On the continuation of an invariant torus in a family with rapid oscillations

A persistence theorem for attracting invariant tori for systems subjected t o rapidly oscillating perturbations is proved. The singular nature of these perturbations prevents the direct application of the standard persistence results for normally hyperbolic invariant manifolds. However, as is illustr ated in this paper, the theory of normally hyperbolic invariant manifolds, when combined with an appropriate continuation method, does apply.