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

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system x˙=J∇xH, where H(x,t,ε)=(1/2)β(x12+x22)+F(x,t,ε) with β≠0,∂xF(0,t,ε)=O(ε) and ∂xxF(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