Approximation Dynamics and the Stability of Invariant Sets

We introduce the concept of a weakly, normally hyperbolic set for a system of ordinary differential equations. This concept includes the notion of a hyperbolic flow, as well as that of a normally hyperbolic invariant manifold. Moreover, it has the property that it is closed under finite set products. Consequently, the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly, normally hyperbolic sets. Our main objective is to show that under a smallC1-perturbation, a weakly, normally hyperbolic setKis preserved by a homeomorphism, where the imageKYis a compact invariant set, with a related hyperbolic structure, for the perturbed equation. In addition, the homeomorphism is close to the identity inC0, 1and the perturbed dynamics onKYare close to the original dynamics onK.