Perturbations of systems with periodic asymptotically stable orbits

Abstract

On an Tz-manifold X, let n be a dynamical system that has an asymptotically stable periodic orbit I: (In our usage, i‘ is asymptotically stable iff inside any neighborhood FV of r, there is an open invariant set V containing r such that, for any other neighborhood Z of r, there is a time T < GO with the property that ?T( b’ x [T, CO)) C Z.) Our principal result is that if U is any neighborhood of r in X, and m& is any dynamical system on X resulting from a. “sufficiently small” perturbation of n, then rLd has a periodic orbit contained in zi. This is a partial extension to dynamical systems of a classical result concerning autonomous differential equations of the form k = f(~, .u) whose first variation satisfies certain conditions [2, p. 3521. In the classical case, the periodic solution of the perturbed system is also asymptotically orbitally stable. We include an example of a dynamical system on E3 with a single asymptotically stable (in the sense of the definirion above) orbit. Small perturbations of this system are constructed whose resulting periodic orbit is not even stable, much less asymptotically stable. A dynamical system on a topological space X is a continuous mapping ‘ir: X x R -+ X (R = real number line) such that z(“Y, 0) = x k’,z E X and “(S, t1 + tz) = T+T(x, t& t3) vx E x and t1 , tz E R. In this paper we will write ~(x, t) as ZH, and use ArrB to denote n(a y B), ip C X, B CR. We will use without reference certain elementary properties of dynamical systems. The reader will find in either [I, 4, or 61 a good introduction to dynamical systems. The set of all dynamical systems on X with the compactopen topology, we term D(X).