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

本项目的目的是发展Comley指数等拓扑方法，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支，并实现计算机算法。我们在2002年和2004年期间取得了以下成果。[A]奇摄动向量场的Conley指标理论及其应用-主要思想是利用Conley指标来研究奇摄动向量场的某些解的存在性。Gedeon-Kokubu-Mischaikow-Oka-Reineck（1999）给出了这种论证的一个方便的公式，并使用它，在模拟浅容器中流体动力学的方程中获得了各种符号动力学描述的解。然后，该框架被扩展到多维慢流形的情况。作为应用，本文给出了Gardner-Smoller（1983）研究的一类反应扩散方程组周期行波存在性的另一种证明，并得到了一组用符号动力学描述的解. [B]全纯向量场的研究--将Poincare-Bendixson型定理推广到余维为1的全纯叶理，得到了这类叶理对边界球的非横截性等结果. [C]Henon族的一致双曲性和非极大熵轨迹--结果之一如下。由参数值（a，B）=（-1.35，0.2）的三次多项式给出的Henon映射不与一致双曲扩张的1维多项式的任何小扰动映射共轭。[D]不动点集的辫型--利用有限不动点圆盘上保向同胚的辫型概念，对不动点集引入了一个等价关系.证明了不动点的辫型完全决定不动点指数。

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.