六種新渾沌系統及三種新型的渾沌同步之研究

Abstract

渾沌系統之研究在物理、化學、生物學、生理學、各種工程等方面皆有日益重要之廣泛應用。非線性Mathieu系統、Duffing系統、van der Pol系統都是典型的重要渾沌系統。Ikeda系統及Mackey-Glass系統是典型的時滯光電及生理系統。本計畫採取適當的耦合方式將它們推廣為兩種雙Mathieu系統、雙Duffing系統、雙van der Pol系統、雙Ikeda系統及雙Mackey-Glass系統，研究其渾沌性質，從而就典型重要渾沌系統而言，既擴大其研究範圍也深化其研究內容。渾沌同步之研究在秘密通訊、神經網路、自組織、物理系統、生態系統、工程系統等方面有長足之應用。本計畫提出三種新型的渾沌同步，具有重要理論及實際意義：一、純誤差穩定的廣義同步法。用以改進目前需先由數值計算預先得出誤差方程中渾沌變量之最大值之有缺陷之方法。二、實用適應廣義同步法。用以糾正目前適應同步法中估值參數趨於未知參數未加證明之盲點。三、不同起始條件下的全同系統渾沌同步。根據渾沌理論之傳統說法，諸渾沌運動對初值極為敏感，隨時間按指數次數迅速分離。吾人發現對兩個全同而無任何連繫的雙Ikeda系統而言，不同的初值可導致二全同系統的延後同步等，對兩個全同而無任何連繫的雙Mackey-Glass系統而言，不同初值可導致各種暫時的延後同步。這些現象與傳統說法不同，極值得深入研究。研究重點為： 1. 兩種雙Mathieu系統之渾沌研究。用相圖、分歧圖、功率譜圖、李雅普諾夫指數、碎形維度等分析渾沌之行為。 2. 採用純誤差穩定理論及精緻之李雅普諾夫函數得出純誤差穩定的廣義同步法。並以對兩種雙Mathieu系統為例實現此種廣義同步。 3. 雙Duffing系統與雙van der Pol系統之渾沌研究。用相圖、分歧圖、功率譜圖、李雅普諾夫指數分析渾沌之行為，包括奇異吸引子之範圍及形狀、超渾沌之行為、碎形之維度等。 4. 應用實用穩定理論實現以實用適應渾沌同步，嚴格證明估值參數必然趨於未知參數，並由數值計算對雙Duffing系統與雙van der Pol系統加以驗證。 5. 雙Ikeda系統及雙Mackey-Glass系統的渾沌研究。 6. 找出各種不同初值之規律性，在此種初值下雙Ikeda系統呈現延後同步、超前同步、反延後同步及反超前同步。找出各種初值下之規律性，在此種初值下雙Mackey-Glass系統呈現暫時之延後同步、超前同步、反延後同步及反超前同步。並試圖解釋其原因。

The study of chaotic system has found wide applications in physics, chemistry, biology, physiology, and various engineerings. Nonlinear Mathieu system, Duffing system, and van der Pol system all are paradigmatic important chaotic systems. Ikeda system and Mackey-Glass system are paradigmatic important electro-optical and physiological time delay systems. In this project, the study is extended to two kinds of double Mathieu system, double Duffing system, double van der Pol system, double Ikeda system, and double Mackey-Glass system by suitable coupling. For these paradigmatic and important systems, the study will be extended and deepened. Chaos synchronizations are applied in various regions, such as secure communication, neural networks, self-organization, physical systems, ecological systems and engineering systems, etc. In this project, three new types of chaos synchronization with theoretical and practical importance are studied: 1. pure error stability synchronization, to improve the present defective method in which the maximum values of state variables appeared in error dynamics must be preliminarily calculated by simulations; 2. pragmatical adaptive generalized synchronization, to correct the absence of proof of that estimated parameters approach the unknown parameters; 3. different initial condition synchronization. By the traditional theory of chaos, the chaotic motions are very sensitive to initial conditions and separate each other exponentially. However, we discover that for two identical double Ikeda systems, lag synchronization, etc can be found for different initial condition, and for two identical double Mackey-Glass systems, various temporary lag synchronizations can be found for different initial conditions. These phenomina are contradictory to traditional theory. These should be studied seriously. The main parts of our study are: 1. The study of chaos of two kinds of double Mathieu system. By phase portraits, bifurcation diagrams, power spectra, Lyapunov exponents, fractal dimensions, the various chaotic behaviors of these systems will be studied. 2. By pure error stability theory and elaborate Lyapunov functions, the pure error generalized synchronization method is given, proved and illustrated by two kinds of double Mathieu systems. 3. The study of chaos of double Duffing system and double van der Pol system. By phase portraits, bifurcation diagrams, power spectra, Lyapunov exponents, the various chaotic behaviors of these systems will be studied. The regions and shapes of the strange attractors, hyperchaotic behaviors and fractal dimensions will also be studied. 4. By pragmatical stability theory, the pragmatical adaptive synchronization of the above systems will be obtained. That the estimated parameters approach the unknown parameters are rigorously proved and illustrated by simulation for double Duffing systems and double van der Pol systems. 5. The study of chaos of double Ikeda system and double Mackey-Glass system. 6. Discover the rule of the initial conditions, for which the double Ikeda systems appear to be in lag-synchronization, anticipated-synchronization, lag-anti-synchronization or anticipated-anti- synchronization, while the double Mackey-Glass systems appear to be in temporary ones. We will try to explain these phenomina.