耦合渾沌系統同步、適應同步及廣義同步之部分穩定性理論方法研究

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(1)國立交通大學機械工程學系. 博士論文. 耦合渾沌系統同步、適應同步及廣義同步之部分穩定性理論方法研究. Synchronization, Adaptive Synchronization and Generalized Synchronization of Coupled Chaotic Systems via Partial Stability Theory. 研究生：陳炎生指導教授：戈正銘教授中華民國九十四年六月.

(2) 耦合渾沌系統同步、適應同步及廣義同步之部分穩定性理論方法研究. Synchronization, Adaptive Synchronization and Generalized Synchronization of Coupled Chaotic Systems via Partial Stability Theory. 研究生：陳炎生. Student：Yen-Sheng Chen. 指導教授：戈正銘. 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 2005 Hsinchu, Taiwan, Republic of China. 中華民國九十四年六月.

(3) 耦合渾沌系統同步、適應同步及廣義同步之部分穩定性理論方法研究學生：陳炎生. 指導老師：戈正銘教授. 國立交通大學機械工程學系. 摘要. 渾沌同步可以由許多方法達成。但是，一般來說，並沒有通用的簡單判據。本論文提出一個一般性的解決方案，透過部分變量穩定性理論來達到渾沌同步，並且可以適用於單向及雙向耦合系統。依照提出方案的程序，首先討論單向耦合系統，共推導出三個判據。一個判據適合系統沒有攝動情況，其他兩個則分別適用於系統存在歸零及非歸零攝動情形。類似於單向耦合系統，針對雙向耦合系統也有三個保證同步發生的定理被推導出來。一個判據適合系統沒有攝動情況，其他兩個則分別適用於系統存在歸零及非歸零攝動情形。在上述的六個判據中，為了確保同步的出現，必須滿足一個矩陣方程式並且事先估算 Lipschitz 常數。特別地，估算 Lipschitz 常數經常太過於保守。為了克服這兩個缺點，矩陣方程式及估算 Lipschitz 常數分別由一個適應耦合增益值及適應估測器所取代。則對於單向及雙向耦合系統，由此法可實現一簡單又方便的適應同步。在先前的結果中，同步指的是全等同步(或是完全同步)。接著本論文探討另一種所謂的廣義同步，其意指在無窮長的迭代時間後，驅迫與被驅迫系統的狀態之間存在一個函數關係。類似地，本文提出一個藉由部分變量穩定性理論來達到廣義同步的方案。依此方案的程序，針對單向耦合系統，證出一個透過線性回授來達到廣義同步的定理。所有在本論文中被推導出的判據都適用於規律及渾沌系統、線性及非線性系統、自治及非自治系統。最後，許多系統被數值模擬用於展示理論分析結果。. i.

(4) Synchronization, Adaptive Synchronization and Generalized Synchronization of Coupled Chaotic Systems via Partial Stability Theory Student：Yen-Sheng Chen. Advisor：Zheng-Ming Ge. Department of Mechanical Engineering National Chiao Tung University. Abstract. Chaos synchronization can be achieved by several methods but there is no easy unified criterion in general. Herein, a general scheme for both unidirectional and mutual coupled systems is proposed to achieve chaos synchronization via stability with respect to partial variables. Follow the procedure of the proposed scheme, the unidirectional coupled systems are discussed first and three sufficient criteria are derived. One of them is suitable for systems without perturbation and the other two are suitable for systems under two kinds of perturbations, vanishing and nonvanishing, respectively. Similar to the unidirectional case, three theorems are proven to ensure occurrence of synchronization for mutual coupled systems. One of them is suitable for systems without perturbation and the other two are suitable for systems under two kinds of perturbations, vanishing and nonvanishing, respectively. In previous six criteria, to guarantee the emergence of synchronization a matrix equation should be satisfied and the estimation of Lipschitz constant is needed. Specifically, the estimate of Lipschitz constant is often conservative. To overcome these two shortcomings, this matrix equation and the estimation of Lipschitz constant are replaced by adopting an adaptive coupling gain and an adaptive estimator, respectively. As a result, a simple and convenient adaptive synchronization is realized ii.

