Dynamics of Nonlinear Differential Equations

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DYNAMICS OF NONLINEAR DIFFERENTIAL EQUATIONS (DMS-0500674) John Mallet-Paret Division of Applied Mathematics Brown UniversityWe propose to study fundamental qualitative properties ofvarious classes of nonlinear dynamical systems, in particular asthey arise from differential and difference equations. The systemsto be considered include those from ordinary and partial differentialequations, systems of lattice differential equations (that is,spatially discrete systems), differential delay equations, andmax-plus operators. Issues such as the existence of equilibria andtheir stability, spontaneous formation of spatial patterns and spatialchaos, the existence and qualitative properties of traveling fronts,and the appearance of spontaneous temporal oscillations will be studiedfor these problems. In addition to the established techniques (boththeoretical and numerical) from differential equations and dynamicalsystems, a significant portion of the proposed research involves thedevelopment and implementation of new tools and techniques with whichto study these systems. Among the techniques to be employed are thoseinvolving singular perturbations, invariant manifolds, exponentialdichotomies, and topological methods.We are developing new mathematical techniques for analyzing andunderstanding differential equations and difference equations. Suchequations typically arise as models in numerous areas of science. Inbroad terms, these types of mathematical systems model time-dependentor evolutionary behavior, as it occurs in a wide range of scientificareas, including biology, chemistry, electrical circuit theory, imageprocessing, and material science. Although this is a very broad scopeof inquiry, the specific problems to be studied exhibit features incommon -- spontaneous formation of patterns, self-sustained oscillations,regulation by internal feedback (often with time delays) -- which can beanalyzed with some of the basic tools of dynamical systems theory. It isexpected the resulting mathematical advances arising from these studieswill increase the knowledge of, and will provide insight into both theabstract theory of dynamical systems, as well as the scientific areas ofinquiry.