September  2011, 16(2): 423-443. doi: 10.3934/dcdsb.2011.16.423

Lyapunov stability for conservative systems with lower degrees of freedom

1. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China

2. 

Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing 100084

3. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received  February 2010 Revised  October 2010 Published  June 2011

It is a central theme to study the Lyapunov stability of periodic solutions of nonlinear differential equations or systems. For dissipative systems, the Lyapunov direct method is an important tool to study the stability. However, this method is not applicable to conservative systems such as Lagrangian equations and Hamiltonian systems. In the last decade, a method that is now known as the 'third order approximation' has been developed by Ortega, and has been applied to particular types of conservative systems including time periodic scalar Lagrangian equations (Ortega, J. Differential Equations, 128(1996), 491-518). This method is based on Moser's twist theorem, a prototype of the KAM theory. Latter, the twist coefficients were re-explained by Zhang in 2003 through the unique positive periodic solutions of the Ermakov-Pinney equation that is associated to the first order approximation (Zhang, J. London Math. Soc., 67(2003), 137-148). After that, Zhang and his collaborators have obtained some important twist criteria and applied the results to some interesting examples of time periodic scalar Lagrangian equations and planar Hamiltonian systems. In this survey, we will introduce the fundamental ideas in these works and will review recent progresses in this field, including applications to examples such as swing, the (relativistic) pendulum and singular equations. Some unsolved problems will be imposed for future study.
Citation: Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423
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show all references

References:
[1]

PhD Thesis, Tsinghua University, Beijing, 2008. Google Scholar

[2]

J. Differential Equations, 247 (2009), 530-542. doi: 10.1016/j.jde.2008.11.013.  Google Scholar

[3]

J. Math. Anal. Appl., 355 (2009), 830-838. doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar

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Abstr. Appl. Anal., 2010 (2010), Art. ID 286040, 12 pp.  Google Scholar

[5]

Discrete Contin. Dyn. Syst. A, 21 (2008), 1071-1094. doi: 10.3934/dcds.2008.21.1071.  Google Scholar

[6]

Z. Angew. Math. Phys., 57 (2006), 183-204. doi: 10.1007/s00033-005-0020-y.  Google Scholar

[7]

Discrete Contin. Dyn. Syst. A, 29 (2011), 169-192. doi: 10.3934/dcds.2011.29.169.  Google Scholar

[8]

Robert E. Krieger Publishing Co., New York, 1980.  Google Scholar

[9]

AMS Translations, Ser. 2, 1 (1955), 163-187.  Google Scholar

[10]

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[12]

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[13]

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[14]

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