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

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

2007-09-27

arxiv.org/abs/0709.4431v1

Mathematics

Classical Analysis and ODEs

To Appear in Commun. Pure Appl. Anal

Scientific paper

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. 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 e>0 tends to 0. Assuming the existence of a T-periodic solution x_e of the perturbed system and its convergence to x_0 as e->0, the paper establishes the existence of d_e->0 as e->0 such that \|x_e(t+d_e)-x_0(t)\|<=eM for some M>0 and any e>0 sufficiently small.

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