[1] Li Jia, Zhu Chunpeng,. On reducibility of a class of nonlinear quasi-periodic systemswith small perturbational parameters near equilibrium [J]. Journal of Southeast University (English Edition), 2012, 28 (2): 256-260. [doi:10.3969/j.issn.1003-7985.2012.02.022]

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On reducibility of a class of nonlinear quasi-periodic systemswith small perturbational parameters near equilibrium()

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

Volumn:

28

Issue:

2012 2

Page:

256-260

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2012-06-30

Info

Title:

On reducibility of a class of nonlinear quasi-periodic systemswith small perturbational parameters near equilibrium

Author(s):

Li Jia¹;  2;  Zhu Chunpeng²

¹Department of Mathematics, Southeast University, Nanjing 211189, China
²Mathematics and Physical Sciences Technology, Xuzhou Institute of Technology, Xuzhou 221008, China

Keywords:

quasi-periodic;  reducible;  non-resonance condition;  non-degeneracy condition;  KAM iteration

PACS:

O175.15

DOI:

10.3969/j.issn.1003-7985.2012.02.022

Abstract:

Consider the reducibility of a class of nonlinear quasi-periodic systems with multiple eigenvalues under perturbational hypothesis in the neighborhood of equilibrium. That is, consider the following system(·overx)=(A+εQ(t))x+εg(t)+h(x, t), where A is a constant matrix with multiple eigenvalues; h=O(x²)(x→0); and h(x, t), Q(t), and g(t)are analytic quasi-periodic with respect to t with the same frequencies. Under suitable hypotheses of non-resonance conditions and non-degeneracy conditions, for most sufficiently small ε, the system can be reducible to a nonlinear quasi-periodic system with an equilibrium point by means of a quasi-periodic transformation.

References:

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[2] Jorba A, Simó C. On the reducibility of linear differential equations with quasiperiodic coefficients [J]. Journal of Differential Equations, 1992, 98(1): 111-124.
[3] Xu Junxiang. On the reducibility of a class of linear differential equations with quasiperiodic coefficients [J]. Mathematika, 1999, 46(2): 443-451.
[4] Eliasson L H. Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation [J]. Communications in Mathematical Physics, 1992, 146(3): 447-482.
[5] Her Hailong, You Jiangong. Full measure reducibility for generic one-parameter family of quasi-periodic linear systems [J]. Journal of Dynamics and Differential Equations, 2008, 20(4): 831-866.
[6] Jorba A, Simó C. On quasi-periodic perturbations of elliptic equilibrium points [J]. SIAM Journal on Mathematical Analysis, 1996, 27(6): 1704-1737.
[7] Wang Xiaocai, Xu Junxiang. On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point [J]. Nonlinear Analysis, 2008, 69(7): 2318-2329.
[8] Bogoljubov N N, Mitropoliski J A, Samoilenko A M. Methods of accelerated convergence in nonlinear mechanics [M]. New York: Springer, 1976.
[9] Whitney H. Analytical extensions of differentiable functions defined in closed sets [J]. Transactions of the American Mathematical Society, 1934, 36(2): 63-89.

Memo

Memo:

Biography: Li Jia(1983—), female, graduate, lijia831112@163.com.
Citation: Li Jia, Zhu Chunpeng.On reducibility of a class of nonlinear quasi-periodic systems with small perturbational parameters near equilibrium[J].Journal of Southeast University(English Edition), 2012, 28(2):256-260.[doi:10.3969/j.issn.1003-7985.2012.02.022]

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