Optimal Subharmonic Entrainment of Weakly Forced Nonlinear Oscillators

Abstract

For many natural and engineered systems, a central function or design goal is the synchronization of one or more rhythmic or oscillating processes to an external forcing signal, which may be periodic on a different time-scale from the actuated process. Such subharmonic synchrony, which is dynamically established when $N$ control cycles occur for every $M$ cycles of a forced oscillator, is referred to as $N$:$M$ entrainment. In many applications, entrainment must be established in an optimal manner, for example, by minimizing control energy or the transient time to phase locking. We present a theory for deriving weak inputs that establish subharmonic $N$:$M$ entrainment of nonlinear oscillators, or of collections of rhythmic dynamical units, while optimizing such objectives. Ordinary differential equation models of oscillating systems are reduced to phase variable representations, each of which consists of a natural frequency and phase response curve. When such reduced models reasonably approximate the oscillatory system dynamics, formal averaging and the calculus of variations are applied to derive optimal subharmonic entrainment waveforms. The optimal entrainment of a canonical model for a spiking neuron is used to illustrate this approach, which can be extended to the class of smooth oscillatory dynamical systems with a strongly attractive limit cycle.