## Abstract and Applied Analysis

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

#### Abstract

Let $a(t),b(t)$ be continuous $T$-periodic functions with $\int_{0}^{T}b(t)dt=0$. We establish one stability criterion for the linear damped oscillator ${x}^{\prime \prime}+b(t){x}^{\prime }+a(t)x=0$. 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 ${x}^{\prime \prime }+b(t){x}^{\prime}+a(t)x+c(t){x}^{2n-1}+e(t,x)=0$, where $n\geq 2,c(t)$ is a continuous $T$-periodic function, $e(t,x)$ is continuous $T$-periodic in $t$ and dominated by the power ${x}^{2n}$ in a neighborhood of $x=0$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 286040, 12 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620760

Digital Object Identifier
doi:10.1155/2010/286040

Mathematical Reviews number (MathSciNet)
MR2680411

Zentralblatt MATH identifier
1204.34073

#### Citation

Chu, Jifeng; Xia, Ting. The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters. Abstr. Appl. Anal. 2010 (2010), Article ID 286040, 12 pages. doi:10.1155/2010/286040. https://projecteuclid.org/euclid.aaa/1288620760