A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

Abstract

The goal of this paper is to study bifurcations of asymptotically stable \(2\pi \)-periodic solutions in the forced asymmetric oscillator \(\ddot{u}+\varepsilon c \dot{u}+u+\varepsilon a u^+=1+\varepsilon \lambda \cos t\) by means of a Lipschitz generalization of the second Bogolubov’s theorem due to the authors. The small parameter \(\varepsilon >0\) is introduced in such a way that any solution of the system corresponding to \(\varepsilon =0\) is \(2\pi \)-periodic. We show that exactly one of these solutions whose amplitude is \(\frac{\lambda }{\sqrt{a^2+c^2}}\) generates a branch of \(2\pi \)-periodic solutions when \(\varepsilon >0\) increases. The solutions of this branch are asymptotically stable provided that \(c>0\).