Periodic solutions of periodically perturbed planar autonomous systems: A topological approach

TL;DR

This paper investigates the existence and properties of periodic solutions in perturbed planar autonomous systems with a focus on rational and irrational period ratios, using topological degree theory.

Contribution

It provides new conditions for the existence and characterization of periodic solutions in perturbed systems with rational period ratios, and shows nonexistence results for irrational ratios.

Findings

Existence of klT_0-periodic solutions when T_0/T_1 is rational

Conditions for these solutions to converge to the limit cycle

Nonexistence of such solutions when T_0/T_1 is irrational

Abstract

Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation e>0 sufficiently small, the existence of klT_0-periodic solutions x_e of the perturbed system which converge to the trajectory x_1 of the limit cycle as e->0. Moreover, we state conditions under which T=klT_0 is the least period of the periodic solutions x_e. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T_0/T_1 is an irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic solutions x_e of the perturbed system converging to x_1. The employed methods are based on the topological degree theory.