I believe the Hamiltonian is actually . The steady states (i.e. critical points) occur at (x,y) = (0,0), (0,1), and (0,-1). Isn't (0,0) the center and (0,1) and (0,-1) the saddles?
The potential function, has the following properties:
Therefore V(0) is a minimum, and hence a stable point in the system, and V(1) and V(-1) are maxima in the potential function, and hence unstable.
Not sure this is getting at your question...