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

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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.