 8 10 12 x 109 0.7 0.75 0.8 0.85 0.9 0.95 Time (sec) x1 (a) 0 2 4 6 8 10 12 x 109 -1 -0.98 -0.96 -0.94 -0.92 -0.9 -0.88 Time (sec) (c) x3

The phase portrait of bounded states x1, x3 is shown in Fig. 2-10.

Fig. 2-10. Phase portrait of x1, x3 for a1=3×10−5, a2=2.3×10−5, ω1=21×10−5 and ω2=40×10−5.

Table 2-1. Sensitivity to initial conditions. One unbounded state x2. Time passage 1.2×1010s.

Initial condition of states Difference between final states of

(a) and (b) case second final states

Case x1 x2 x3 x4 Df1 Df2 Df3 Df4 (a) 0.29 -0.47 -0.99 0.77 (b) 0.29-1×10-7 -0.47+1×10-7 _-0.99+1×10-7 _0.77-1×10-7 0.2045 -304.67 -0.133 -0.9875 (a) 0.8 -0.77 -0.99 -0.57 (b) 0.8-1×10-7 -0.77+1×10-7 _-0.99+1×10-7 _-0.57-1×10-7 0.036 8.967 -0.004 -0.232 (a) 0.19 -0.27 -0.98 0.57 (b) 0.19-1×10-7 -0.27+1×10-7 _-0.98+1×10-7 _0.57-1×10-7 -0.068 -202.46 -0.11 0.738 (a) 0.29 -0.27 -0.987 _0.27 (b) 0.29-1×10-7 -0.27+1×10-7 _-0.98+1×10-7 _0.27-1×10-7 0.062 45.41 -0.039 3 0.337 (a) 0.29 -0.47 -0.99 0.27 (b) 0.29-1×10-7 -0.74+1×10-7 _-0.99+1×10-7 _0.27-1×10-7 -0.342 254.6 -0.109 -0.189 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 -1 -0.998 -0.996 -0.994 -0.992 -0.99 -0.988 -0.986 x3 x1

Table 2-2. Sensitivity to initial conditions. Two unbounded states x2 and x4. Time passage 1.2×

1010s.

Initial condition of states Difference between final states of (a) and

(b) case second final states

Case x1 x2 x3 x4 Df1 Df2 Df3 Df4 (a) 0.8 -0.87 -0.29 -0.42 (b) 0.8+1×10-7 -0.87+1×10-7 _-0.29+1×10-7 _-0.42+1×10-7 -0.036 49.36 -0.0017 -63.95 (a) 0.75 -0.57 -0.85 -0.42 (b) 0.75+1×10-7 -0.57+1×10-7 _-0.85+1×10-7 _-0.42+1×10-7 0.109 246.67 0.0184 -467.04 (a) 0.09 -0.27 -0.45 -0.57 (b) 0.09+1×10-7 -0.27+1×10-7 _-0.45+1×10-7 _-0.57+1×10-7 0.271 557.57 0.336 788.61 (a) 0.19 -0.27 -0.95 -0.75 (b) 0.19+1×10-7 -0.27+1×10-7 _-0.95+1×10-7 _-0.75+1×10-7 0.286 -374.79 0.0199 778.72 (a) 0.39 -0.27 -0.98 -0.75 -0.135 -1.49 -0.0285 2.91

Table 2-1 and Table 2-2 show the sensitivity to initial conditions.

The current chaos theory connot explain that the unbounded chaos violates the fundamental definition of chaos.

Chapter 3

Chaos Control and Chaotization

3-1 Preliminary

Chaos control exploits the sensitivity to initial conditions and to perturbations that is inherent in chaos as a means to stabilize unstable periodic orbits within a chaotic attractor. The control can operate by altering system variables or system parameters, and either by discrete corrections or by continuous feedback. Many methods of chaos control have been derived and tested.

A commonly applied method for control of chaotic dynamical systems was discovered by Ott, Grebogi and Yorke (OGY) [78]. The OGY method requires an analytical description of the linear map describing the behavior near the fixed point. This map is used to determine small perturbations that, when periodically applied, use the system’s own dynamics to send the system towards an unstable fixed point. Continued application of these perturbations keeps the system near the fixed point, thereby stabilizing the unstable fixed point even in a noisy system. The OGY method generally is inadequate when the system is far from the fixed point and the linear map is no longer valid. The original OGY method is also limited to controlling only one- or two-dimensional maps. However, the method has been extended to higher dimensions and to cases where multiple control parameters are available.

