Copyright © 2010 Jifeng Chu and Ting Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let a(t),b(t) be continuous T-periodic functions with ∫0Tb(t)dt=0. We establish one stability criterion for the linear damped oscillator x′′+b(t)x′+a(t)x=0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator x′′+b(t)x′+a(t)x+c(t)x2n-1+e(t,x)=0, where n≥2,c(t) is a continuous T-periodic function, e(t,x) is continuous T-periodic in t and dominated by the power x2n in a neighborhood of x=0.