(5) for both unidirectional and mutual coupled systems. In the foregoing results, the synchronization discussed indicates the identical synchronization (or complete synchronization). Another kind of synchronization called generalized synchronization which means that there is a functional relation between the states of driving and response systems as time goes to infinity are studied in the chapter 5. Similar, a scheme to achieve chaos generalized synchronization via partial stability is proposed. Follow the procedure of this scheme, one theorem is proven to ensure generalized synchronization for a general kind of unidirectional coupled systems by linear feedback. All the criteria derived in this dissertation work for regular and chaotic systems, linear and nonlinear systems, autonomous and nonautonomous systems. Finally, several systems are simulated numerically to illustrate the theoretical analyses.. iii.

(6) 誌謝從 88 年進入交大機械系就讀碩士乃至 94 年取得博士學位，六個寒暑秋冬不啻是一段漫長的歲月，隨著時空的流轉，我也在此成長。首先我要感謝指導教授戈正銘教授的悉心指導，由於老師因材施教，使我得以發揮潛能。透過老師嚴謹的治學精神以及不斷追求創新的理念，使我明白做學問無法朝夕竟功，而在於長時間功夫的累積。在此，我要對於老師致上最高的敬意。我也要感謝幾位口試委員，陳文華教授、張家歐教授、周傳心教授、董必正教授、成維華教授及邵錦昌教授。諸位老師給我許多精闢的建議，使得我的博士論文能夠更加完善，再次致上萬分謝意。在交大的這幾年，我要感謝研究室所有的學長、同學以及學弟。特別感謝張時明、張萬坤、徐華均、鐘國展、江博容、林俊吉、鮑東昇及王宣智幾位同學對我的幫忙，尤其是時明對我的照顧最多。還有許許多多朋友的支持，無法一一列名，在此一併致謝。另外，我也要感謝我的母親、父親、兄嫂以及所有的家人，在這段期間他們是我背後的支柱，沒有他們的支持與鼓勵，我無法順利完成博士學位。最後，我要感恩師父及十方諸佛菩薩，總是在我最困頓的時候，適時地伸出手來幫助我、引導我、給我希望、給我光明，從此不再行走於黑暗中。當然，還有許多可愛的師兄姐們，就像一家人般的互相扶持。言至於此，內心的感動久久無法平復。僅以此文獻給所有曾經護持我的一切人事物。. iv.

(7) CONTENTS 摘要 ···············································································································i ABSTRACT································································································ii 誌謝 ·············································································································iv CONTENTS································································································v LIST OF FIGURES ·················································································vii Chapter 1 Introduction···········································································1 Chapter 2 A General Scheme and Synchronization of Unidirectional Coupled Systems ···································································4 2.1. A General Scheme ······················································································4. 2.2. Unidirectional Coupled Systems without Perturbation ·························5. 2.3. Unidirectional Coupled Systems with Two Kinds of Perturbations······7. 2.4. Numerical Illustrated Examples·······························································8 2.4.1. Unidirectional Coupled Systems without Perturbation ···········10. 2.4.2. Unidirectional Coupled Systems with Perturbation Δf < K e ·······································································································11. 2.4.3. Unidirectional Coupled Systems with Perturbation Small on the Average ···················································································11. Chapter 3 Synchronization of Mutual Coupled Systems ·················33 3.1. Mutual Coupled Systems without Perturbation ···································33. 3.2. Mutual Coupled Systems with Two Kinds of Perturbations················34. 3.3. Numerical Illustrated Examples·····························································36 3.3.1. Mutual Coupled Systems without Perturbation ·······················37. 3.3.2. Mutual Coupled Systems with Perturbation Δf < K e ·······38. 3.3.3. Mutual Coupled Systems with Perturbation Small on the Average··························································································39. Chapter 4 Adaptive Synchronization of Unidirectional and Mutual Coupled Systems ·································································61 4.1. Introduction······························································································61 v.

(8) 4.2. Adaptive Synchronization of Unidirectional Coupled Systems···········61. 4.3. Adaptive Synchronization of Mutual Coupled Systems·······················62. 4.4. Numerical Illustrated Examples·····························································64. Chapter 5 Generalized Synchronization of Coupled Systems ·········78 5.1. Introduction······························································································78. 5.2. Theoretical Analysis·················································································78. 5.3. Numerical Illustrated Examples·····························································80. Chapter 6 Conclusions ··········································································94 APPENDIX ·······························································································96 REFERENCES·······················································································100 PAPER LIST···························································································106. vi.

