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

Abstract

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。

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 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.