In some nontraditional applications chaos can be useful and beneficial which led to research on the task of purposely making a nonchaotic dynamical system chaotic and this process is called ‘‘anti-control of chaos or chaotization’’. The chaotization technique is investigated which can be applied to generate chaos irrespective of all the parameter values and states of the system.

3-2 Chaos Control of the Quantum-CNN Systems

feedback. The Routh-Hurwitz theorem is used to derive the conditions of stability of controlled Quantum-CNN systems.

3-2-1 Model of Two Quantum-CNN Oscillator Systems

Recently developed Quantum Cellular Neural Networks (Quantum-CNN) chaotic oscillator is used. Quantum-CNN oscillator equations are derived from a Schrödinger equation taking into account quantum dots cellular automata structures to which in the last decade a wide interest has been devoted with particular attention towards quantum computing. For a two-cell Quantum-CNN, the following differential equations are obtained:

2 1 1 1 2 1 2 1 1 3 1 ₂ 2 1 2 3 2 3 4 3 4 2 3 1 2 ₂ 4 3 2 1 sin ( ) 2 cos 1 2 1 sin ( ) 2 cos 1 x a x x x x x x a x x x a x x x x x x a x x ω ω ⎧ = − − ⎪ ⎪ = − − + ⎪ ₋ ⎪ ⎨ = − − ⎪ ⎪ ⎪ = − − + ⎪ ₋ ⎩ (3-2-1)

where x1 and x3 are polarizations, x2 and x4 are quantum phase displacements, a1 and a2 are

proportional to the inter-dot energy inside each cell and ω1 and ω2 are parameters that weigh

effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional CNNs. Let a1＝a2＝0.3, ω1＝0.33 and ω2＝0.31. The initial values of

linear coupled Quantum-CNN systems are taken as x1(0)＝0.8, x2(0)＝−0.77, x3(0)＝−0.72 and x4(0)＝0.57 respectively. Figs. 3-1~3-2 are phase portrait, and power spectra density diagram

respectively.

3-2-2 Controlling Quantum-CNN attractor to equilibrium O (0,0,0,0)

Case I Linear feedback control

2 1 1 1 2 1 1 1 2 1 1 3 1 ₂ 2 2 2 1 2 3 2 3 4 3 3 3 4 2 3 1 2 ₂ 4 4 4 3 2 1 sin ( ) 2 cos 1 2 1 sin ( ) 2 cos 1 x a x x k x x x x x a x k x x x a x x k x x x x x a x k x x ω ω ⎧ = − − + ⎪ ⎪ = − − + + ⎪ ₋ ⎪ ⎨ = − − + ⎪ ⎪ ⎪ = − − + + ⎪ ₋ ⎩ (3-2-2)

Expand the right hand sides of Eq. (3-2-1) into power series:

2 3 1 1 1 2 2 2 1 1 2 3 2 1 1 3 1 1 1 2 1 2 2 2 3 3 2 3 4 4 4 3 3 2 3 4 2 3 1 2 3 3 4 3 4 4 1 1 2 ( ) 2 6 1 1 ( ) 2 ( ) 2 2 1 1 2 ( ) 2 6 1 1 ( ) 2 ( ) 2 2 x a x x x x k x x x x a x x x x k x x a x x x x k x x x x a x x x x k x ω ω ⎧ = − − + − + + ⎪ ⎪ ⎪ = − − + − + + + ⎪ ⎨ ⎪ = − − + − + + ⎪ ⎪ ⎪ = − − + − + + + ⎩  "  "  "  " (3-2-3)

whose Jacobian matrix is

1 1 1 1 2 1 3 2 2 2 2 4 2 0 0 2 0 0 0 2 0 2 k a a k k a a k ω ω ω ω − − ⎡ ⎤ ⎢ _{− −} ⎥ ⎢ ⎥ ⎢ − − ⎥ ⎢ _{− −} ⎥ ⎣ ⎦ .

