Just wondering if you have

dx/dt = y^3 - y

dy/dt = -x

then you can find the Hamiltonian which is H(x,y) = (x^2)/2 +(y^4)/2 -(y^2)/2

My question is what does the hamiltonian say about the nature of the steady states i know that (0,0) is a saddle and (0,1) is a centre

My second question is how can you describe the nature of the critical points of H(x,y). Im assuming it might help to draw a phase plane

thanks