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Global bifurcations of homoclinic orbits in vector fields

矢量场中同宿轨道的全局分岔

基本信息

批准号：
10640115
负责人：
KOKUBU Hiroshi
金额：
$ 2.11万
依托单位：
KYOTO 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 1999
项目状态：
已结题

项目摘要

The results obtained in the project are as follows:1. In relation to a topological invariant of dynamical systems called the Conley index, the transition matrix, which plays an important role in the study of bifurcations of connecting orbits, is generalized for multi-parameter families. As the first step toward developing the Conley index theory for certain types of singularly perturbed vector fields called slow-fast systems, a topological-algebraic condition for the existence of periodic and heteroclinic orbits is obtained when the slow manifold is normally hyperbolic and one-dimensional.2. It is proved that certain type of heteroclinic cycles bifurcates from a codimension three degenerate singularity of vector fields. As a result, the occurrence of chaotic attractors from the degenerate singularity follows.3. Stability of stationary solutions (Couette flows) in fluid motion in rotating cylinders is numerically studied when the cylinders rotate in the same/opposite directions. The problem can be reduced to ODE systems and the validated numerical simulation can be applied. As a result, the critical Taylor number is determined and, by making use of the local bifurcation theory, the existence of Taylor vortex (stationary solution) and bifurcations of periodic solutions (in some cases, wavy Taylor vortex) is rigorously proved.4. Transition phenomena among semi-stable states through chaos in coupled systems of chaotic oscillators is discovered and named as "chaotic itinerancy" by Kaneko, Tsuda and Ikeda. In this project, the creation mechanism of chaotic itinerancy in globally coupled maps (GCM) is studied. From the invariance of GCM with respect to permutations of oscillators, the hiererchical structure of invariant subspaces is determined and is used to prove that the chaotic itinerancy in GCM occurs through crisis in attractors lying in lower dimensional subspaces which destabilize in the complementary directions.

1.对于动力系统的一种称为Conley指数的拓扑不变量，将在连通轨道分叉研究中起重要作用的转移矩阵推广到多参数族中。作为发展某些称为慢快系统的奇摄动向量场的Conley指数理论的第一步，当慢流形是正常双曲的一维时，得到了周期轨道和异宿轨道存在的拓扑-代数条件。证明了从向量场的余维三次退化奇点分叉出某些类型的异宿环。因此，混沌吸引子的出现跟从了简并奇点。数值研究了旋转圆柱体中流体运动的定常解(Couette流)在圆柱体同向和反向旋转时的稳定性。该问题可以归结为常微分方程组的问题，并且可以应用经过验证的数值模拟。确定了临界泰勒数，并利用局部分叉理论，严格证明了泰勒涡(定常解)的存在性和周期解(某些情况下为波浪型泰勒涡)分支的存在性。金子、津田和池田发现了混沌振子耦合系统中半稳态之间的混沌跃迁现象，并将其命名为“混沌巡游”。在本项目中，研究了全局耦合映射(GCM)中混沌游程的产生机制。从GCM关于振子排列的不变性出发，确定了GCM中不变子空间的层次结构，并证明了GCM中的混沌游荡是通过位于低维子空间中的吸引子的危机而发生的，这些吸引子在互补方向上失稳。