Relative equilibria and bifurcations in the generalized Van der Waals 4-D 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 previous work of the authors. A description is given of the lower dimensional relative equilibria in such normalized systems for which we introduce the, in this context, new concept of moment polytope. In addition bifurcations of relative equilibria corresponding to three dimensional tori are studied in some particular cases where we focus on Hamiltonian Hopf bifurcations and bifurcations in the 3-D van der Waals and Zeeman systems. Keywords: Nonlinear dynamical system; Hamiltonian system; Bifurcation; Reduction; Symmetry; Hamiltonian Hopf bifurcation; Zeeman effect; van der Waals system