SEPARATRIX SURFACES AND INVARIANT-MANIFOLDS OF A CLASS OF INTEGRABLE HAMILTONIAN-SYSTEMS AND THEIR PERTURBATIONS

The study is made of the foliations of the energy levels of a class of integrable Hamiltonian systems by the sets of constant energy and ang ular momentum, including a classification of the topological bifurcati ons and a dynamical characterization of the critical leaves (separatri x surfaces) of the foliation. Then, Hamiltonian perturbations of this class of integrable Hamiltonians are considered, and conditions are gi ven for the persistence of the separatrix structure of the foliations, and for the existence of transversal ejection-collision orbits of the perturbed system. Finally, we consider a class of non-Hamiltonian per turbations of a family of integrable systems of the type studied befor e, and we prove the persistence of ''almost all'' the tori and cylinde rs that foliate the energy levels of the unperturbed system as a conse quence of KAM theory.