The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters
type: Item
instance of: scholarly article
title: The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters
zbMATH Open document ID: 1204.34073
DOI: 10.1155/2010/286040
author: Jifeng Chu
author: Ting Xia
published in: Abstract and Applied Analysis
publication date: 2010-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/232592
review text: Summary: Let \(a(t),b(t)\) be continuous \(T\)-periodic functions with \(\int^T_0 b(t)\,dt = 0\). We establish a 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 \[ x''+b(t)x'+a(t)x+c(t)x^{2n-1}+e(t,x)=0, \] where \(n\geq 2\), \(c\) 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 \(x=0\).
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 5791613
MaRDI profile type: Publication
OpenAlex ID: W2059716137
Wikidata QID: Q58649808
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description: scientific article; zbMATH DE number 5791613