Abstract and Applied Analysis

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

Jifeng Chu and Ting Xia

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Abstract

Let a ( t ) , b ( t ) be continuous T -periodic functions with 0 T b ( t ) d t = 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 ) x 2 n - 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 x 2 n 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

Permanent link to this document
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


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