Small parameter perturbations of nonlinear periodic systems

TL;DR

This paper studies how small parameter perturbations affect the existence of periodic solutions in nonlinear periodic systems, using topological degree theory and linearized system analysis.

Contribution

It introduces new conditions based on boundary behavior of linearized systems to guarantee periodic solutions in perturbed nonlinear systems.

Findings

Conditions for existence of periodic solutions are established.

Application demonstrated on the van der Pol equation.

Method leverages topological degree theory for nonlinear analysis.

Abstract

In this paper we consider a class of nonlinear periodic differential systems perturbed by two nonlinear periodic terms with multiplicative different powers of a small parameter $e>0$. For such a class of systems we provide conditions which guarantee the existence of periodic solutions of given period $T>0$. These conditions are expressed in terms of the behaviour on the boundary of an open bounded set $U$ of $R^n$ of the solutions of suitably defined linearized systems. The approach is based on the classical theory of the topological degree for compact vector fields. An application to the existence of periodic solutions to the van der Pol equation is also presented.