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

© 2009 Society for Industrial and Applied Mathematics. In this paper we study the existence, uniqueness, and asymptotic stability of the periodic solutions of the Lipschitz system x = εg(t, x, ε), where ε > 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 2π-periodic solutions of the nonsmooth van der Pol oscillator ü + ε (|u| - 1) u + (1 + aε)u = ελ 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 λ. Finally we compare such curves with the resonance curves of the classical van der Pol oscillator ü + ε (u2 - 1) u + (1 + aε)u = ελ sin t.