The characteristic equation of J is

1 1 1 1 2 1 3 2 2 2 2 4 1 1 1 1 4 1 1 2 1 2 1 1 2 1 3 2 2 2 4 3 1 2 3 4 1 2 3 4 3 4 1 2 0 0 2 0 0 0 2 0 2 2 0 2 0 ( ) 2 2 2 0 0 0 2 ( ) (( )( ) k a a k k a a k k a k a k a k a a k k a k k k k k k k k k k k k λ ω λ ω λ ω ω λ λ λ λ ω λ ω ω λ ω λ ω ω λ λ − − − ⎡ ⎤ ⎢ _{− − −} ⎥ ⎢ ⎥ ⎢ − − − ⎥ ⎢ _{− − −} ⎥ ⎣ ⎦ − − − − − − ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = − − _⎢ − − − _⎥+ _⎢ − − − _⎥ ⎢ − − ⎥ ⎢ − ⎥ ⎣ ⎦ ⎣ ⎦ = + + + + + + + + + 2 2 2 1 2 1 1 2 2 2 2 2 1 2 3 4 2 2 2 3 4 1 2 1 1 1 1 2 3 4 2 2 2 2 1 3 4 2 1 2 1 1 3 4 2 2 1 2 1 2 1 2 1 2 2 1 4( ) 2( )) (( )( 4 2 ) ( )( 4 2 )) ( 4( ) 2( ) 16 8 ( )). a a a a k k k k a a k k k k a a k k k k a k k a k k a k k a k k a a a a a a ω ω λ ω ω λ ω ω ω ω + + − + + + + − + + + − + + + − − + + + (3-2-4) Take 2 2 1 2 3 4 1 2 3 4 3 4 1 2 1 2 1 1 2 2 ( ), ( )( ) 4( ) 2( ), B= k + + +k k k C= k +k k +k +k k +k k + a +a − aω +aω

2 2 1 2 3 4 2 2 2 3 4 1 2 1 1 1 2 2 2 2 1 2 3 4 1 3 4 2 1 2 1 1 3 4 2 2 1 2 1 2 1 2 1 2 2 1 ( )( 4 2 ) ( )( 4 2 ), ( 4( ) 2( ) 16 8 ( )). D k k k k a a k k k k a a E k k k k a k k a k k a k k a k k a a a a a a ω ω ω ω ω ω = + + − + + + − = + + − − + + + If 0 B> , BC D 0 B − > , B CD BE( ) D2 0 BC D − − > − and E>0 (3-2-5) then all the eigenvalues of the Jacobian matrix have negative real parts. So we choose k1=1, k2=1, k3=1 and k4=1, and find B=4, BC D 4.18

B − ₌ , 2 ( ) 4.22 B CD BE D BC D − − ₌ − and E=4.62.

Proposition 1. According to Routh-Hurwitz theorem and Lyapunov asymptotically stability

theorem, when ki’s satisfy (3-2-5), the controlled Quantum-CNN system (3-2-2) is locally

asymptotically stable at the equilibrium O (0, 0, 0, 0) in spite of the higher order terms.

Numerical simulations are used to investigate the controlled Quantum-CNN system (3-2-1) using fourth-order Runge-Kutta scheme with time step 0.005. After 100 second, the states (x1, x2, x3, x4) of the controlled Quantum-CNN system (3-2-2) approach zeros in Fig. 3-3.

Case II Nonlinear sine function feedback control

We design controller is u_i =k_isinx_i

2 1 1 1 2 1 1 1 2 1 1 3 1 ₂ 2 2 2 1 2 3 2 3 4 3 3 3 4 2 3 1 2 ₂ 4 4 4 3 2 1 sin sin ( ) 2 cos sin 1 2 1 sin sin ( ) 2 cos sin 1 x a x x k x x x x x a x k x x x a x x k x x x x x a x k x x ω ω ⎧ = − − + ⎪ ⎪ = − − + + ⎪ ₋ ⎪ ⎨ = − − + ⎪ ⎪ ⎪ = − − + + ⎪ ₋ ⎩ (3-2-6)

Expand the right hand sides of Eq. (3-2-6) into power series:

