多維度動態函數的分歧與混沌現象

Abstract

在[1]中，我們考慮一大類能夠轉成差分方程的多項式動態系統，且證明此類的動態系統的非遊蕩集是有界的。在[2]中，我們研究非退化性的同臨相切，證明同臨相切點會被一序列的一致雙曲不變集所聚集。在[3]中，我們探討奇異差分方程，證明在擾動後會有拓樸馬蹄出現，所以具有混沌現象。在[4]中，我們找出3維ACT函數的穩定區域及分歧現象，並證明雙曲不變集的存在。 在此研究計畫中，我們打算結合上述的數學結論與想法，進一步考慮有趣並具代表性的動態系統函數族，可視為Henon函數的高維度推廣，也是同臨相切點附近複雜動態的主要範例。希望延續Palis和Takens專書[5]裡的主題，進一步探討高維度的動態函數的混沌行為。

In the previous work [1], we have considered a wild class of polynomial maps which turns into a class of difference equations and showed that for any dynamical systems from this class, the nonwandering set is bounded. In [2], we also consider nondegenerate homoclinic tangency and shows that the homoclinic point is accumulated by a sequence of uniformly hyperbolic invariant sets. In [3], we investigate singular difference equations and show the existence of topological horseshoe after perturbations. In [4], we study the stability region and bifurcations of the ACT map and show the possession of chaotic behaviors. In this project, we intend to combine the above mathematical ideas. We will investigate a class of normal forms which can be considered as generalization of the famous Henon maps as well as derived from the first return map near a homoclinic tangency in planar or spatial dynamical systems. We will continues the theme of Palis and Takens [5] and study the chaotic phenomuna of the normal forms. [1] M.-C. Li and M. Malkin, 2004, Bounded nonwandering sets for polynomial maps, Journal of Dynamical and Control Systems, 10, pp.377-389. [2] M.-C. Li, 2003, Nondegenerate homoclinic tangency and hyperbolic sets, Nonlinear Analysis, 52, pp. 1521-1533. [3] M.-C. Li and M. Malkin, 2006, Topological horseshoes for perturbations of singular difference equations, Nonlinearity, 19, 795-811. [4] B.-S. Du, M.-C. Li and M. Malkin, 2006, Topological horseshoes for Arneodo-Coullet-Tresser maps, Regular and Chaotic Dynamics, 11, 181-190. [5] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Fractal dimensions and infinitely many attractors. Cambridge Studies in Advanced Mathematics, 35. Cambridge University Press, Cambridge, 1993.