Long Range Behavior in Dynamical Systems and Partial Differential Equations

University Of Texas At Austin, Austin TX

Investigators

Abstract

We want to investigate several projects in Dynamical Systems and in Partial Differential equations. We will also consider multi-particle systems that share properties with both/ One unifying thread is that we would like to understand the relation between variational approaches and more geometric ones. In particular, we would like to produce proofs of diffusion that use geometric and variational methods and to extend results in dynamics obtained by variational methods to partial differential equations. An important tool for geometric methods is the theory of normally hyperbolic manifolds and we would like to extend it to Partial differential equations and infinite particle systems. Sometimes, small short range causes may have large long term effects. For example, small periodic forces may build up large changes in energy. Some small changes in the local properties of a material may lead to the emergence of patterns that cover large distances. In other situations, however, local effects just average out. We would like to device a broad based array of methods (including numerical studies and geometric techniques) that can be used to decide whether build up or averaging occurs. We would also like to pay special attention to some concrete models appearing in technological applications.