Hamiltonian four fold 1:1 resonance with two rotational symmetries

This paper deals with the analysis of Hamiltonian Hopf bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1-1-1-1 resonance), in the presence of two quadratic symmetries I1 and I2. The model is a generalization of the classical models obtained from regularized Kepler systems describing the parallel Stark and Zeeman effects. After normalization the truncated normal form gives rise to an integrable system which is analyzed using reduction to a one degree of freedom system. The Hamiltonian Hopf bifurcations are found using the ‘geometric method’ set up by one of the authors.