Chaotic Transport in Dynamical Systems

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.

In Chapter 1 we introduced a variety of applications for which some of the questions relevant to the applications could be phrased in terms of a phase space transport problem. These phase space transport problems were motivated by considering systems that could be expressed as perturbations of integrable one-degree-of-freedom Hamiltonian systems. This was instructive because the unperturbed systems possessed qualitatively different motions, bounded by separatrices, that could be easily characterized in the context of the application. When the system was perturbed, it was natural to discuss transitions between these regions of qualitatively different motions.

Over the past ten years much enthusiasm has arisen over the application of the methods of dynamical systems to problems concerned with mixing and transport in fluids; for a recent survey, see Ottino [1989]. The general setting for these problems is as follows.

In this chapter we will study transport in two-dimensional vector fields having a quasiperiodic time dependence (note: quasiperiodicity will be precisely defined shortly). In generalizing the time dependence of the vector fields from the periodic case many new difficulties arise, both conceptual and technical. We now want to examine these difficulties in the context of a general discussion of the construction of discrete time maps from the trajectories of time-dependent vector fields.

MacKay et al. [1984, 1987] and Meiss and Ott [1986] were the first to consider transport between regions in phase space separated by partial barriers such as cantori and segments of stable and unstable manifolds of periodic orbits of two-dimensional, area-preserving maps. They proposed a model for transport which requires certain assumptions on the underlying dynamics that result in a description of transport as a Markov process. In this chapter we will describe the Markov model of Mackay, Meiss, Ott, and Percival and compare it with the exact methods for two-dimensional area-preserving maps developed by Rom-Kedar and Wiggins and described in Chapter 2. The material in this chapter is derived from joint work with Rom-Kedar (see Rom-Kedar and Wiggins [1990]) and Camassa (see Camassa and Wiggins [1991]).

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.”