Periodic solutions for a class of singulary perturbated systems

TL;DR

This paper establishes conditions for the existence of periodic solutions in singularly perturbed differential systems using topological degree and averaging theories, relaxing previous stricter conditions.

Contribution

It introduces weaker conditions for guaranteeing periodic solutions in singularly perturbed systems compared to prior work, expanding applicability.

Findings

Existence of periodic solutions for small perturbation parameter e

Conditions based on topological degree and averaging theory

Relaxation of previous restrictive conditions

Abstract

In this paper we provide conditions to ensure the existence, for $e>0$ sufficiently small, of periodic solutions of given period $T>0$ in a prescribed domain $U$ for a class of singularly perturbed first order differential systems. Here $e>0$ is the perturbation parameter. Our approach, based on the topological degree theory and the averaging theory, permits to weaken the conditions in [K.R. Schneider, Vibrational control of singularly perturbed systems, in "Lectures Notes in Control and Information Science", 259, 397-408, Springer, London, 2001, Theorem 2] under which the existence of periodic solutions is proved.