Higher order analysis on the existence of periodic solutions in continuous differential equations via degree theory

TL;DR

This paper extends higher order analysis for periodic solutions to continuous, non-Lipschitz differential equations using degree theory, filling a gap between Lipschitz and discontinuous cases.

Contribution

It develops a higher order analysis framework for continuous non-Lipschitz differential equations using coincidence degree theory, providing new existence conditions.

Findings

Derived sufficient conditions for periodic solutions in non-Lipschitz systems

Applied results to higher order perturbations of harmonic oscillators

Extended degree theory methods to a broader class of differential equations

Abstract

Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.