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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 354063, 17 pages
http://dx.doi.org/10.1155/2011/354063
Research Article

On the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter

Department of Mathematics, Southeast University, Nanjing 210096, China

Received 1 December 2010; Revised 18 April 2011; Accepted 25 May 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 Jia Li and Junxiang Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system ̇ 𝑥 = 𝐽 𝑥 𝐻 , where 𝐻 ( 𝑥 , 𝑡 , 𝜀 ) = ( 1 / 2 ) 𝛽 ( 𝑥 2 1 + 𝑥 2 2 ) + 𝐹 ( 𝑥 , 𝑡 , 𝜀 ) with 𝛽 0 , 𝜕 𝑥 𝐹 ( 0 , 𝑡 , 𝜀 ) = 𝑂 ( 𝜀 ) and 𝜕 𝑥 𝑥 𝐹 ( 0 , 𝑡 , 𝜀 ) = 𝑂 ( 𝜀 ) as 𝜀 0 . Without any nondegeneracy condition with respect to ε, we prove that for most of the sufficiently small ε, by a quasi-periodic symplectic transformation, it can be reduced to a quasi-periodic Hamiltonian system with an equilibrium.