應用GYC部分區域穩定理論於Froude-van der Pol系統之廣義同步與控制,以Legendre函數為參數的Lorenz 系統之超渾沌,歷史Chen系統渾沌的技巧性控制及藉實用漸進穩定理論的陰-陽適應渾沌同步

Abstract

本篇論文以GYC部分區域穩定理論來研究新渾沌Froude-van der Pol系統其廣義同步及渾沌控制。藉由GYC部分區域穩定理論，Lyapunov函數及控制器皆較傳統系統設計簡單，且因為系統階數低於傳統系統，所以可得到較小的模擬誤差。本篇以相圖、Lyapunov指數、分歧圖、參數圖等數值模擬方法，研究帶有Legendre函數為參數的Lorenz系統其超渾沌現象。此外本篇首次研究歷史Chen系統的渾沌行為，當代Chen系統皆以被詳盡研究，但至今尚未有文章對歷史Chen系統深入著墨，因此，接下來研究討論Chen系統中“陰”參數其歷史渾沌行為。本篇藉由線性耦合的方式，選取一合適的耦合參數，來探討“陽”和“陰” Chen系統其廣義同步渾沌現象，並藉由實用漸進穩定理論，其原理是兩實用適應同步渾沌系統，方程式中參數將其一設為不確定的，其餘設為估測的參數，加以利用實用漸進穩定理論來研究 “陰-陽”適應渾沌同步。

In this thesis, a new chaotic Froude-van der Pol system is studied. A new strategy of achieving chaos generalized synchronization and chaos control by GYC partial region stability is proposed. Using the GYC partial region stability theory, the Lyapunov function used becomes a simple linear homogeneous function of error states and the controllers are simpler than traditional controllers, and give less simulation error because they are in lower order than that of traditional controllers. The chaotic behaviors of a Lorenz system with Legendre function parameters is firstly studied numerically by time histories of states, phase portraits, Poincaré maps, bifurcation diagram, Lyapunov exponents and parameter diagrams. Abundance of hyperchaos and of chaos is found, which offers the potential for many applications. In this thesis, the behavior of historical Chen system is firstly studied. To our best knowledge, most of contemporary Chen system are researched in detail, but there are no articles in investigating a thorough inquiry about the history of Chen system so far. Therefore, the historical chaos of Chen system with “Yin parameters” is introduced. In this thesis, we employ an applicable coupling parameters by linear coupling strategy to complete the goal of generalized synchronization of Yin and Yang Chen systems and take advantage of using an adaptive Yin-Yang chaos synchronization of Yin and Yang Chen system by pragmatical asymptotically stability theorem. This pragmatical adaptive synchronization of two chaotic systems of which one has uncertain parameters the another has estimated parameters, is achieved by pragmatical asymptotically stability theorem.