Asymptotic Stability of Periodic Solutions for Nonsmooth Differential Equations with Application to the Nonsmooth van der Pol Oscillator

Abstract

In this paper we study the existence, uniqueness, and asymptotic stability of the periodic solutions of the Lipschitz system $\dot{x}=\varepsilon g(t,x,\varepsilon)$, where $\varepsilon>0$ is small. Our results extend the classical second Bogoliubov theorem for the existence of stable periodic solutions to nonsmooth differential systems. As an application we prove the existence of asymptotically stable $2\pi$-periodic solutions of the nonsmooth van der Pol oscillator $\ddot{u}+\varepsilon\,(|u|-1)\,\dot{u}+(1+a\varepsilon)u=\varepsilon\lambda\sin t$. Moreover, we construct the so-called resonance curves that describe the dependence of the amplitude of these solutions as a function of the parameters a and $\lambda$. Finally we compare such curves with the resonance curves of the classical van der Pol oscillator $\ddot{u}+\varepsilon\,\bigl(u^2-1\bigr)\,\dot{u}+(1+a\varepsilon)u=\varepsilon\lambda\sin t$.