KAM aspects of the quasiperiodic Hamiltonian Hopf bifurcation

In this work we consider a 1:-1 non semi-simple resonant periodic orbit of a three-degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, we prove the branching o ® a two-parameter family of two-dimensional invariant tori of the normalized system, whose normal behavior depends intrinsically on the coe±cients of its low-order terms. Thus, only elliptic or elliptic together with parabolic and hyperbolic tori may detach from the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and inverse, respectively, quasiperiodic Hopf bifurcation. In this paper we focus on the direct case, which has many applications in several ¯elds of science. Our target is to prove, in the framework of KAM theory, the persistence of most of the (normally) elliptic tori of the normal form, when the whole Hamiltonian is taken into account, and to give a very precise characterization of the parameters labelling them, which can be selected with a very clear dynamical meaning. Furthermore, we give sharp quantitative estimates on the \density " of surviving tori, when the distance to the resonant periodic orbit goes to zero, and show that the 4-dimensional Cantor manifold holding them admits a Whitney-C1 extension. Due to the strong degeneracy of the problem, some standard KA