On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes

We consider an autonomous system in R-n having a limit cycle x(0) of period T > 0 which is nondegenerate in a suitable sense, ( see Definition 2.1). We then consider the perturbed system obtained by adding to the autonomous system a T- periodic, not necessarily differentiable, term whose amplitude tends to 0 as a small parameter epsilon > 0 tends to 0. Assuming the existence of a T-periodic solution x(epsilon) of the perturbed system and its convergence to x(0) as epsilon -> 0, the paper establishes the existence of Delta(epsilon) -> 0 as epsilon -> 0 such that parallel to x(epsilon)(t + Delta(epsilon)) - x(0)(t)parallel to <= epsilon M for some M > 0 and any epsilon > 0 sufficiently small. This paper completes the work initiated by the authors in [ ] and [ ]. Indeed, in [ ] the existence of a family of T-periodic solutions x(epsilon) of the perturbed system considered here was proved. While in [ ] for perturbed systems in R-2 the rate of convergence was investigated by means of the method considered in this paper.