Study of chaotic dynamical systems by means of the Conley index theory

本研究计划的目的是发展描述动力系统拓扑结构的理论。这是通过将康利指数理论（最初是为类梯度系统制定的）扩展到更广泛的动力系统，特别是那些表现出混沌的动力系统，它们的动态中有周期性和复杂的行为。本项目的主要成果总结为以下三个方面：1.研究成果。适用于奇异摄动矢量场的Conley指标理论的发展及其在混沌动力系统分析中的应用：当奇异摄动矢量场具有一维慢流形时，其相空间结构可以分解为管-盒-帽集合的形式，通过对分解的各个部分的分析，可以得到整个相空间结构的Conley指标信息。因此，我们可以得到更多的特征轨道的性质，如周期轨道和连接轨道。作为应用，通过分析浅水波浪的不规则振荡行为的模型微分方程，对该理论进行了验证，得出该行为是混沌的结论。多参数系统转移矩阵理论的推广：重新考虑了转移矩阵的概念，得到了一个新的公理化表述，即转移矩阵是由Morse分解给出的同调Conley指标得到的链复上的链映射。这个新公式可以很自然地推广动力系统多参数族的转移矩阵的概念。其他相关结果：对于一个分段线性一维映射，我们研究了拓扑熵，它可以作为一种测量动力系统复杂性的方法。与此相关的是，一项利用计算机辅助对混沌吸引子存在性进行严格证明的研究正在进行中。少

The purpose of this research project was to develop the theory describing the topological structure of the dynamical systems. This was done by extending the Conley index theory, which was originally formulated for gradient-like systems, to a broader class of dynamical systems especially the ones exhibiting chaos, a recurrent and complex behavior in their dynamics. The main results of this project are summarized in the following three items:1. Development of the Conley index theory adapted for singularly perturbed vector fields, and its application to the analysis of some chaotic dynamical systems: In case that the singularly perturbed vector filed has a one-dimensional slow manifold, its phase space structure can be decomposed into the form of the tube-box-cap collection, which enables us to obtain the Conley index information of the entire phase space structure from the analysis of the individual peices of the decomposition. As a result, one can obtain the properties of the characteri … More stic orbits, such as periodic and connecting orbits. As an application, the theory was tested by analyzing the model differential equation of a irregular oscillatory behavior of a shallow water wave, and concluded that the behavior is chaotic.2. Extension of the transition matrix theory for multi-parameter systems: The notion of transition matrix is re-considered, which resulted in a new axiomatic formulation, namely, the transition matrix is a chain map on the chain complex obtained from the homology Conley indices given by the Morse decomposition. This new formulation can be used to naturally extend the notion of transition matrix for multi-parameter families of dynamical systems.3. Other related results: For a piecewise linear one dimensional maps, we studied topoplogical entropy which can be a kind of measurement of the complexity of the dynamical systems. Related to this argument, a study of rigorous proof for the existence of chaotic attracter with computer aid is now in progress. Less