Global bifurcations of homoclinic orbits in vector fields

在本工程中获得的结果如下：1。针对动力系统的拓扑不变量Conley指数，将在连接轨道分岔研究中起重要作用的转移矩阵推广到多参数族。作为发展Conley指标理论的第一步，对于某些类型的称为慢-快系统的奇异摄动向量场，当慢流形通常是双曲的和一维的，得到了周期轨道和异斜轨道存在的拓扑代数条件。证明了一类异斜环是由矢量场的余维三维简并奇点分叉的。因此，从简并奇点产生混沌吸引子。用数值方法研究了旋转圆柱中固液（Couette流）在相同/相反方向旋转时的稳定性。该问题可以简化为ODE系统，验证后的数值模拟可以应用。结果，确定了临界泰勒数，并利用局部分岔理论，严格证明了泰勒涡（平稳解）和周期解分岔（在某些情况下为波浪泰勒涡）的存在性。Kaneko， Tsuda和Ikeda发现了混沌振荡耦合系统中半稳定状态通过混沌的过渡现象，并将其命名为“混沌巡回”。本课题研究了全局耦合地图（GCM）中混沌流动的产生机制。从GCM对振子置换的不变性出发，确定了不变性子空间的层次结构，并证明了GCM中的混沌流动是通过低维子空间中的吸引子在互补方向上失稳而发生的。

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.