Existence of quasi-periodic solutions of the real pendulum equation

被引：5

作者：

Lu, Lin ^{[1
]}

Li, Xuemei ^{[1
]}

机构：

[1] Hunan Normal Univ, Dept Math, Changsha 410081, Hunan, Peoples R China

来源：

CHAOS SOLITONS & FRACTALS | 2014年 / 62-63卷

关键词：

INVARIANT TORI; COMPLEX DYNAMICS; KAM THEOREM; PERSISTENCE; REDUCIBILITY;

D O I：

10.1016/j.chaos.2014.03.003

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

The pendulum equation x = -delta x -delta x (1 + f(o) cos omega(1)t) sinx +f(1), sin omega(2)t is considered in this paper, where f(o),f(1) and delta are small real parameters, the ratio of oh and oh is irrational, and frequencies omega(1), and omega(2) satisfy the Diophantine condition. The unperturbed system (f(o) = f(1) = delta = 0) has several fixed points for different parameter alpha. We use KAM theory to prove that the perturbed system possesses quasi-periodic solutions in neighborhoods of those fixed points. (c) 2014 Elsevier Ltd. All rights reserved.

引用

页码：23 / 33

页数：11

共 33 条

[11] Rotating solutions of the parametrically excited pendulum [J].

Garira, W ;

Bishop, SR .

JOURNAL OF SOUND AND VIBRATION, 2003, 263 (01) :233-239

[12]

Haro A, 2006, DISCRETE CONT DYN-B, V6, P1261

[13] Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems [J].

Her, Hai-Long ;

You, Jiangong .

JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 2008, 20 (04) :831-866

[14]

Housner G. W., 1963, Bulletin of the Seismological Society of America, V53, P403, DOI DOI 10.1785/BSSA0530020403

[15] On the persistence of lower-dimensional invariant hyperbolic tori for smooth Hamiltonian systems [J].

Huang, DB ;

Liu, ZR .

NONLINEARITY, 2000, 13 (01) :189-202

[16] Complex dynamics in pendulum equation with parametric and external excitations II [J].

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China ;

不详 ;

不详 .

Int. J. Bifurcation Chaos, 2006, 10 (3053-3078)

[17] Complex dynamics in pendulum equation with parametric and external excitations I [J].

Jing, Zhujun ;

Yang, Jianping .

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2006, 16 (10) :2887-2902

[18] On quasi-periodic perturbations of elliptic equilibrium points [J].

Jorba, A ;

Simo, C .

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 1996, 27 (06) :1704-1737

[19] ON THE REDUCIBILITY OF LINEAR-DIFFERENTIAL EQUATIONS WITH QUASI-PERIODIC COEFFICIENTS [J].

JORBA, A ;

SIMO, C .

JOURNAL OF DIFFERENTIAL EQUATIONS, 1992, 98 (01) :111-124

[20] On the persistence of lower dimensional invariant tori under quasi-periodic perturbations [J].

Jorba, A ;

Villanueva, J .

JOURNAL OF NONLINEAR SCIENCE, 1997, 7 (05) :427-473

← 1 2 3 4 →