Minimum energy subharmonic synchronization of an uncertain nonlinear oscillator

The synchronization of one or more rhythmic or oscillating processes to an external forcing signal is a central design goal for many engineered systems, as well as a notable function of many natural processes. Such phenomena may involve subharmonic synchrony, in which the stimulating input is periodic on a different time-scale from the actuated process. Specifically, the establishment of a synchronized state in which N control input cycles occur for every M cycles of a forced oscillator is referred to as N∶M entrainment. By applying phase variable reduction, formal averaging, and the calculus of variations, we derive minimum-energy inputs for N∶M entrainment of weakly forced nonlinear oscillators with parameter uncertainty. The Van der Pol oscillator and the Hodgkin-Huxley neuron model are examined as examples.