(9) List of Figures Fig. 2.1. Chaotic attractor of the Rössler system···················································· 13. Fig. 2.2. Chaotic attractor of the Duffing-van der Pol System ······························· 14. Fig. 2.3. State errors versus time of unidirectional coupled Rössler systems without perturbation ····························································································· 15. Fig. 2.4. Projections of synchronized manifold for unidirectional Rössler systems without perturbation ················································································ 16. Fig. 2.5. Lyapunov spectra of unidirectional coupled Rössler systems without perturbation ····························································································· 17. Fig. 2.6. State errors versus time of unidirectional coupled Rössler systems without perturbation while γ = 0.09 ···································································· 18. Fig. 2.7. State errors versus time of unidirectional coupled Duffing-van der Pol system without perturbation ···································································· 19. Fig. 2.8. Projections. of. synchronized. manifold. of. unidirectional. coupled. Duffing-van der Pol system without perturbation ···································· 20 Fig. 2.9. State errors versus time of unidirectional coupled Rössler systems with perturbation Δf1 = z2 − z1 and Δf 2 = sin t ⋅ ( x1 − x2 ) ······························· 21. Fig. 2.10 Projections of synchronized manifold of unidirectional coupled Rössler systems with perturbation Δf1 = z2 − z1 and Δf 2 = sin t ⋅ ( x1 − x2 ) ·········· 22 Fig. 2.11 State errors versus time of unidirectional coupled Duffing-van der Pol system with perturbation Δf1 = cos t sin( y2 − y1 ) ····································· 23 Fig. 2.12 Projections. of. synchronized. manifold. of. unidirectional. coupled. Duffing-van der Pol system with perturbation Δf1 = cos t sin( y2 − y1 ) ···· 24 Fig. 2.13 State errors versus time of unidirectional coupled Rössler systems with perturbation Δf1 = randn(t ) and Δf 2 = 5cos 30t ··································· 25. Fig. 2.14 Projections of synchronized manifold of unidirectional coupled Rössler systems with perturbation Δf1 = randn(t ) and Δf 2 = 5cos 30t ············· 26 Fig. 2.15 State errors versus time of unidirectional coupled Rössler systems with Δf1 = randn(t ) , Δf 2 = 5cos 30t and γ = 80 ·········································· 27. Fig. 2.16 Projections of synchronized manifold of unidirectional coupled Rössler vii.

(10) systems with Δf1 = randn(t ) , Δf 2 = 5cos 30t and γ = 80 ··················· 28 Fig. 2.17 State errors versus time of unidirectional coupled Duffing-van der Pol system with perturbations Δf1 = cos 20π t and Δf 2 = sin 30π t ·············· 29 Fig. 2.18 Projections. of. synchronized. manifold. of. unidirectional. coupled. Duffing-van der Pol system with perturbations Δf1 = cos 20π t and Δf 2 = sin 30π t ························································································· 30. Fig. 2.19 State errors versus time of unidirectional coupled Duffing-van der Pol system with Fig. 2.20 Projections. Δf1 = cos 20π t , Δf 2 = sin 30π t and γ = 100 ·················· 31. of. synchronized. manifold. of. unidirectional. coupled. Duffing-van der Pol system with Δf1 = cos 20π t , Δf 2 = sin 30π t and. γ = 100 ···································································································· 32 Fig. 3.1. Chaotic attractor of the Lorenz system ···················································· 41. Fig. 3.2. Chaotic attractor of the Ueda system. ······················································ 42. Fig. 3.3. State errors versus time of mutual coupled Lorenz system without perturbation ····························································································· 43. Fig. 3.4. Projections of synchronized manifold for mutual coupled Lorenz system without perturbation ················································································ 44. Fig. 3.5. Lyapunov spectra of mutual coupled Lorenz systems without system perturbation ····························································································· 45. Fig. 3.6. State errors versus time of mutual coupled Lorenz systems without perturbation while γ = 0.6 ······································································ 46. Fig. 3.7. State errors versus time of mutual coupled Ueda system without perturbation ····························································································· 47. Fig. 3.8. Projections of synchronized manifold of mutual coupled Ueda system without perturbation ················································································ 48. Fig. 3.9. State errors versus time of mutual coupled Lorenz systems with perturbations Δf1 = cos t ⋅ ( y1 − y2 ) and Δf 6 = x1 − x2 ···························· 49. Fig. 3.10 Projections of synchronized manifold of mutual coupled Lorenz systems with perturbations Δf1 = cos t ⋅ ( y1 − y2 ) and Δf 6 = x1 − x2 ···················· 50 Fig. 3.11 State errors versus time of mutual coupled Ueda systems with perturbations Δf 2 = cos t sin( x2 − x1 ) and Δf 3 = y2 − y1 ························· 51. viii.

