Accurate Phase After Slow Passage Through Subharmonic Resonance

Abstract

A strongly nonlinear oscillator with $O(\epsilon)$ damping and $O(\epsilon)$ sinusoidal forcing is considered. The frequency is energy dependent, permitting energy levels corresponding to subharmonic resonance. Before and after subharmonic resonance, equations for the energy and phase of the linebreak nonlinear oscillator are derived using multiphase averaging. The average energy and phase are shown to satisfy to sufficiently high order the same differential equations as occur without periodic forcing. The slow passage through a subharmonic resonance is analyzed. By matching the energy and phase to sufficiently high order, an $O(\epsilon)$ additional jump in the average energy across the subharmonic resonance layer is computed in addition to the previously known $O(\epsilon^{1 {/2}})$ jump in the average energy and the previously known O(1) jump in phase. The more accurate jump in energy is used to obtain an asymptotic approximation (whose error is small) of the phase of the nonlinear oscillator after a subharmonic resonance layer. A time shift for the average energy is computed which is equivalent to the entire jump in energy across a subharmonic resonance layer. The time shift accounts for the averaged energy after resonance. This time shift is shown to yield the correct phase of the nonlinear oscillator after resonance with an elementary constant phase adjustment chosen to be consistent with the jump in phase. After the subharmonic resonance, the average energy and phase are shown to be the same as the average energy and phase that would occur without the periodic forcing if the time shift (delay) and phase adjustment are included.