2 3 3 1 1 1 2 2 2 1 1 1 2 3 3 2 1 1 3 1 1 1 2 1 2 2 2 2 3 3 3 2 3 4 4 4 3 3 4 2 3 3 4 2 3 1 2 3 3 4 3 4 4 4 1 1 1 2 ( ) ( ) 2 6 6 1 1 1 ( ) 2 ( ) ( ) 2 2 6 1 1 1 2 ( ) ( ) 2 6 6 1 1 1 ( ) 2 ( ) ( ) 2 2 6 x a x x x x k x x x x x a x x x x k x x x a x x x x k x x x x x a x x x x k x x ω ω ⎧ = − − + + − + + − + ⎪ ⎪ = − − + − + + + − + ⎨ = − − + + − + + − + = − − + − + + + − +  " "  " "  " "  " " ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (3-2-7)

1 1 1 1 2 1 3 2 2 2 2 4 2 0 0 2 0 0 0 2 0 2 k a a k k a a k ω ω ω ω − − ⎡ ⎤ ⎢ _{− −} ⎥ ⎢ ⎥ ⎢ − − ⎥ ⎢ _{− −} ⎥ ⎣ ⎦ .

The characteristic equation of J is

1 1 1 1 2 1 3 2 2 2 2 4 1 1 1 1 4 1 1 2 1 2 1 1 2 1 3 2 2 2 4 3 1 2 3 4 1 2 3 4 3 4 1 2 0 0 2 0 0 0 2 0 2 2 0 2 0 ( ) 2 2 2 0 0 0 2 ( ) (( )( ) k a a k k a a k k a k a k a k a a k k a k k k k k k k k k k k k λ ω λ ω λ ω ω λ λ λ λ ω λ ω ω λ ω λ ω ω λ λ − − − ⎡ ⎤ ⎢ _{− − −} ⎥ ⎢ ⎥ ⎢ − − − ⎥ ⎢ _{− − −} ⎥ ⎣ ⎦ − − − − − − ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = − − _⎢ − − − _⎥+ _⎢ − − − _⎥ ⎢ − − ⎥ ⎢ − ⎥ ⎣ ⎦ ⎣ ⎦ = + + + + + + + + + 2 2 2 1 2 1 1 2 2 2 1 2 3 4 2 2 2 3 4 1 2 1 1 1 2 2 1 2 3 4 2 2 2 2 1 3 4 2 1 2 1 1 3 4 2 2 1 2 1 2 1 2 1 2 2 1 4( ) 2( (( )( 4 2 ) ( )( 4 2 )) )) ( 4( ) 2( ) 16 8 ( )). a a a k k k k a a k k k k a a a k k k k a k k a k k a k k a k k a a a a a a ω ω ω λ ω λ ω ω ω ω + + − + + + − + + + − + + + + − − + + + (3-2-8) Take 2 2 1 2 3 4 1 2 3 4 3 4 1 2 1 2 1 1 2 2 2 2 1 2 3 4 2 2 2 3 4 1 2 1 1 1 2 2 2 2 1 2 3 4 1 3 4 2 1 2 1 1 3 4 2 2 1 2 1 2 1 2 1 2 2 1 ( ), ( )( ) 4( ) 2( ) ( )( 4 2 ) ( )( 4 2 ), ( 4( ) 2( ) 16 8 ( )). B k k k k C k k k k k k k k a a a a D k k k k a a k k k k a a E k k k k a k k a k k a k k a k k a a a a a a ω ω ω ω ω ω ω ω = + + + = + + + + + + − + = + + − + + + − = + + − − + + + If 0 B> , BC D 0 B − > , B CD BE( ) D2 0 BC D − − > − and E>0 (3-2-9) then all the eigenvalues of the Jacobian matrix have negative real parts. So we choose k1=1, k2=1, k3=1 and k4=1, and find B=4, BC D 4.18

B − ₌ , 2 ( ) 4.22 B CD BE D BC D − − ₌ − and E=4.62.

Proposition 2. According to Routh-Hurwitz theorem and Lyapunov asymptotically stability

theorem, when ki’s satisfy (3-2-9), the controlled Quantum-CNN system (3-2-6) is locally

asymptotically stable at the equilibrium O (0, 0, 0, 0) in spite of the higher order terms.

Numerical simulations are used to investigate the controlled Quantum-CNN system (3-2-1) using fourth-order Runge-Kutta scheme with time step 0.005. After 100 second, the states (x1, x2, x3, x4) of the controlled Quantum-CNN system (3-2-6) approach zeros in Fig. 3-4.

