Towards a Global Theory of Singularly Perturbed Dynamical Systems

Dynamical systems with multiple time scales arise naturally in many domains. Models of neural systems provide the principal motivation for this paper. Most of the previous mathematical analysis of qualitative properties of multiple time scale systems has dealt with local phenomena that occur in low dimensions. The neural system models raise questions that lie beyond the scope of existing work. Our aim here is to outline a theory that extends the local theory to a description of qualitative features of the global dynamics for systems with two time scales. We fill in portions of this outline within the context of systems that have two slow variables and two fast variables. Even within this low dimensional setting, most basic questions about global properties remain unanswered. The setting within which we work is a system of differential equations in R of the form