A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

© 2014, Foundation for Scientific Research and Technological Innovation. The goal of this paper is to study bifurcations of asymptotically stable $$2\pi $$2π-periodic solutions in the forced asymmetric oscillator $$\ddot{u}+\varepsilon c \dot{u}+u+\varepsilon a u^+=1+\varepsilon \lambda \cos t$$u¨+εcu˙+u+εa^u+=1+ελcost by means of a Lipschitz generalization of the second Bogolubov’s theorem due to the authors. The small parameter $$\varepsilon >0$$ε>0 is introduced in such a way that any solution of the system corresponding to $$\varepsilon =0$$ε=0 is $$2\pi $$2π-periodic. We show that exactly one of these solutions whose amplitude is $$\frac{\lambda }{\sqrt{a^2+c^2}}$$λ^a2+^c2 generates a branch of $$2\pi $$2π-periodic solutions when $$\varepsilon >0$$ε>0 increases. The solutions of this branch are asymptotically stable provided that $$c>0$$c>0.