(11) Fig. 3.12 Projections of synchronized manifold of mutual coupled Ueda systems with perturbations Δf 2 = cos t sin( x2 − x1 ) and Δf 3 = y2 − y1 ················· 52 Fig. 3.13 State errors versus time of mutual coupled Lorenz systems with perturbations Δf 2 = 2sin(20π t ) , Δf 4 = randn(t ) and Δf 6 = 5cos(30π t ) ················································································································ 53 Fig. 3.14 Projections of synchronized manifold of mutual coupled Lorenz systems with. perturbations. Δf 2 = 2sin(20π t ). ,. Δf 4 = randn(t ). and. Δf 6 = 5cos(30π t ) ···················································································· 54. Fig. 3.15 State errors versus time of mutual coupled Lorenz systems with Δf 2 = 2sin(20π t ) , Δf 4 = randn(t ) , Δf 6 = 5cos(30π t ) and γ = 130 ···· 55. Fig. 3.16 Projections of synchronized manifold of mutual coupled Lorenz systems with Δf 2 = 2sin(20π t ) , Δf 4 = randn(t ) , Δf 6 = 5cos(30π t ) and γ = 130 ················································································································ 56 Fig. 3.17 State errors versus time of mutual coupled Ueda systems with perturbations Δf 2 = cos(25π t ) and f3 = 5sin(15π t ) ····························· 57 Fig. 3.18 Projections of synchronized manifold of mutual coupled Ueda systems with perturbations Δf 2 = cos(25π t ) and f3 = 5sin(15π t ) ····················· 58 Fig. 3.19 State errors versus time of mutual coupled Ueda systems with Δf 2 = cos(25π t ) , f3 = 5sin(15π t ) and γ = 100 ···································· 59. Fig. 3.20 Projections of synchronized manifold of mutual coupled Ueda systems with Δf 2 = cos(25π t ) , f3 = 5sin(15π t ) and γ = 100 ··························· 60 Fig. 4.1. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 0.1 of unidirectional coupled Lorenz systems·································· 66 Fig. 4.2. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 25 and. ε = 0.1 of unidirectional coupled Lorenz systems·································· 67 Fig. 4.3. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 20 of unidirectional coupled Lorenz systems ·································· 68 Fig. 4.4. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 0.1 of unidirectional coupled Duffing systems································· 69 Fig. 4.5. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 5 and ix.

(12) ε = 0.1 of unidirectional coupled Duffing systems································· 70 Fig. 4.6. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 8 of unidirectional coupled Duffing systems···································· 71 Fig. 4.7. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 0.1 of mutual coupled Lorenz systems············································· 72 Fig. 4.8. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 20 and. ε = 0.1 of mutual coupled Lorenz systems············································· 73 Fig. 4.9. State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 18 of mutual coupled Lorenz systems·············································· 74 Fig. 4.10 State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 0.1 of mutual coupled Duffing systems ··········································· 75 Fig. 4.11 State errors and Estimated Lipschitz constant versus time for Lˆ0 = 5 and. ε = 0.1 of mutual coupled Duffing systems ··········································· 76 Fig. 4.12 State errors and Estimated Lipschitz constant versus time for Lˆ0 = 1 and. ε = 3 of mutual coupled Duffing systems ·············································· 77 Fig. 5.1. e1 , e2 and e3 versus time······································································· 83. Fig. 5.2. Projections of synchronized manifold······················································ 84. Fig. 5.3. Phase portrait of the driving system························································· 85. Fig. 5.4. Phase portrait of the response system ······················································ 86. Fig. 5.5. e1 , e2 and e3 versus time······································································· 87. Fig. 5.6. Projections of synchronized manifold······················································ 88. Fig. 5.7. Phase portrait of the response system ······················································ 89. Fig. 5.8. e1 and e2 versus time ··········································································· 90. Fig. 5.9. Projections of synchronized manifold······················································ 91. Fig. 5.10 Phase portrait of the driving system························································· 92 Fig. 5.11 Phase portrait of the response system ······················································ 93. x.