Case III Nonlinear state cross cosine nonlinear function feedback control We design controller is u_i =k x_{i i}cosx_i

2 1 1 1 2 1 1 1 1 2 1 1 3 1 ₂ 2 2 2 2 1 2 3 2 3 4 3 3 3 3 4 2 3 1 2 ₂ 4 4 4 4 3 2 1 sin cos ( ) 2 cos cos 1 2 1 sin cos ( ) 2 cos cos 1 x a x x k x x x x x x a x k x x x x a x x k x x x x x x a x k x x x ω ω ⎧ = − − + ⎪ ⎪ = − − + + ⎪ ₋ ⎪ ⎨ = − − + ⎪ ⎪ ⎪ = − − + + ⎪ ₋ ⎩ (3-2-10)

Expand the right hand sides of Eq. (3-2-10) into power series:

2 3 3 1 1 1 2 2 2 1 1 1 2 3 3 2 1 1 3 1 1 1 2 1 2 2 2 2 3 3 3 2 3 4 4 4 3 3 4 2 3 3 4 2 3 1 2 3 3 4 3 4 4 4 1 1 1 2 ( ) ( ) 2 6 6 1 1 1 ( ) 2 ( ) ( ) 2 2 6 1 1 1 2 ( ) ( ) 2 6 6 1 1 1 ( ) 2 ( ) ( ) 2 2 6 x a x x x x k x x x x x a x x x x k x x x a x x x x k x x x x x a x x x x k x x ω ω ⎧ = − − + + − + + − + ⎪ ⎪ = − − + − + + + − + ⎨ = − − + + − + + − + = − − + − + + + − +  " "  " "  " "  " " ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (3-2-11)

whose Jacobian matrix is

1 1 1 1 2 1 3 2 2 2 2 4 2 0 0 2 0 0 0 2 0 2 k a a k k a a k ω ω ω ω − − ⎡ ⎤ ⎢ _{− −} ⎥ ⎢ ⎥ ⎢ − − ⎥ ⎢ _{− −} ⎥ ⎣ ⎦ .

The characteristic equation of J is

1 1 1 1 2 1 3 2 2 2 2 4 1 1 1 1 4 1 1 2 1 2 1 1 2 1 3 2 2 2 4 3 1 2 3 4 1 2 3 4 3 4 1 2 0 0 2 0 0 0 2 0 2 2 0 2 0 ( ) 2 2 2 0 0 0 2 ( ) (( )( ) k a a k k a a k k a k a k a k a a k k a k k k k k k k k k k k k λ ω λ ω λ ω ω λ λ λ λ ω λ ω ω λ ω λ ω ω λ λ − − − ⎡ ⎤ ⎢ _{− − −} ⎥ ⎢ ⎥ ⎢ − − − ⎥ ⎢ _{− − −} ⎥ ⎣ ⎦ − − − − − − ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = − − _⎢ − − − _⎥+ _⎢ − − − _⎥ ⎢ − − ⎥ ⎢ − ⎥ ⎣ ⎦ ⎣ ⎦ = + + + + + + + + + 2 2 2 1 2 1 1 2 2 2 2 2 1 2 3 4 2 2 2 3 4 1 2 1 1 1 1 2 3 4 2 2 2 2 1 3 4 2 1 2 1 1 3 4 2 2 1 2 1 2 1 2 1 2 2 1 4( ) 2( )) (( )( 4 2 ) ( )( 4 2 )) ( 4( ) 2( ) 16 8 ( )). a a a a k k k k a a k k k k a a k k k k a k k a k k a k k a k k a a a a a a ω ω λ ω ω λ ω ω ω ω + + − + + + + − + + + − + + + − − + + + (3-2-12)

2 2 1 2 3 4 1 2 3 4 3 4 1 2 1 2 1 1 2 2 2 2 1 2 3 4 2 2 2 3 4 1 2 1 1 1 2 2 2 2 1 2 3 4 1 3 4 2 1 2 1 1 3 4 2 2 1 2 1 2 1 2 1 2 2 1 ( ), ( )( ) 4( ) 2( ), ( )( 4 2 ) ( )( 4 2 ), ( 4( ) 2( ) 16 8 ( )). B k k k k C k k k k k k k k a a a a D k k k k a a k k k k a a E k k k k a k k a k k a k k a k k a a a a a a ω ω ω ω ω ω ω ω = + + + = + + + + + + − + = + + − + + + − = + + − − + + + If 0 B> , BC D 0 B − > , B CD BE( ) D2 0 BC D − − > − and E>0 (3-2-13) then all the eigenvalues of the Jacobian matrix have negative real parts. So we choose k1=1, k2=1, k3=1 and k4=1, and find B=4, BC D 4.18

B − ₌ , 2 ( ) 4.22 B CD BE D BC D − − ₌ − and E=4.62.

