A General Theory of Resonance Between Weakly Coupled Oscillatory Systems

Resonant oscillations are investigated between several fullynon-linear oscillatory systems which are subject to a weak coupling.The equations of motion are obtained from a Hamiltonian whichis first expressed in terms of angle action variables, and thenaveraged. It is shown that this results in a reduced set ofequations, the reduction depending on the number of resonancesbetween the various systems. For example a resonance in a systemwith two degrees of freedom results in equations which are mathematicallyequivalent to those for one degree of freedom. The theory isillustrated by application to the forced oscillations of a simplependulum and to the resonant interaction between the two modesof oscillation of a double pendulum.