[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()

一类带有非线性阻尼项的二阶周期系统的Lagrangian稳定性

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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

一类带有非线性阻尼项的二阶周期系统的Lagrangian稳定性

Author(s):

Jiang Shunjun^{1, 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

江舜君^{1, 2}, 方芳³

¹东南大学数学系, 南京 211189; ²南京工业大学理学院, 南京 211816; ³南京工程学院基础部, 南京 211167

Keywords:

reversible system;  KAM theorem;  boundedness of solutions

可逆系统;  KAM定理;  解的有界性

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.

用KAM迭代方法研究了下列二阶微分方程:(Φ_pp(x′))′+F(x, x′, t)+ω^ppΦ_pp(x′)+α|x|^ll+e(x, t)=0, 其中, Φ_pp(s)=|s|^p-2s, p>1, α>0, ω>0为正常数, l满足-1<ω<p+2.当F(x, x′, t)与e(x, t)的导数满足一定条件时, 利用可逆映射的小扭转定理得到拟周期解的存在性与所有解的有界性.

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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