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Topological and computational methods for chaos and global structure of dynamical systems

动力系统混沌和全局结构的拓扑和计算方法

基本信息

批准号：
14540219
负责人：
OKA Hiroe
金额：
$ 2.11万
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：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540219/
关键词：
dynamical system computational method chaos topological method topological entropy symbolic dynamical system Conley Index singularly perturbaed vector field 大域的構造特異摂動周期軌道 heteroclinic軌道 Lyapunov数計算機支援証明微分方程式

项目摘要

The purpose of this project is to develop the topological methods like Comley Index in order to study the global structure and the bifurcations of dynamical systems given by differential equations and/or mappings, moreover to realize such methos as computer algorithm. We obtained the following results during 2002 and 2004.[A] Conley index theory for singularly perturbed vector fields and its applications----The principal idea is to use Conley index for the existence of certain solutions in singularly perturbed vector fields. A convenient formulation of such argument is given by Gedeon-Kokubu-Mischaikow-Oka-Reineck (1999), and using this, a variety of solutions described in terms of symbolic dynamics are obtained in an equations modeling fluid dynamics in a shallow container. The framework is then extended to the case of multi-dimensional slow manifold. As an application, an alternative proof for the existence of periodic traveling waves in some reaction-diffusion system studied by Gardner-Smoller (1983), and furthermore, a set of solutions described in terms of symbolic dynamics as above is also obtained there.[B] Study of holomorphic vector fields-Aiming at the extension of Poincare-Bendixson type theorem for codimension one holomorphic foliations, several results such as non-transversality of such foliations to the boundary sphere, are obtained.[C] Uniform hyperbolicity and non-maximal entropy locus of Henon Family-One of the results is the following. Henon mappings given by cubic polynomials at the parameter value (a, b)=(-1.35, 0.2) is not conjugate to any small perturbed mappings of 1-dimensiomal polynomial which is uniformly hyperbolic and expanding.[D] Braid type of the set of fixed points-An equivalence relation is introduced to the set of the fixed points using notion of the braid type for orientaion preserving homeomorphisms on discs with finite fiexed points. It is shown that the braid type of the fixed points determines the fixed point index completetly.

该项目的目的是开发像Comley指数这样的拓扑方法，以研究由微分方程和/或映射给出的动力系统的全局结构和分岔，并实现计算机算法等方法。我们在2002年和2004年取得了如下成果。[A]奇异摄动向量场的康利指数理论及其应用----其主要思想是利用康利指数来证明奇异摄动向量场中某些解的存在性。 Gedeon-Kokubu-Mischaikow-Oka-Reineck (1999) 给出了此类论证的便捷表述，并使用该表述，在浅容器中的流体动力学建模方程中获得了用符号动力学描述的各种解。然后将该框架扩展到多维慢流形的情况。作为应用，Gardner-Smoller(1983)研究的某些反应扩散系统中周期性行波的存在性得到了另一种证明，并得到了上述用符号动力学描述的一组解。得到了这种叶状结构对边界球的非横向性。[C] Henon族的均匀双曲性和非最大熵轨迹-结果之一如下。由参数值 (a, b)=(-1.35, 0.2) 处的三次多项式给出的 Henon 映射不与一致双曲且展开的一维多项式的任何小扰动映射共轭。[D] 定点集的辫子类型 - 使用辫子类型的概念将等价关系引入到定点集以进行定向在具有有限固定点的圆盘上保持同胚。结果表明，定点的编织类型完全决定了定点指数。