(13) Chapter 1 Introduction 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 [4-16]. In particular, it was pointed out that chaos synchronization has the potential in secure communication. Many engineers and scientists were attracted to this discipline [17-36]. Synchronization means that the states of response system approach eventually to the ones of driving system. Two kinds of chaos synchronization are discussed the most often. (1) Duplication (or master-slave): the first kind introduced by Pecora and Carroll [3] consists of a driving system and a response system. The former one evolves chaotic orbits and the latter is identical to the driving system except some partial states replaced by that of the driving 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 response system will behave the same orbit with the driving 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 driving and response systems as time evolves. This function need 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]. Zero crossing of Lyapunov exponent which is used widely as a criterion of chaos synchronization is derived from the variational equation. There is a drawback that we can only calculate finite evolution time in computer simulation but infinite evolution 1.

(14) time is needed by definition of Lyapunov exponent. The variational equation itself is also used to ensure the occurrence of synchronism. But its stability is in the sense of Lyapunov first method. Especially, the domain of attraction is infinitesimal, as a result that the stability of synchronization guaranteed by the variational equation is not robust. On other hand, it is difficult to use traditional Lyapunov direct method since the state error equation is not a pure function of state error in general. In this dissertation, we propose a general scheme to achieve chaos synchronization via partial stability due to Rumjantsev [78]. The previous obstacles will be overcome by our method and it serves as a criterion for chaos synchronization by control methods. Follows the procedure of proposed scheme, the unidirectional coupled systems are discussed and three sufficient criteria are derived in chapter 2. One of them is suitable for systems without perturbation and the other two are suitable for systems under two kinds of perturbations, vanishing and nonvanishing, respectively. In chapter 3, the effort is concentrated on synchronization of mutual coupled systems. Similar to the unidirectional case, three theorems are proven to ensure the occurrence of synchronization. One of them is suitable for systems without perturbation and the other two are suitable for systems under two kinds of perturbations, vanishing and nonvanishing, respectively. In previous six criteria, to guarantee the emergence of synchronization a matrix equation should be satisfied and the estimation of Lipschitz constant is needed. Moreover, the estimate of Lipschitz constant is often conservative. To overcome these two shortcomings, this matrix equation and the estimation of Lipschitz constant are replaced by adopting an adaptive coupling gain and an adaptive estimator, respectively. As a result, a simple and convenient adaptive synchronization of chaotic systems is realized for both unidirectional and mutual coupled systems in chapter4. It is easier and more convenient to use this method for synchronization of both unidirectional and mutual coupled systems than the six theorems in chapter 2 and 3. Furthermore, to increase the convergent rate of state error dynamics we only need to set a larger initial condition of the adaptive equation. The synchronization discussed indicates the identical synchronization (or complete synchronization) in the foregoing results. Another kind of synchronization called generalized synchronization which means that there is a functional relation between the states of driving and response systems as time goes to infinity are studied in the chapter 5. This function can increase the complication of synchronization. 2.

(15) Similar to the chapter 2, a scheme to achieve generalized synchronization of chaos via partial stability is proposed. One theorem is proven to guarantee the occurrence generalized synchronization for a general kind of unidirectional coupled nonautonomous systems by linear feedback. Furthermore, the function between the states of the two coupled systems can be arbitrary assigned. Superficially, the order of the error dynamic equation is enlarged since it is replaced by an extended equation in this scheme. But only partial variables are manipulated in actual. Furthermore, many control techniques can be applied to synchronize coupled systems in this scheme. All the criteria derived in this dissertation work for regular and chaotic, linear and nonlinear systems, autonomous and nonautonomous systems. Finally, several examples are simulated numerically to illustrate the theoretical analyses.. 3.

