The Conley index and the method of averaging

Abstract

In particular, one useful theorem (see, e.g., Hale [S] for details) states roughly that if f satisfies certain uniformity hypotheses, and the averaged equation has x = 0 as a hyperbolic equilibrium point, then the timedependent equation has a unique almost periodic solution in a neighborhood of zero for sufficiently small E > 0. This solution is uniformly asymptotically stable if, in addition, x = 0 is asymptotically stable for the averaged equation. The hyperbolicity of x = 0 is clearly crucial, even for autonomous equations. For example, consider f (x, E) = -x3 +&x2. We obtain fo(x) = -x3,