耦合混沌系統中黎阿普諾夫指數的變化

Abstract

本篇論文致力於了解兩個相同或相異的混沌系統，經由某些變數耦合所組成的混沌系統裡黎阿普諾夫指數［Lyapunov exponent］的行為。我們研究黎阿普諾夫指數如何隨著耦合參數強度增大而產生變化，當然在這樣的研究下，振幅同步化［Amplitude synchronization］與廣義同步化［Generalized synchronization］的概念自然會被考慮進來。另外，探索黎阿普諾夫指數與耦合參數之間的函數關係也是一份令人關注的工作，透過數值實驗，我們發現到許多有趣的現象，其中，我們可以利用數學分析去說明部分現象的原因。針對這份研究的目的，我們除了整理相關文獻裡出現的黎阿普諾夫指數定義外，也討論它們之間的關聯性。

In this thesis, we aim at understanding the behaviors of Lyapunov exponents for the system of two identical or two distinct chaotic oscillators coupled through some or all of their components. We investigate how the Lyapunov exponents vary as the coupling strength (parameter) increases from zero. Such a consideration naturally involves the notions of synchronization and generalized synchronization for chaotic dynamical systems. In addition, it is an interesting task to formulate a functional relation between Lyapunov exponents and coupling parameter in the coupled system. Several interesting phenomena are derived through extensive numerical experiments. Some of them can be justified by analytical arguments. For the purpose of this study, we also summarize various definitions of Lyapunov exponent in the literatures and discuss connections among them.