The stability of the elliptic equilibrium of planar quasi-periodic Hamiltonian systems

In this paper, we study the planar Hamiltonian system x = J(A(theta)x + del f(x,theta)), theta = omega, x is an element of R-2, theta is an element of T-d, where f is real analytic in x and theta, A(theta) is a 2 x 2 real analytic symmetric matrix, J = ((1) (-1)) and omega is a Diophantine vector. Under the assumption that the unperturbed system is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.