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

TL;DR

This paper investigates how periodic solutions of autonomous systems behave under small periodic perturbations, establishing bounds on their convergence as the perturbation amplitude approaches zero.

Contribution

It provides a new quantitative estimate on the convergence rate of perturbed periodic solutions to the original limit cycle as perturbations vanish.

Findings

Existence of a small shift d_e approaching zero as perturbation diminishes.

Bound on the difference between perturbed and unperturbed solutions involving the perturbation size.

Convergence of the perturbed solution to the original limit cycle with an explicit error bound.

Abstract

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.