[1] Jiang Shunjun, Fang Fang,. Lagrangian stability of a class of second-order periodic systemswith nonlinear damping term [J]. Journal of Southeast University (English Edition), 2012, 28 (1): 130-134. [doi:10.3969/j.issn.1003-7985.2012.01.022]

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Lagrangian stability of a class of second-order periodic systemswith nonlinear damping term()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

28

Issue:

2012 1

Page:

130-134

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2012-03-30

Info

Title:

Lagrangian stability of a class of second-order periodic systemswith nonlinear damping term

Author(s):

Jiang Shunjun¹;  2;  Fang Fang³

¹Department of Mathematics, Southeast University, Nanjing 211189, China
²College of Sciences, Nanjing University of Technology, Nanjing 211816, China
³Department of Basic Course, Nanjing Institute of Technology, Na

Keywords:

reversible system;  KAM theorem;  boundedness of solutions

PACS:

O193

DOI:

10.3969/j.issn.1003-7985.2012.01.022

Abstract:

By the iteration of the KAM, the following second-order differential equation(Φ_pp(x′))′+F(x, x′, t)+ω^ppΦ_pp(x′)+α|x|^ll+e(x, t)=0 is studied, where Φ_pp(s)=|s|^p-2s, p>1, α>0 and ω>0 are positive constants, and l satisfies -1<ω<p+2. Under some assumptions on the parities of F(x, x′, t)and e(x, t), by a small twist theorem of reversible mapping, the existence of quasi-periodic solutions and boundedness of all the solutions are obtained.

References:

[1] Morris G R. A case of boundedness of Littlewood’s problem on oscillatory differential equations [J]. Bulletin of the Australian Mathematical Society, 1976, 14(1):71-93.
[2] Dieckerhoff R, Zehnder E. Boundedness of solutions via the twist theorem[J]. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1987, 14(4):79-95.
[3] Levi M. Quasiperiodic motions in superquadratic time-periodic potential[J].Communications in Mathematical Physics, 1991, 143(1):43-83.
[4] Liu B. Boundedness for solutions of nonlinear Hill’s equations with periodic forcing terms via Moser’s twist theorem [J]. Journal of Differential Equations, 1989, 79(2):304-315.
[5] Kunze M, Kupper T, Liu B. Boundedness and unboundedness of solutions for reversible scillatorsat resonance [J]. Nonlinearity, 2001, 14(5):1105-1122.
[6] Liu B. Quasi-periodic solutions of semilinear Liénard reversible oscillators [J]. Discrete and Continuous Dynamical Systems, 2005, 12(1):137-160.
[7] Liu B, Song J. Invariant curves of reversible mappings with small twist[J]. Acta Mathematics Sinic, 2004, 20(1):15-24.

Memo

Memo:

Biography: Jiang Shunjun(1979—), male, doctor, lecturer, jiangshunjun@yahoo.com.cn.
Foundation items: The National Natural Science Foundation of China(No.11071038), the Natural Science Foundation of Jiangsu Province(No.BK2010420).
Citation: Jiang Shunjun, Fang Fang. Lagrangian stability of a class of second-order periodic systems with nonlinear damping term[J].Journal of Southeast University(English Edition), 2012, 28(1):130-134.[doi:10.3969/j.issn.1003-7985.2012.01.022]

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