一类同宿缠结的动力学性质
On the dynamics of certain homoclinic tangles
摘要：
本文研究了二阶周期扰动方程中{ \it非共振耗散}同宿鞍点的稳定流形与不非稳定流形的横截相交所形成的同宿缠结.我们证明了这些同宿缠结的动力学是{ \it无限包裹马蹄映射}的动力学。用$μ$作为表示扰动大小的参数，证明了：（a）存在无穷多个$μ$的不相交开区间，在$μ= 0$处累积，使得扰动方程的整个同宿缠结由一个具有无穷多个符号的马蹄形组成，（b）在每个参数区间之间存在参数，使得同宿缠结包含吸引周期解，以及（c）在同宿缠结允许某些耗散双曲周期解的非退化横截同宿相切的参数之间也存在参数.特别地，（c）意味着存在具有SRB测度的奇异吸引子。
英文摘要：
In this paper we study homoclinic tangles formed by transversal intersections of the stable and the unstable manifold of a {\it non-resonant, dissipative} homoclinic saddle point in periodically perturbed second order equations. We prove that the dynamics of these homoclinic tangles are that of {\it infinitely wrapped horseshoe maps}. Using $μ$ as a parameter representing the magnitude of the perturbations, we prove that (a) there exist infinitely many disjoint open intervals of $μ$, accumulating at $μ= 0$, such that the entire homoclinic tangle of the perturbed equation consists of one single horseshoe of infinitely many symbols, (b) there are parameters in between each of these parameter intervals, such that the homoclinic tangle contains attracting periodic solutions, and (c) there are also parameters in between where the homoclinic tangles admit non-degenerate transversal homoclinic tangency of certain dissipative hyperbolic periodic solutions. In particular, (c) implies the existence of strange attractors with SRB measures for a positive measure set of parameters.
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