Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator

TL;DR

This paper investigates the existence, uniqueness, and stability of periodic solutions in nonsmooth differential equations, specifically applying the findings to analyze the resonance behavior of a nonsmooth van der Pol oscillator.

Contribution

It extends classical results to nonsmooth systems and constructs amplitude dependence curves for stable periodic solutions in the nonsmooth van der Pol oscillator.

Findings

Constructed amplitude dependence curves for stable solutions

Compared resonance curves of nonsmooth and classical van der Pol oscillators

Established asymptotic stability results under Lipschitz conditions

Abstract

In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for a Lipschitz system with a small right hand side. Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our ones. By means of the results established we construct the curves of dependence of the amplitude of asymptotically stable $2\pi$--periodic solutions of the nonsmooth van der Pol oscillator on the detuning parameter and the amplitude of the perturbation. After, we compare the resonance curves obtained, with the resonance curves of the classical van der Pol oscillator which were first constructed by Andronov and Witt.