Proposition 3. According to Routh-Hurwitz theorem and Lyapunov asymptotically stability

theorem, when ki’s satisfy (3-2-13), the controlled Quantum-CNN system (3-2-10) is locally

asymptotically stable at the equilibrium O(0, 0, 0, 0) in spite of the higher order terms.

Numerical simulations are used to investigate the controlled Quantum-CNN system (3-2-1) using fourth-order Runge-Kutta scheme with time step 0.005. After 100 second, the behaviors of the states (x1, x2, x3, x4) of the controlled Quantum-CNN system (3-2-10) approach zeros in Fig.

3-5.

Fig. 3-1. Phase portraits of the Quantum-CNN System. -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x3 x1 x4 (c) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x3 x2 x4 (d) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x2 _x 1 x4 (b) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a) x1 x2 x3

Fig. 3-2. Power Spectrum Density diagram of the Quantum-CNN System.

Fig. 3-3. Time histories of the states of the Quantum-CNN System (3-3-2). Time (sec) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x1 (a) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x2 (b) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x3 (c) 0 20 40 60 80 100 120 140 160 180 200 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Time (sec) x4 (d) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 (a) Frequency (rad/sec) Log 10 S ( x1 ) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 (b) Frequency (rad/sec) Log 10 S (x2 ) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 (c) Frequency (rad/sec) Log 10 S (x3 ) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 (d) Frequency (rad/sec) Log 10 S (x4 )

Fig. 3-4. Time histories of the states of the Quantum-CNN System (3-2-6).

Fig. 3-5. Time histories of the states of the Quantum-CNN System (10).

0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x3 (c) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x1 (a) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x3 (c) 0 20 40 60 80 100 120 140 160 180 200 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Time (sec) x4 (d) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x2 (b) 0 20 40 60 80 100 120 140 160 180 200 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x1 (a) 0 20 40 60 80 100 120 140 160 180 200 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x4 (d) Time (sec) 0 20 40 60 80 100 120 140 160 180 200 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Time (sec) x2 (b)

Fig. 3-6. Performances of states of the system with linear, sine or state cross cosine function 108 109 110 111 112 113 114 115 116 0 0.2 0.4 0.6 0.8 1x 10 -3 xicosxi sinxi xi Time (sec) x1 (b) 102 103 104 105 106 107 108 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 xicosxi sinxi xi Time (sec) x1 (a)

3-3 Chaos Control of Quantum-CNN Oscillators Chaotic System by

Variable Structure Control

Assume that the aim is to control system x Ax f x= + ( )+u t( ) tracking a given desired state vector [ ,_{1 2}, , ]T

n

y= y y ⋅⋅⋅ y , where y= Ay. Let e=x−y be the tracking error vector. The tracking error dynamics is:

( ) ( )

e x y= − = Ae f x+ −u t (3-3-1) where f(x) is nonlinear item vector.

The controller is designed as u(t)=H(t)+f(x) in which H(t)=Kw(t). The newly defined control signal w(t) is determined through the sliding mode approach,

( ) ( ) 0 ( ) ( ) ( ) 0 w t s e w t w t s e + − ⎧ > ⎪ ⎨ < ⎪⎩ (3-3-2)

s(e) is the switching surface and is considered as

( )

s e =Ce. (3-3-3) The reaching law assumed to be s= − ⋅q sgn( )s − . This design results in the following control rs

signal.

[

]

1

( ) ( ) ( ) sgn( )

w t _{= −} CK − C rI A e q_{+ + ⋅} s _(3-3-4)

It can be shown that the closed loop system will be stable for positive r and q parameters.

