Nonhyperbolic homoclinic chaos

Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a Melnikov-type condition plus an additional assumption, the negatively and positively asymptotic sets persist under periodic perturbations, together with their infinitely many intersections on the Poincar\'e section. We also examine, by means of essentially the same procedure, the case of (heteroclinic) orbits tending to the infinity; this case includes in particular the classical Sitnikov 3--body problem.