Transport in к-Degree-of-Freedom Hamiltonian Systems, 3 ≤ к < ∞: The Generalization of Separatrices to Higher Dimensions and Their Geometrical Structure

Abstract

The goal in this chapter is to generalize many of the concepts developed in the previous chapters for lower-dimensional dynamical systems to higher dimensions. We will consider only Hamiltonian systems, although further generalizations to non-Hamiltonian systems are possible (these will be briefly discussed later). We will begin by considering the types of structures that can arise in the phase space of a Hamiltonian system and the potential of these structures for providing barriers to transport. In particular, we are looking for an appropriate generalization of the notion of a “separatrix” to higher dimensions. First, however, let us consider the essential characteristics that define what we mean by the term “separatrix.”