(16) Chapter 2 A General Scheme and Synchronization of Unidirectional Coupled Systems. 2.1 A General Scheme Consider the following coupled nonautomonous systems x 1 = f (t , x1 , x 2 ), x 2 = g (t , x1 , x 2 ),. (2.1). where x1 , x 2 ∈ R n are the states variables and Ω is a domain containing the origin.. Assume that the solution of Eq. (2.1) exist for infinite time. That is, for given (t 0 , x10 , x 20 ) ∈ Ω the solution [ϕ1T (t ; t0 , x0 , xˆ 0 ) ϕ 2T (t ; t0 , x0 , xˆ 0 )]T. of Eq. (2.1). exists for t ≥ t 0 . At the first, we recall the definition of identical synchronization (or complete synchronization). Definition The system (2.1) is (identical) synchronized if there is an invariant manifold S ⊂ R × R 2n for the solution [ϕ1T (t ; t0 , x0 , xˆ 0 ) ϕ 2T (t ; t0 , x0 , xˆ 0 )]T of Eq.. (2.1) s.t. lim ϕ1 (t ; t0 , x10 , x 20 ) − ϕ 2 (t ; t0 , x10 , x 20 ) = 0 with (t 0 , x10 , x 20 ) ∈ Ω . t →∞. For convenience, rewrite Eq. (2.1) in a form which contains a coupling term to enhance synchronization x 1 = f (t , x1 ) + G1 (t , x1 , x 2 ), x 2 = f (t , x 2 ) + G 2 (t , x1 , x 2 ), where. f : Ω ⊂ R × R2n → Rn. f (t , x1 ) − f (t , x 2 ) ≤ L x1 − x 2. satisfy. (2.2) the. Lipschitz. condition. in x for all (t , x1 ) and (t , x 2 ) in Ω with. Lipschitz constants L and G 1 , G 2 are the coupling functions. Assume that. G1 (t , x1 , x 2 ) = 0. and G 2 (t , x1 , x 2 ) = 0. for. x1 (t ) = x 2 (t ), ∀t ≥ t0 . That is the. synchronized sub-manifold of Eqs. (2.2) agrees with the original uncoupled one while synchronization occurs.. In order to discuss the transversal stability of. synchronization manifold, define e = x 2 − x1 to be the state error. Error equations can be written as 4.

(17) e = f (t , x 2 ) − f (t , x1 ) + [G 2 (t , x1 , x 2 ) − G 1 (t , x1 , x 2 )] .. (2.3). Notice that the right hand side of Eqs. (2.3) is not a pure function of e , as a result that the traditional Lyapunov direct method might hardly be used. The variational equation and zero crossing of Lyapunov exponent are used to clarify transversal stability locally. Moreover, Josić [81] analyzed that synchronization manifolds will persist under perturbation if such manifolds posses a property of k-hyperbolicity. In our method, the upper half (lower half also works) of Eq. (2.2) is added into Eq. (2.3) with x 2 replaced by x 2 = e + x 1 , then an extended equation is obtained as follows x 1 = f (t , x1 ) + G1 (t , x1 , e + x1 ),. e = f (t , e + x1 ) − f (t , x1 ) + [G 2 (t , x1 , e + x1 ) − G1 (t , x1 , e + x1 ) ].. (2.4). If the partial variable e in Eq. (2.4) are asymptotically stable about e = 0 , the synchronization manifold is stable in transversal directions. This means that the system in the form of Eq. (2.2) is synchronized. The determination of whether e is asymptotically stable can be done via stability with respect to partial variables. The theory of partial stability can be found in appendix or in [78-80]. Note that the same procedure can be developed for Eq. (2.1). But this form of system might too general to be used. The scheme proposed in this section not only satisfies the case of mutual coupled nonlinear systems but also satisfies the unidirectional case. Actually, it works for the case of unidirectional coupled nonlinear systems if G1 = 0 . The rest mission is to choose appropriate controllers G1 and G 2 to guarantee the occurrence of synchronization. In the remainder of this chapter, we will adopt this scheme to develop some criteria of synchronization for unidirectional coupled systems and give some simulated illustrations.. 2.2 Unidirectional Coupled Systems without Perturbation In this section, a theorem will be given for unidirectional coupled nonautonomous system which is a special case of Eq. (2.2). This theorem is suitable for the case without perturbation and will be applied to two examples, the Rössler system and the Duffing-van der Pol system. Choose G1 = 0 and G 2 = Γ(x1 − x 2 ) , then the Eq. (2.2) becomes. 5.