Finally, let us consider our dynamic system Quantum-CNN system. The equation considered is 2 1 1 1 2 1 2 1 1 3 1 ₂ 2 1 2 3 2 3 4 3 4 2 3 1 2 ₂ 4 3 2 1 sin ( ) 2 cos 1 2 1 sin ( ) 2 cos 1 x a x x x x x x a x x x a x x x x x x a x x ω ω ⎧ = − − ⎪ ⎪ = − − + ⎪ − ⎪ ⎨ = − − ⎪ ⎪ ⎪ = − − + ⎪ ₋ ⎩ (3-3-5)

Let a1＝a2＝4.9, ω1＝1.13 and ω2＝0.85. The initial values of Quantum-CNN systems are

taken as x1(0)＝0.8, x2(0)＝−0.77, x3(0)＝−0.72 and x4(0)＝0.97 respectively. The result is show

Fig. 3-7. Phase portraits of Quantum-CNN by variable structure control. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x1 x3 x4 (c) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x2 x3 x4 (d) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x1 x2 x3 (a) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x1 x2 x4 (b)

3-4 Chaos Control of Quantum-CNN Oscillators Chaotic System by

Impulse Control

A technique for suppressing chaos is to apply a periodic impulse input to the system. Consider the system of the form (3-2-1) and assume that the system is controlled by a periodic impulse input 0 ( _d) i u ρ δ τ∞ iT = =

∑

− (3-4-1) where ρ is a constant impulse intensity, Td is the periodic between two consecutive impulses, and

δ is the standard delta function. With different values of ρ and Td the controlled system can be

stabilized at different periodic orbits or fix point.

Finally, let us consider our dynamic system Quantum-CNN system with periodic impulse of linear feedback. The equation considered is

2 1 1 1 2 1 1 1 2 1 1 3 1 ₂ 2 2 2 1 2 3 2 3 4 3 3 3 4 2 3 1 2 ₂ 4 4 4 3 2 1 sin , ( ) 2 cos , 1 2 1 sin , ( ) 2 cos . 1 x a x x u x x x x x a x u x x x a x x u x x x x x a x u x x ω ω ⎧ = − − − ⎪ ⎪ = − − + − ⎪ ₋ ⎪ ⎨ = − − − ⎪ ⎪ ⎪ = − − + − ⎪ ₋ ⎩ (3-4-2)

Let a1＝a2＝2.47, ω1＝1 and ω2＝1. The initial values of Quantum-CNN systems are taken

as x1(0)＝0.8, x2(0)＝−0.77, x3(0)＝−0.72 and x4(0)＝0.57 respectively.

The result is show in Fig. 3-8.

Fig. 3-8. Time histories of states by impulse control.

0 5 10 15 20 25 30 35 40 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x1 x2 x3 x4 Time (sec) x 1 , x 2 , x 3 , x4

3-5 Chaotization of Quantum-CNN Chaotic System by Optimum

Control

Optimal control is a well-established engineering control strategy, and is useful for both linear and nonlinear system with linear or nonlinear controllers [98]. Now, we use a typical optimal control for the chaotization of chaos of Quantum-CNN system. Let a1＝a2＝2.47 and ω1＝ω2＝1.

The initial values of linear coupled Quantum-CNN systems are taken as x1(0)＝0.8, x2(0)＝−0.77, x3(0)＝−0.72 and x4(0)＝0.57 respectively. Consider the system (3-2-1) with a controller u and

define the Hamilton function:

1 2 3 4 1 2 3 4 1 2 3 4

( , , , , , ) T ( , , , , , ); T [ ]

H x x x x u p = p F x x x x u p p = p p p p (3-5-1)

where p is a Lagrange multiplier, called a co-state vector, F is the right hand side of Eq. (3-2-1). Following the variation principle of optimal control, we can obtain

2 1 2 1 1 3 1 ₂ 2 3 2 3 4 4 2 3 1 1 3 2 ₂ 4 3 ( ( ) 2 cos ) ( 2 1 sin ) ( ( ) 1 2 cos ) 0 1 x p x x a x p a x x p x x x x a x x ω ω − − + + − − + − − − + = − (3-5-2) 1 2 ₂ 2 1 2 sin 0 1 a p x x − = − (3-5-3) This yield a non-trivial solution for (p2, p3, p4) if and only if

1 2 2 1 2 sin 0 1 a x x − ₌ − (3-5-4) It gives an optimal surface singularly in the state space. This type of control assumes values on the two allowable boundaries (3-5-3) and (3-5-4) alternatively according to a switching surface. Locating system trajectories on the surface, a typical feedback control in the form

1 2 2 1 2 sgn[ sin ] 1 b a u k x x − = − − (3-5-5) can be used. By adjusting the value of kb from zero initial value to kb=1.6×10−4 in the above

1 if 0 sgn[ ] 0 if 0 1 if 0 v v v v > ⎧ ⎪ =_⎨ = ⎪− < ⎩ (3-5-6)

the chaotic motion with one positive Lyapunov exponent can be controlled to chaotic motion with two positive Lyapunov exponents as shown by the simulation result in Fig. 3-9.

