奇異差分方程擾動後的混沌動態現象(I)

Abstract

在[1]中，我們已經考慮了一大類能夠轉成差分方程的多項式動態系統，而且證明了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，證明了同臨相切點會被一串一致雙曲不變集所聚集。在[3]中，我們證明高維度的Henon函數和ACT函數有混沌現象。 在此研究計畫中，我們打算結合上述的數學結論與想法，進一步考慮一類能夠轉化成差分方程的動態系統函數族，使用即將建立的推廣性隱函數定理，我們希望證明當參數接近奇異極限時，動態函數會具有馬蹄結構所以有混沌現象。

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 have shown that the generalized high-dimensional Henon maps and the ACT map have chaotic behaviors. In this project, we intend to combine the above mathematical ideas. First, we consider a wild family of dynamical systems which can be derived into a family of difference equations. Then using a generalized version of the implicit function theorem which we will establish, we want to show that for any dynamical systems near 「singular limit」, there exists a horseshoe structure and hence chaotic phenomena occur.