Introduction, Motivation, and Background

Abstract

In this chapter we want to describe and motivate some aspects of the theory of invariant manifolds which we will explore throughout the rest of this book, as well as give a brief survey of the field and the breadth of applications of invariant manifold methods and ideas throughout science and engineering. Roughly speaking, an invariant manifold is a surface contained in the phase space of a dynamical system that has the property that orbits starting on the surface remain on the surface throughout the course of their dynamical evolution in one or both directions of time; i.e., an invariant manifold is a collection of orbits which form a surface. If the surface has a boundary, then trajectories can leave the surface by crossing the boundary. Additionally, under certain conditions, the set of orbits which approach or recede from an invariant manifold M asymptotically in time are also invariant manifolds which are called the stable and unstable manifolds, respectively, of the invariant manifold. Moreover, the stable and unstable manifolds may be foliated or fibered by lower dimensional submanifolds having the property that trajectories starting on these fibers satisfy certain asymptotic growth rate conditions. Many of these features persist under perturbation, a property which is useful for the development of many types of global perturbation methods. We begin by considering areas of applications where ideas from invariant manifold theory have played an important role.