Two-state Markovian theory of input–output frequency and phase synchronization

A Markovian dichotomic system driven by a deterministic time-periodic force is analyzed in terms of the statistical properties of the switching events between the states. The consideration of the counting process of the switching events leads us to define a discrete phase. We obtain expressions for the instantaneous output frequency and phase diffusion associated to the dichotomic process, as well as for their cycle averages. These expressions are completely determined by the rates of escape from both states. They are a convenient starting point for the study of the stochastic frequency and phase synchronization in a wide range of situations (both classical and quantum) in which two metastable states are involved.