Introduction and Examples

Abstract

Dynamics is the study of how systems change in time. Current research trends in dynamics place much emphasis on understanding the nature of the attractors of a system. Justification is often given for this by noting that since attractors capture the asymptotic behavior of a system their study will shed light on the observable motions of the system. This is certainly true; however, many important observable dynamical phenomena are not asymptotic in nature, but rather transient. Indeed, one could take the practical, but rather extreme, point of view that everything we observe in nature is transient, and that therefore transient, as opposed to asymptotic, dynamics is of much more importance in mathematical descriptions of natural phenomena. Moreover, a very important class of dynamical systems, the Hamiltonian systems, do not have attractors by any reasonable and practical definition of the concept. Therefore, it is important from the point of view of applications to have a framework for studying these issues. In this monograph we want to motivate many of these issues from the viewpoint of problems of phase space transport.