接近反可積分極限之可微分方程的混沌軌跡

Abstract

在這篇論文中，我們研究可以被轉變為下列這種差分方程式的動態系統，ϵg(x_(t-n_2 ),…,x_t,…,x_(t+n_1 );ϵ)+g_0 (x_t )=0 其中參數 ϵ 為實數，常數n_1和n_2皆為非負整數，函數g從R^(n_1+n_2+1)映射到R，函數g_0從R映射到R，兩函數之一階偏導數皆存在且連續。對於這類的動態系統，在特定的假設條件之下，我們證明了某些混沌軌跡的存在，並且明確的說明如何建構出這些軌跡，而且針對每一種混沌軌跡，我們都可以造出無窮多組不同的軌跡。此外，我們還提出一個判別法則，可以由差分方程式直接看出某些混沌軌跡的存在。最後我們以下列四種映射為例， 1. Hénon map. 2. Modified Mira map. 3. Arneodo-Coullet-Tresser map. 4. Quadratic volume preserving map. 說明如何將其轉變為差分方程，以及針對不同特性的映射，在什麼條件之下可以建構出特定的混沌軌跡，包括 1. Transversal homoclinic orbits. 2. Transversal heteroclinic orbits. 3. Snap-back repellers. 4. Heteroclinic repellers.

In this work, we study the dynamical system which can be transformed into a difference equation of the form ϵg(x_(t-n_2 ),…,x_t,…,x_(t+n_1 );ϵ)+g_0 (x_t )=0 where ϵ∈R is a parameter, g∶R^(n_1+n_2+1)→R and g_0 ∶R→R are both C^1 functions and n_1, n_2 are both nonnegative integers. Suppose that the function g_0 has k simple zeros where k≥2. We give criteria of existence of some kinds of chaotic orbits, and construct infinite ones explicitly. We prove that some kinds of chaotic orbits exist and point out how to construct those orbits explicitly. Moreover, we can build infinite different ones for each kind of chaotic orbits. As applications of these results, we establish snap-back repellers and heteroclinical repellers for the modified Mira maps and transversal homoclinic orbits or transversal heteroclinic orbits for the Hénon maps, the Arneodo-Coullet-Tresser maps and the quadratic volume preserving maps.