Dynamical Systems: Basic Theory

The basic concept underlying the study of dynamics in infinite dimensional spaces is that of a semiflow, or as it is sometimes called, a semigroup. This semiflow is a time-dependent action on the ambient space, which we assume to be a complete metric space W, for example, a Banach space or a Fréchet space. One should think of the semiflow as a mechanism for describing the solutions of an underlying evolutionary equation. This evolutionary equation is oftentimes the abstract formulation of a given partial differential equation or, sometimes, an ordinary differential equation with time delays. In this chapter we will examine some basic properties of semiflows. Our principal objective is to describe the longtime dynamics in terms of the invariant sets, the limit sets, and the attractors of the semiflow. A comprehensive theory of global attractors is included here. Later in this volume, we will develop the connections between the semiflow and the underlying evolutionary equation.