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

利用康利指数理论研究混沌动力系统

基本信息

批准号：
10640220
负责人：
OKA Hiroe
金额：
$ 2.24万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2001
项目状态：
已结题

项目摘要

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

该研究项目的目的是开发描述动态系统拓扑结构的理论。这是通过将最初针对类似梯度的系统制定的Conley指数理论扩展到更广泛的动力学类别，尤其是进行混乱的动力学，这是其动力学中经常且复杂的行为来完成的。该项目的主要结果总结在以下三个项目中：1。康利指数理论的开发适用于奇异扰动的向量领域，及其在某些混乱动态系统的分析中的应用：如果提交的单一扰动矢量具有一维慢速流形，则可以将其相位空间结构分解为单个空间结构，从而使我们的阶段置于conleex的形式，以使我们的阶段置于串联的分析中，该阶段是conley的整个阶段信息的信息分解。结果，可以获得特征的属性……更多的棍棒轨道，例如周期性和连接轨道。作为一种应用，通过分析浅水波的不规则振荡行为的模型微分方程来测试该理论，并得出结论认为该行为是混乱的2。多参数系统的过渡矩阵理论的扩展：重新考虑了过渡矩阵的概念，这导致了新的公理公式，即，过渡矩阵是从Morse Decompostions给出的同源性conley Indices获得的链贴中的链映射。该新公式可用于自然扩展动态系统多参数家族的过渡矩阵的通知3。其他相关结果：对于分段线性的一维图，我们研究了拓扑熵，这可以是对动态系统复杂性的一种测量。与这一论点相关，现在正在进行一项针对与计算机援助的混乱攻击者存在的严格证明的研究。较少的