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Abstract and Applied Analysis
Volume 2010, Article ID 286040, 12 pages
Research Article

The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China

Received 27 May 2010; Accepted 11 August 2010

Academic Editor: Stefan Siegmund

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.


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 n2,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.