A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

Abstract

The goal of this paper is to study bifurcations of asymptotically stable 2π-periodic solutions in the forced asymmetric oscillatorü + εcu + u + εau + = 1+ελ cos t by means of a Lipschitz generalization of the second Bogolubov's theorem due to the authors. The small parameter ε > 0 is introduced in such a way that any solution of the system corresponding to ε = 0 is 2π-periodic. We show that exactly one of these solutions whose amplitude is λ √ a 2 +c 2 generates a branch of 2π-periodic solutions when ε > 0 increases. The solutions of this branch are asymptotically stable provided that c > 0.