Bifurcations of the Hamiltonian fourfold 1:1 resonance with toroidal symmetry

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-centre bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries ¿ and L1. When we nor- malize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a 3-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing with the Hamiltonian Hopf bifurcation the `geometric approach' is used, following the steps set up by one of the authors in the context of 3-DOF systems. In order to see the interplay between integrals and physical param- eters in the analysis of bifurcations, we consider as perturbation a one-parameter family, which in particular includes one of the classical Stark-Zeeman models (par- allel case) in 3 dimension