基于拓扑马蹄的混沌动力学研究进展

PROGRESSES ON CHAOTIC DYNAMICS STUDY WITH TOPOLOGICAL HORSESHOES

拓扑马蹄理论是混沌研究的重要分支,是迄今为止能够达到数学严格性的核心混沌研究方法之一.基于简明的空间几何化思想,拓扑马蹄为非线性系统复杂行为的数值与理论研究搭建了一座桥梁,从而人们可以进行混沌行为的不变集刻画、存在性证明、拓扑熵估计、以及内在机制揭示等一系列研究. 本文希望通过综述当今基于拓扑马蹄的混沌研究最新进展,使研究者深入了解这套功能强大的方法体系,并能加以应用. 本文首先从人们所熟知的 Smale 马蹄开始介绍现代拓扑马蹄理论的发展历程; 然后介绍了当今的拓扑马蹄引理,讨论了拓扑马蹄存在条件和相应的搜寻方法; 最后,利用拓扑马蹄研究的工具箱 HsTool,以经典离的散 Hénon 映射和著名的 Saito 电路为例,展示其拓扑马蹄的研究过程.

The topological horseshoe theory is one of the most important ways to study chaos rigorously. With striking geometric clarity, the method bridges the gap between chaotic theory and numerical computation, and has been extensively used in the study of chaotic invariant sets, computer assisted proofs, topological entropy estimation, etc. In order to make more researchers understand this powerful method, we presents a brief review of to- pological horseshoe this paper. We first introduce the history from Smale＇ s horseshoe to topological horseshoes, showing their essential feature; and then present some useful theorems, the corresponding conditions and numerical methods, as well as a toolbox called HsTool; then give two examples, the Henon map and the hyperchaotic Saito circuit, to show how to find horseshoes in practical systems.