Dynamics of Nonlinear Differential Equations

Brown University, Providence RI

Investigators

Abstract

PI: John Mallet-Paret, Brown University DMS-0200178 Abstract: We propose a fundamental study of qualitative properties of various classes of nonlinear dynamical systems, in particular as they arise from differential and difference equations. Among the systems to be considered are those from ordinary and partial differential equations, systems of lattice differential equations (that is, spatially discrete systems), differential delay equations, and nonlinear maps on cones. Issues such as existence of equilibria and their stability, spontaneous formation of spatial patterns and spatial chaos, and existence and qualitative properties of traveling fronts, will be studied for these equations. In addition to established techniques (both theoretical and numerical) from differential equations and dynamical systems, a significant portion of the proposed research involves the development and implementation of new tools and techniques with which to study these systems. Among the techniques to be employed are those involving singular perturbations, invariant manifolds, exponential dichotomies, and topological methods (Morse decompositions, fixed point theorems, degree theory). We are developing new mathematical techniques for analyzing and understanding differential equations and difference equations which arise as models in various areas of science. Very broadly, these types of mathematical systems model time-dependent or evolutionary behavior, as it occurs in a wide range of scientific areas, including biology, chemistry, electrical circuit theory, image processing, and material science. While this is a very broad scope of inquiry, the specific problems to be studied exhibit features in common -- spontaneous formation of patterns, self-sustained oscillations, regulation by internal feedback (often with time delays) -- which can be analyzed with some of the basic tools of dynamical systems theory. It is expected the resulting mathematical advances arising from these studies will increase the knowledge of and will provide insight into both the abstract theory of dynamical systems and the scientific areas of inquiry.