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.
