# A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

A. Buică, J. Llibre, O. Makarenkov

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

## Abstract

© 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+εau+=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 $$\frac{\lambda }{\sqrt{a^2+c^2}}$$λa2+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.
Original language English 415-421 Differential Equations and Dynamical Systems 23 4 https://doi.org/10.1007/s12591-014-0199-5 Published - 26 Oct 2015

## Keywords

• Asymptotic stability
• Jumping nonlinearity
• Method of averaging
• Periodic solutions