Dynamics Reported

The title refers to a theory which is based on independent research in three different fields—differential geometry, dynamical systems and solid state physics—and which has attracted growing interest and research activity in the last few years. The objects of this theory are respectively:

We consider the flow of a one-dimensional reaction diffusion equation with Dirichlet boundary conditions Let v, w denote stationary, i.e. t-independent solutions. We say that v connects to w, if there exists an orbit u(t, x) of (1.1), (1.2) such that i.e. u(t, ·) is a heteroclinic orbit connecting v to w. In this report we address the following question:

(*) Given v, which stationary solutions w does it connect to?

This paper is a self-contained exposition of the theory of averaging for periodic and quasiperiodic systems, with the emphasis being on the author’s research (part of it joint work with Clark Robinson) on qualitative aspects of nonlinear resonance. Many topics in averaging theory are not covered, among them: averaging for systems more general than quasiperiodic; relations between averaging and multiple time-scale methods; Eckhaus’s approach to averaging; combinations of averaging with matching of asymptotic expansions. The principal question which is addressed is: when does averaging (to first or higher order) lead to an accurate qualitative description of the solutions of the original (unaveraged) equation? By qualitative description we mean both locally (existence and stability of certain invariant sets such as periodic orbits, or almost invariant ‘lingering’ and globally (connecting orbits between invariant sets, or claims that in certain large regions all orbits drift in a certain direction).

There is a vast literature on singular perturbations both from the point of view of applications as well as of results concerning the theoretical foundations. For a general survey the reader is referred to the books by Cole [3], Eckhaus [5], [6], Kaplun [15], O’Malley [27], Van Dyke [32], Wasow [35] and to the articles by Fraenkel [10], Hoppensteadt [14], Kevorkian [16], Lagerstrom and Casten [17] and Vasil’eva [33], [34]. A good deal of work in singular perturbations is devoted to boundary value problems. In this paper, however, we will restrict ourselves to initial value problems (IVP’s) although we believe that the ideas derived may be useful for boundary value problems as well.

Smale [12,13] studied diffeomorphisms with transversal homoclinic points and showed that the dynamics are chaotic in the neighbourhood of the orbit of such a point, in the sense that there is a compact invariant set on which the action of some iterate of the diffeomorphism is topologically conjugate to the action of the Bernoulli shift. One immediate consequence of this is Birkhoff’s result that the diffeomorphism has infinitely many periodic points. It also turns out that nearby diffeomorphisms must also have transversal homoclinic points and hence also infinitely many periodic points. Thus the property of having infinitely many periodic points cannot, in general, be perturbed away.