Invariant manifolds with asymptotic phase

Abstract

THIS PAPER is concerned with invariant manifolds allowing a generic approach to solutions (of the underlying autonomous differential system) only. This means. that any solution approaching such an invariant manifold M ultimately behaves like a particular solution on J4, in the sense that the difference n-(t) i(l) of the two solutions tends to 0 as t + CQ. In other words, only solutions on the stable manifold of M can approach M as f+ cc. Invariant manifolds with that property are said to have an asymptotic phase. The simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose Floquet multipliers lie all but one in the open unit disc of the complex plane. Malkin [12] proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold. More generally, a k-dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k Floquet multipliers have modulus less than 1 (see Hale & Stokes [9]). Conditions for a torus to be asymptotically stable with asymptotic phase are given by Samoilenko [13] and Coppel[5]. Invariant manifolds without any specified structure are considered by Fenichel [6,7] and in Kirchgraber’s note [ 111. Moreover, each center manifold has an asymptotic phase if the corresponding equilibrium is stable (see Carr & Al-Amood [4] and Henry [lo]). In any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold. In each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability. A negative answer to this question is given by the author in [l, 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase. In [l] the manifold is supposed to be compact while the noncompact case is treated in [2] under the restriction that the generating family of periodic solutions has an amplitude independent period. Recently Hale & Massatt [8] extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations. In this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result: A compact invariant manifold has an asymptotic phase if the manifold is hyperbolic (assumption Hl) and carries a flow parallel in the sense of our assumption H2. This includes most of the aforementioned results.