The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters
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 equil...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/286040 |
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| author | Jifeng Chu Ting Xia |
| author_facet | Jifeng Chu Ting Xia |
| author_sort | Jifeng Chu |
| collection | DOAJ |
| description | 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 n≥2,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. |
| format | Article |
| id | doaj-art-ccd45a2651ed440796706d850bcfb84e |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-ccd45a2651ed440796706d850bcfb84e2025-02-03T01:11:35ZengWileyAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/286040286040The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic ParametersJifeng Chu0Ting Xia1Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, ChinaDepartment of Mathematics, College of Sciences, Hohai University, Nanjing 210098, ChinaLet 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 n≥2,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.http://dx.doi.org/10.1155/2010/286040 |
| spellingShingle | Jifeng Chu Ting Xia The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters Abstract and Applied Analysis |
| title | The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters |
| title_full | The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters |
| title_fullStr | The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters |
| title_full_unstemmed | The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters |
| title_short | The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters |
| title_sort | lyapunov stability for the linear and nonlinear damped oscillator with time periodic parameters |
| url | http://dx.doi.org/10.1155/2010/286040 |
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