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

Abstract

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system $\dot{x}=J{\nabla }_{x}H$, where $H(x,t,ϵ)=(1/2)\beta (x_1^2+x_2^2)+F(x,t,ϵ)$ with $\beta \ne 0,{\partial }_{x}F(0,t,ϵ)=O\left(ϵ\right)$ and ${\partial }_{xx}F(0,t,ϵ)=O\left(ϵ\right)$ as $ϵ\to 0$. 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.