Stability and Gradient Dynamical Systems

Abstract

The objective in these notes is to present an approach to dynamical systems in infinite dimensions. It does not seem reasonable to make a comparison of all of the orbits of the dynamics of two systems on non locally compact infinite dimensional spaces. Therefore, we choose to compare them on the set of globally defined bounded solutions. Fundamental problems are posed and several important results are stated when this set is compact. We then give results on the dynamical system which will ensure that this set is compact. Many applications are give to partial differential equations of parabolic and hyperbolic type as well as functional differential equations. If X is a metric space, then any continuous map T that takes X to X is a dynam-ical system with the dynamics being described by the iterates of T. Consider an autonomous evolutionary equation on X for which the initial value problem is well defined. If we define T (t)x, t ≥ 0, as the solution through x, then the family of mappings {T (t), t ≥ 0} is a dynamical system on X. For the case in which X is a compact manifold (or even locally compact), there is an extensive qualitative theory of dynamical systems associated with the stability and bifurcation of the orbit structure.