A note on stochastic phase lockings in a relaxation oscillator forced by a sine input with additive noise

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. Using the probability density functions of the first-passage time, the operator which governs a transition of an input phase density after one cycle of the oscillator is defined. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.