Removal of resonances by rotation in linearly degenerate two-dimensional oscillator systems

A system of two nonlinearly interacting, resonant harmonic oscillators is investigated, seeking transformation to approximate action-angle variables in the vicinity of the equilibrium via the canonical perturbation theory. A variety of polynomial perturbations dependent on parameters is considered. The freedom of choice of the zero-order approximation characteristic of a linearly degenerate (resonant) system is used to cancel lower-order resonant terms in the canonical perturbation series. It is found that the cancellation of the resonant terms is only possible for particular values of parameters of the interaction term. These special sets of parameters include all the cases with the Panlev́ property.