Homoclinic orbits to invariant tori near a homoclinic orbit to center–center–saddle equilibrium

We consider a perturbation of an integrable Hamiltonian vector field with three degrees of freedom with a center–center–saddle equilibrium having a homoclinic orbit or loop. With the help of a Poincaré map (chosen based on the unperturbed homoclinic loop), we study the homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic KAM tori, on the center manifold near the equilibrium. If the perturbation is such that the homoclinic loop is preserved (i.e. the perturbation also has a homoclinic loop inherited from the unperturbed one), we establish that, in general, the manifolds intersect along 8, 12 or 16 transverse homoclinic orbits. On the other hand, in a more generic situation (the loop is not preserved; a condition for this fact is obtained by means of a Melnikov-like method), the manifolds intersect along four transverse homoclinic orbits, though a small neighborhood of the loop has to be excluded. In a first approximation, those homoclinic orbits can be detected as nondegenerate critical points of a Melnikov potential defined on the 2-torus. The number of homoclinic orbits is given by Morse theory applied to the Melnikov potential.