The Stability of the Elliptic Equilibrium of Planar Quasi-periodic Hamiltonian Systems

In this paper, we study the planar Hamiltonian systemwhere where f is real analytic in x and θ,A（θ） is a 2×2real analytic symmetric matrix,J=（1^-1） and w is a Diophantine vector. Under the assumption that the unperturbed system reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.