On Perturbed Oscillators in 1 - 1 - 1 Resonance : The Case of Axially Symmetric Cubic

Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. We use a special set of action-angle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced ow of the normal form yields information on the original system. We illustrate the results by analysing the family of (arbitrary) axially symmetric cubic potentials.