In my study of dynamical systems I just met this specific problem:
If a 2D autonomous system is governed by the Hamiltonian $ H(x,y)= \frac{A}{k}sin(kx)sin(\pi y) $ where $ A,k $ are non zero constants. We are asked to find the homoclinic, heteroclinic and closed (periodic) orbits.
Just to be clear here is the system:
$ \dot{x} = -\pi\frac{A}{k}sin(kx)cos(\pi y) $
$ \dot{y} = Acos(kx)sin(\pi y) $
And I was wondering if there an analytic method to do this? I cannot seem to find such a method online or in the literature, so I am in need of help on this. I thank all helpers.