On perturbed oscillators in 1-1-1 resonance: The case of axially symmetric cubic potentials
From authors' 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. To this end, we use a special set of action-angle variables, as well as conveniently chosen generators of the ring of invariant functions. Also, we analyse the 2-parameter family of (arbitrary) axially symmetric cubic potentials. This family has rich dynamics, displaying all local bifurcations of co-dimension one. With the exception of six ratios of the parameter values, the dynamic behavior close to the origin turns out to be completely determined by the normal form of order 1.
scientific article; zbMATH DE number 1758073
scientific article