U.S.-Japan Cooperative Science: Topological Methods in Nonlinear Dynamics

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0089631MischaikowThis award supports a three-year collaborative research project between Professor Konstantin Mischaikow of the Georgia Institute of Technology and Professor Hiroe Oka of Ryukoku University in Japan. A perennial challenge in science and engineering is the determination and description of coherent behaviors in systems with large (perhaps infinite) numbers of degrees of freedom. Given the ubiquity of dynamical systems as models for physical processes, it is perhaps not surprising that the primary obstacle to a coherent, effective program of dynamical systems applications is the deeply entangled manner in which low-dimensional phenomena are embedded within high-dimensional systems. The goal is to extend and develop new and effective topological techniques for extracting coherent low-dimensional dynamical phenomena from potentially high-dimensional systems. The investigators' primary tool is the index theory of Conley, which collates the topological features of the expanding portion of the dynamics into algebraic-topological data structures (e.g., homology and cohomology). The three areas of cooperative research for the theory are: 1) construction of an effective topological singular perturbation theory for reducing the dynamics of a fast-slow system to the slow manifold; 2) adaptation of the index theory to the context of complicated flows with an emphasis on particle paths and embeddings; and 3) reconstruction of dynamical information from time-series data, particularly as a tool for rigorous numerical integration of systems with noise.The project brings together the efforts of two laboratories that have complementary expertise and research capabilities. Through the exchange of ideas and technology, this project will broaden our base of basic knowledge and promote international understanding and cooperation. The researchers plan to publish results of their research in scientific journals and report on the findings at scientific meetings.