離散漢米爾頓系統中二階差分方程的廣域吸引子及拓樸混沌

Abstract

於本篇論文中，我們討論一個差分方程式兩種不同的動態： ∆[p∆x(t-1)]+qx(t)=f(x(t-1))或f(x(t)), t∈Z， 其中∆x(t-1)=ax(t)-bx(t-1)。此兩種動態行為分別為廣域吸引子與拓樸混沌。我們做出了多樣的結果。在參數a,b,p與q的某種條件之下，此方程任意解的軌跡最終都將會收斂到一個廣域吸引子。請參照定理2.2與2.3。在某個特定的參數值之下，若存在一個與f有關的函數且此函數擁有不只一個簡單根或者正拓樸熵，則限制在此方程式解集合上的轉移映射會具有拓樸混沌。請參照定理2.6、2.7、2.8及2.9。最後，我們將此方程式經由變數變換轉變成參數化的連續函數。我們也可將之表示成離散漢米爾頓系統的形式。針對f(x(t))的情況，定理2.10表示會存在一個與f有關的函數且此函數擁有正拓樸熵使得對應函數具有拓樸混沌。針對f(x(t-1))的情況，若滿足前面的條件並且此與f有關的函數值域被局部性地限制住範圍，則定理2.11表示此對應函數也會擁有拓樸混沌。

In this thesis, we discuss two distinct dynamics of the difference equation ∆[p∆x(t-1)]+qx(t)=f(x(t-1)) or f(x(t)), t∈Z, where ∆x(t-1)=ax(t)-bx(t-1). These two dynamics are the behavior of globally attracting and topological chaos. We have several results. Under some conditions of a, b, p and q, every orbit of the equation asymptotically converges to a global attractor. See theorems 2.2 and 2.3. If there exists a function relating to f which has more than one simple zeros or positive topological entropy at an expected parametric value, then the shift map restricted to the set of solutions of this equation has topological chaos. See theorems 2.6, 2.7, 2.8 and 2.9. Finally, we transform this equation into a parameterized continuous function by changing variables. We can also write it as the form of a discrete Hamiltonian system. For the case f(x(t)), theorem 2.10 says that there exists a function relating to f which has positive topological entropy such that the corresponding function has topological chaos. For the case f(x(t-1)), with an additional assumption that the function relating to f is locally trapping, theorem 2.11 says that the corresponding function has also topological chaos.