Describing the Flow on the Attractor of One Dimensional Reaction Diffusion Equations by Systems of ODE

Abstract

Many systems of reaction diffusion equations, delay equations, damped wave equations define nonlinear semigroups T(t):X → X, t ≥ 0, (X a function space) which admit a compact attractor i.e. a compact connected invariant set A ⊂ X which is uniformly asymptotically stable and attracts all points in X [1]. In this situation understanding the dynamic of the semigroup T(t) may be considered equivalent to the description of the flow on A. Since from abstract theorems on dissipative semigroups [2],[3] it follows that the attractor is a finite dimensional set, once this point of view has been adopted, it is natural to ask if the infinite dimensional semigroup T(t) can, in some sense, be considered equivalent to the finite dimensional dynamical system generated by a suitable system of ODE. The most direct way one can try for constructing such a system of ODE is to show that the attractor is contained in a finite dimensional invariant manifold M. When this is the case one obtains the sought finite dimensional dynamical system simply by restricting T(t) to M. In general this construction is only formal because M cannot be computed explicitly and therefore the dynamical system describing the flow on M is only approximatly known and the problem of structural stability arises.Therefore instead of asking for a finite dimensional dynamical system which coincides with T(t) on A it may be more natural to require only topological equivalence in the sense of the following definition [4].