Compact Global Attractors

Abstract

The second chapter of this book is dedicated to the study of different kinds of dissipativity for dynamical systems (both autonomous and nonautonomous): point, compact, local, bounded, and weak. Criteria for point, compact, and local dissipativity are given. It is shown that for dynamical systems in locally compact spaces, the three types of dissipativity are equivalent. Examples are given showing that in the general case, the notions of point, compact, and local dissipativity are different. The notion of Levinson’s center (the maximal compact invariant set), which is an important characteristic of compact dissipative systems, is introduced.