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

该奖项支持乔治亚理工学院的Konstantin Mischaikow教授和日本琉球大学的hiroo Oka教授之间为期三年的合作研究项目。科学和工程领域的一个长期挑战是确定和描述具有大量（也许是无限）自由度的系统中的相干行为。考虑到动力系统作为物理过程模型的普遍性，对于连贯有效的动力系统应用程序的主要障碍是低维现象嵌入高维系统的深度纠缠方式，这可能并不奇怪。目标是扩展和发展新的有效的拓扑技术，从潜在的高维系统中提取相干的低维动态现象。研究者的主要工具是Conley的索引理论，它将动力学扩展部分的拓扑特征整理成代数拓扑数据结构（例如，同调和上同调）。该理论的三个合作研究领域是：1)构造一个有效的拓扑奇异摄动理论，将快慢系统的动力学化简为慢流形；2)指标理论适应于复杂流动的环境，强调粒子路径和嵌入；3)从时间序列数据中重建动态信息，特别是作为带噪声系统的严格数值积分工具。该项目汇集了两个具有互补专业知识和研究能力的实验室的努力。通过思想和技术的交流，该项目将扩大我们的基础知识基础，促进国际了解与合作。研究人员计划在科学期刊上发表他们的研究结果，并在科学会议上报告他们的发现。

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.