Relative equilibria and bifurcations in the generalized van der Waals 4D oscillator

A uniparametric 4-DOF family of perturbed Hamiltonian oscillators in 1:1:1:1 resonance is studied as a generalization for several models for perturbed Keplerian systems. Normalization by Lie transforms (only first order is considered here) as well as geometric reduction related to the invariants associated to the symmetries is used based on the previous work of the authors. A description is given for the lower dimensional relative equilibria in such normalized systems for which we introduce, in this context, the new concept of moment polytope. In addition bifurcations of relative equilibria corresponding to 3D tori are studied in some particular cases where we focus on Hamiltonian–Hopf bifurcations and bifurcations in the 3D van der Waals and Zeeman systems.