Matching across subharmonic resonances

A strongly nonlinear oscillator is analyzed with an O(e) dissipative perturbation and an O(e) periodic forcing. Multiphase averaging determines the modulations of the energy and phase, but is known to fail near subharmonic resonance layers. We include an O(e) jump in the energy across each subharmonic resonant layer which is needed to determine the phase after resonance. This is in addition to the usual O(e 1/2 ) jump in energy determined by the method of matched asymptotic expansions. We introduce a time shift for the energy after subharmonic resonance which is equivalent to the jump in energy. We show that the phase after a subharmonic resonance is described by the same time shift if an elementary phase adjustment is made.