(18) x 1 = f (t , x1 ), x 2 = f (t , x 2 ) + Γ(x1 − x 2 ), where f satisfies Lipschitz condition with. (2.5) f (t , x1 ) − f (t , x 2 ) ≤ L x1 − x 2. in x. for all (t , x1 ) and (t , x 2 ) in domain Ω with Lipschitz constant L and Γ ∈ M n×n is a constant matrix whose entries represent the coupling strength of the linear feedback term (x1 − x 2 ) . The index of entry γ ij means that the j-th component of (x1 − x 2 ) exerts on the i-th component of x 2 . Follow the procedure stated in section 2.1. Eq. (2.5) can be rephrased in the form of an extended equation as. x 1 = f (t , x1 ), e = f (t , e + x1 ) − f (t , x1 ) − Γe .. (2.6). where e = x 2 − x1 .. Theorem 2.1 The partial state e is uniformly asymptotically to 0 in Eq. (2.6) if LI n − Γ is negative definite, i.e. the system in the form of Eq. (2.5) is synchronized if LI n − Γ is negative definite.. 1 Proof Choose a function V (x1 , e) = eT e which is positive definite function with 2 respect to e and with infinitesimal upper bound. Then its time derivative along the solution of Eq. (2.6) is V = eT e = eT [f (t , e + x1 ) − f (t , x1 ) − Γe] ≤ e ⋅ L e − eT Γe 2. ≤ L e − eT Γe = eT ( LI n − Γ)e . The state error e uniformly asymptotically approaches 0 if LI n − Γ is negative definite by Theorem A2 in appendix. The Cauchy-Schwarz inequality and the Lipschitz condition were used in the derivation. Remark 2.1 From the matrix theory, we know that LI n − Γ is negative definite if and only if all its eigenvalues are negative. For the case Γ = diag (γ 1 , γ 2 ," , γ n ) with. γi > 0. for. i = 1," , n ,. synchronization. occurs. if. γ min > L ,. where. γ min ≤ γ i , i = 1," , n . This is because the time derivative of V (x1 , e) can be written as 6.

(19) 2 V (x1 , e) ≤ ( L − γ min ) e . Moreover, the result is global by Theorem A4 if f is. globally Lipschitzian.. 2.3 Unidirectional Coupled Systems with Two Kinds of Perturbations The criterion given in section 2.2 is suitable for the case without system perturbation. If the system possesses a vanishing perturbation, similar result can be obtained. Consider unidirectional coupled nonautonomous systems with perturbation in the form of. x 1 = f (x1 ), x 2 = f (x 2 ) + Δf (t , x1 , x 2 ) + Γ(x1 − x 2 ), where f satisfies Lipschitz condition with. f (t , x1 ) − f (t , x 2 ) ≤ L x1 − x 2. (2.7) in x. for all (t , x1 ) and (t , x 2 ) in domain Ω with Lipschitz constant L and Γ ∈ M n×n is a constant matrix whose entries represent the coupling strength of the linear feedback term (x1 − x 2 ) . The term Δf (t , x1 , x 2 ) is a vanishing perturbation which means that Δf (t , x1 , x 2 ) = 0 with x1 (t ) = x 2 (t ), ∀ t . Δf (t , x1 , x 2 ) can be rephrased to be Δf (t , x1 , e) while e = x 2 − x1 . Then, an extended equation can be obtained as. x 1 = f (x1 ), e = f (e + x1 ) − f (x1 ) + Δf (t , x1 , e) − Γe .. (2.8). Theorem 2.2 Assume that ∃ K > 0 ⇒ Δf < K e . Then the Eq. (2.8) is uniformly asymptotically e-stable if ( L + K )I n − Γ is negative definite, i.e. the system in the form of Eq. (2.7) is synchronized if ( L + K )I n − Γ is negative definite. 1 Proof Choose a function V (x1 , e) = eT e which is positive definite function with 2 respect to e and with infinitesimal upper bound. Then its time derivative along the solution of Eq. (2.6) is V = eT e ≤ ( L + K ) e 2 − eT Γe ≤ eT [ ( L + K )I n − Γ ] e . Hence, the Eq. (2.8) is uniformly asymptotically e-stable if ( L + K )I n − Γ is negative definite. 7.

