On Hamiltonian systems with homoclinic orbit to a saddle-center

We consider a real analytic two degrees of freedom Hamiltonian system possessing a homoclinic orbit to a saddle-center equilibrium (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such the system and show that in nonresonance case there are countable sets of multi-round homoclinic orbits to a saddle-center. We also find families of periodic orbits, accumulating at homoclinic orbits.