Geometric singular perturbation theory for reaction-diffusion systems

Abstract

Geometric singular perturbation theory (GSPT) has proven to be an invaluable tool for the study of multiple time scale ordinary differential equations (ODEs). The extension of the theory to partial differential equations (PDEs) presents significant challenges as it implies a passage from finite-dimensional to infinite-dimensional dynamics. As a first step in this direction, we restrict ourselves to reaction-diffusion systems and generalise several aspects of GSPT for ODEs to such systems. Firstly, we explore the non-hyperbolic case, where we treat dynamic fold and Hopf bifurcations of fast-slow PDE-PDE systems with slowly varying bifurcation parameter. Our ap- proach is to work with the Galerkin discretisation and employ standard GSPT along with estimates controlling the contribution of higher order modes. The method is not limited to the studied examples and can be generalised to a large class of prob- lems. Secondly, we treat a fast-slow PDE-ODE system exhibiting bifurcation delay by constructing slow manifolds and combine this with application of a center manifold theorem. This provides a framework for dealing with such problems and although the results are not new, they have not been applied to the particular case of multiple scale systems.