Non-Integrability and Infinite Branching of Solutions of 2DOF Hamiltonian Systems in Complex Plane of Time

It has been proved by S.L.Ziglin, for a large class of 2-degree-of-freedom (d.o.f) Hamiltonian systems, that transverse intersections of the invariant manifolds of saddle fixed points imply infinite branching of solutions in the complex time plane and the non-existence of a second analytic integral of the motion. Here, we review in detail our recent results, following a similar approach to show the existence of infinitely-sheeted solutions for 2 d.o.f. Hamiltonians which exhibit, upon perturbation, subharmonic bifurcations of resonant tori around an elliptic fixed point. Moreover, as shown recently, these Hamiltonian systems are non-integrable if their resonant tori form a dense set. These results can be extended to the case where the periodic perturbation is not Hamiltonian.