Stable Patterns and Synchronization in Dynamical Systems

9704853 Hale拟议的研究涉及由格上常微分方程组、偏微分方程组和时滞微分方程定义的动力系统中稳定模式的产生和同步。主要的兴趣在于扩散耦合、区域形状或边界层如何影响解的动力学，特别是稳定性。数学工具包括奇异摄动理论、不变流形和分支。在化学动力学、物理、生物、生物化学和生态学等各个领域，理解负责自组织、图案形成和同步的处理机是至关重要的。人们已经知道，单个粒子(或子系统)的扩散相互作用的性质起着基本的作用。这个项目的主旨是试图理解这些现象及其对耦合、区域形状和边界条件的依赖。

9704853 Hale The proposed research is concerned with the generation of stable patterns and synchronization in dynamical systems defined by ordinary differential equations on lattices, partial differential equations and delay differential equations. The primary interest is in the manner in which diffusive coupling, domain shape or boundary layer affect the dynamics and especially the stability of solutions. The mathematical tools involve singular perturbation theory, invariant manifolds and bifurcation. It is of paramount importance to understand the processor responsible for self organization, pattern formation and synchronization in various fields such as chemical kinetics, physics, biology, biochemistry and ecology. It has been known for some time that the nature of diffusive interaction of individual particles (or subsystems) plays a fundamental role. The main thrust of this project is to attempt to understand these phenomena and their dependence upon the coupling, domain shape and boundary conditions.