Subharmonic oscillations in stochastic systems under periodic driving

Subharmonic response is a well known phenomena in, e.g., deterministic nonlinear dynamical systems. We investigate the conditions under which such subharmonic oscillations can persist for a long time in open systems with stochastic dynamics due to thermal fluctuations. In contrast to stochastic autonomous systems in a stationary state, for which the number of coherent oscillations is fundamentally bounded by the number of states in the underlying network, we demonstrate that in periodically driven systems, subharmonic oscillations can, in principle, remain coherent forever, even in networks with a small number of states. We also show that, {\sl inter alia}, the thermodynamic cost rises only logarithmically with the number of coherent oscillations in a model calculation and that the possible periods of the persistent subharmonic response grow linearly with the number of states. We argue that our results can be relevant for biochemical oscillations and for stochastic models of time-crystals.