(20) Remark 2.2 ( L + K )I n − Γ is negative definite if and only if all its eigenvalues are negative. When Γ = diag (γ 1 , γ 2 ," , γ n ) with γ i > 0 for i = 1," , n , synchronization occurs if γ min > L + K , where γ min is the minimum one in γ i . Furthermore, this result is global by Theorem A4 if f and Δf (t , x1 , x 2 ) are globally Lipschitzian. If Δf (t , x1 , x 2 ) is not a vanishing perturbation, the origin 0 is no longer a trivial solution. It is difficult to design a controller to guarantee the occurrence of asymptotically partial stability like Theorem 2.2. What we called the stable under constantly acting perturbation small on the average will take it over. Theorem 2.3 Assume that the functions f. and Df (x) are continuous and. bounded in Q . The the Eq. (2.7) is uniformly e-stable under constantly acting perturbation small on the average if LI n − Γ is negative definite.. Proof From theorem 2.1, the partial state e is uniformly asymptotically to 0 in Eq. (2.6) if LI n − Γ is negative definite. By corollary A1, the Eq. (2.7) is uniformly. e-stable under constantly acting perturbation small on the average if LI n − Γ is negative definite with the assumption that f and Df (x) are continuous and bounded in Q . This completes the proof.. Remark 2.3 Theorem 2.3 means that the coupled structure perturbed systems (2.7) are practical synchronized [82]. If. Γ = diag (γ 1 , γ 2 ," , γ n ). with. γi > 0. for. i = 1," , n , practical synchronization occurs if γ min > L , where γ min ≤ γ i , i = 1," , n . Moreover, the larger γ min is, the smaller bounds of the state errors are. This result is global if f is globally Lipschitzian.. 2.4 Numerical Illustrated Examples In this section, the Rössler system and the Duffing-van der Pol system are adopted to demonstrate the results given in section 2.2 and 2.3. They are simulated for the cases with and without system perturbation, respectively. The system equation of the Rössler system is as following. 8.

(21) x = − y − z f1 (x), y = x + ay f 2 (x), z = b + z ( x − c) f 3 (x), where a = b = 0.2 and c=5.7 ensure that there exists chaotic behavior. The chaotic attractor is shown in Fig. 2.1. To apply the theorem given in this chapter, one needs to estimate the Lipschitz constant at the beginning. By Cauchy-Schwarz inequality, it can be derived for any x 2 = [ x2. z2 ]T , x1 = [ x1. y2. z1 ]T , we have. y1. f1 (x 2 ) − f1 (x1 ) ≤ [0 − 1 − 1] x 2 − x1 , f 2 (x 2 ) − f 2 (x1 ) ≤ [1 a 0] x 2 − x1 , f3 (x 2 ) − f 3 (x1 ) = z2 x2 − z1 x1 − ce3 = z2 x2 − z2 x1 + z2 x1 − z1 x1 − ce3 = z2 e1 + x1e3 − ce3 ≤. [ B3. 0 B1 − c ] x 2 − x1 ,. where xi (t ) ≤ B1 , yi (t ) ≤ B2 , zi (t ) ≤ B3 , ∀t > t0 , i = 1, 2 . Hence, a Lipschitz constant can be obtained as. L=. [0 − 1 − 1]. 2. + [1 a 0]. 2. [ B3. +. 0 B1 − c ]. 2. .. From numerical simulation, B1 = 12, B2 = 8, B3 = 23 , then L = 23.55 . The governing equation of the Duffing-van der Pol system is x = y f1 (x), y = μ (1 − γ x 2 ) y − x 3 + A sin Ωt f 2 (x).. The chaotic behavior exists while μ = 0.2, γ = 8, A = 5 and Ω = 1.02 . The chaotic attractor is shown in Fig. 2.2. Apply the Cauchy-Schwarz inequality to estimate the Lipschitz constant. For any x 2 = [ x2. y2 ]T , x1 = [ x1. y1 ]T , it can be derived. f1 (x 2 ) − f1 (x1 ) ≤ x 2 − x1 , f 2 (x 2 ) − f 2 (x1 ) ≤ μ e1 − γ x22 y2 + γ x12 y1 − x23 + x13 ≤ ⎡⎣ 2γ B1 B2 + 3B12. μ + γ B12 ⎤⎦ x 2 − x1 ,. where xi (t ) ≤ B1 , yi (t ) ≤ B2 , ∀t > t0 , i = 1, 2 . Hence, one Lipschitz constant can be obtained as L = 1 + ⎡⎣ 2γ B1 B2 + 3B12. 9. μ + γ B12 ⎤⎦. 2. ..