Global bifurcations of homoclinic orbits in vector fields

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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.