Sebastián Ferrer, Heinz Hanßmann, Jesús Palacián, Patricia Yanguas
On Perturbed Oscillators in 1-1-1 Resonance :
The Case of Axially Symmetric Cubic Potentials
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ABSTRACT.  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 flow of the
normal form yields information on the original system. We illustrate the
results by analysing the family of (arbitrary) axially symmetric cubic
potentials.