Fig. 3-9. Lyapunov exponent of Quantum-CNN system.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10-4 -8 -6 -4 -2 0 2 4 6 8 10x 10 -5 kb L yapunov exponent.

Chapter 4

Generalized and Symplectic Synchronizations

4-1 Preliminary

In recent years, the synchronization of chaotic systems has been studied in various fields. For a particular chaotic system, a master, together with an identical or a different system, a slave system, our goal is to synchronize them via coupling or other methods.

Among many kinds of synchronizations, the generalized synchronization is investigated. It means there exists a functional relationship between the states of the master and those of the slave. In this Section 4-2, a special kind of generalized synchronizations

y=x+F(t) (4-1-1) is studied, where x, y are the state vectors of the master and the slave respectively, F(t) is a given vector function of time, which may take various forms, either regular or chaotic functions of time. The generalized synchronization developed may be applied to the design of secure communication. When F(t)=0, it reduces to a complete synchronization.

In this Section 4-3, a special kind of generalized synchronizations

y_i = ± +x_i F t_i( ), i=1, 2, , 2n (4-1-2) is studied, where xi, yi are the states of the master and the slave respectively, Fi(t) is a given

function of time, which may take various forms, either a regular or a chaotic functions of time. When Fi(t)=0, it reduces to a complete synchronization when the signs before xi are all positive;

it reduces to an anti-synchronization when Fi(t)=0 and the signs before xi are all negative; it

reduces to a partial complete synchronization and a partial anti-synchronization when the signs before xi are partly positive and partly negative.

Many approaches have been presented for the synchronization of chaotic systems. There are a chaotic master system and either an identical or a different slave system. Our goal is the synchronization of the chaotic master and the chaotic slave by coupling or by other methods.

exists a functional relationship between the states of the master and that of the slave. In this Section 4-4, a new synchronization

y H x y t= ( , , )+F t( ) (4-1-3) is studied, where x, y are the state vectors of the ‘‘master’’ and of the ‘‘slave’’, respectively, F(t) is a given function of time in different form, such as a regular or a chaotic function. When H(x, y,

t)=x, Eq. (4-1-3) reduces to the generalized synchronization given in Section 4-2.

In Eq. (4-1-3), the final desired state y of the ‘‘slave’’ system not only depends upon the ‘‘master’’ system state x but also depends upon the ‘‘slave’’ system state y itself. Therefore the ‘‘slave’’ system is not a traditional pure slave obeying the ‘‘master’’ system completely but plays a role to determine the final desired state of the ‘‘slave’’ system. In other words, it plays an ‘‘interwined’’ role, so we call this kind of synchronization ‘‘symplectic synchronization’’*, and call the ‘‘master’’ system partner A, the ‘‘slave’’ system partner B.

When H(x, y, t)=H(x, t)+F(t), Eq. (4-1-3) becomes

y=H(x, t)+F(t) (4-1-4)

which reduces to generalized synchronization. Therefore generalized synchronization is a special case of the symplectic synchronization. There exists great potential of the application of the symplectic synchronization. For instance, when the symplectically synchronized chaotic signal is used as a signal carrier, the secure communication is more difficult to be deciphered.

The variable structure control technique is a discontinuous control strategy that involves, first, selecting a switching surface for the desired dynamics and, secondly, designing a discontinuous control law such that the system trajectory first reaches the surface and then stays in it forever.

4-2 The Generalized Synchronization of a Quantum-CNN Chaotic

Oscillator with Different Order Systems

4-2-1 Generalized Synchronization Scheme

*The term ‘‘symplectic’’ comes from the Greek for ‘‘interwined’’. H. Weyl first introduced the term in 1939 in his book “The Classical Groups” (P. 165 in both the first edition, 1939, and second edition, 1946, Princeton University Press).