A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

A. Buică, J. Llibre, O. Makarenkov

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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 languageEnglish
Pages (from-to)415-421
JournalDifferential Equations and Dynamical Systems
Volume23
Issue number4
DOIs
Publication statusPublished - 26 Oct 2015

Keywords

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

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