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

Jifeng Chu and Ting Xia

Abstract and Applied Analysis, 2010, vol. 2010, 1-12

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 .

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:286040

DOI: 10.1155/2010/286040

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