[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()

一类具有小扰动参数的非线性拟周期系统 在平衡点附近的可约化性

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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

一类具有小扰动参数的非线性拟周期系统 在平衡点附近的可约化性

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

李佳¹;  2;  朱春鹏²

¹东南大学数学系, 南京 211189; ²徐州工程学院数学与物理科学学院, 徐州 221008

Keywords:

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

拟周期;  可约化性;  非共振条件;  非退化条件;  KAM迭代

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.

考虑一类有重特征值的非线性拟周期系统在小扰动下平衡点附近的可约化性问题, 也就是研究(·overx)=(A+εQ(t))x+εg(t)+h(x, t), 其中A可以是具有重特征值的常数矩阵;h=O(x²)(x→0);h(x, t), Q(t)和g(t)关于t是解析拟周期的, 且有相同的频率.在某些非共振条件及非退化条件下, 对充分小的大多数ε, 通过仿线性拟周期变换, 系统可约化为具有平衡点的非线性拟周期系统.

References:

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[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.
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[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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