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, it is proved
the branching off a two-parameter family of two-dimensional invariant tori of the normalised system,
whose normal behaviour depends intrinsically on the coefficients of its low-order terms. Thus, only
elliptic or elliptic together with parabolic and hyperbolic tori may detach form the resonant periodic
orbit. Both patterns are mentioned in the literature as the direct and, respectively, inverse quasiperiodic
Hopf bifurcation. In this report we focus on the direct case, which has many applications in
several fields of science. Thus, we present here a summary of the results, obtained in the framework
of KAM theory, concerning 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 characterisation of
the parameters labelling them, which can be selected with a very clear dynamical meaning. These
results include sharp quantitative estimates on the “density” of surviving tori, when the distance to
the resonant periodic orbit goes to zero, and state that the 4-dimensional Cantor manifold holding
these tori admits a Whitney-C^\infty extension. In addition, an application to the Circular Spatial
Three-Body Problem (CSRTBP) is reviewed.Peer Reviewe
