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 $c>0$.