Internal Resonances In Nonlinearly-Coupled Oscillators

The present paper gives an analytical perturbation theory to treat the problem of internal resonances of higher orders in systems of nonlinearly-coupled oscillators. The problem of two nonlinearly-coupled oscillators is treated first. It is found even for higher-order internal resonances that whereas the actions of the individual oscillators are nearly constant when the system does not show an internal resonance, the total action of the uncoupled system is nearly constant when the system undergoes an internal resonance and allows for a significant redistribution of action between the two oscillators — a result which is known to hold for the lowest-order internal resonance. We will then use these results to make a brief discussion of a possible relation between energy sharing and the onset of ergodicity for non-integrable systems