## Abstract and Applied Analysis

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

#### Abstract

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system $\dot{x}=J{\nabla }_{x}H$, where $H\left(x,t,\epsilon \right)=\left(1/2\right)\beta \left({x}_{1}^{2}+{x}_{2}^{2}\right)+F\left(x,t,\epsilon \right)$ with $\beta \ne 0,{\partial }_{x}F\left(0,t,\epsilon \right)=O\left(\epsilon \right)$ and ${\partial }_{xx}F\left(0,t,\epsilon \right)=O\left(\epsilon \right)$ as $\epsilon \to 0$. 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2011, Number 1 (2011), Article ID 354063, 17 pages.

Dates
First available in Project Euclid: 15 March 2012

https://projecteuclid.org/euclid.aaa/1331818381

Digital Object Identifier
doi:10.1155/2011/354063

Mathematical Reviews number (MathSciNet)
MR2819767

Zentralblatt MATH identifier
1222.37049

#### Citation

Li, Jia; Xu, Junxiang. On the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter. Abstr. Appl. Anal. 2011 (2011), no. 1, Article ID 354063, 17 pages. doi:10.1155/2011/354063. https://projecteuclid.org/euclid.aaa/1331818381