Stochastic phase transition operator.

In this study a Markov operator is introduced that represents the density evolution of an impulse-driven stochastic biological oscillator. The operator's stochastic kernel is constructed using the asymptotic expansion of stochastic processes instead of solving the Fokker-Planck equation. The Markov operator is shown to successfully approximate the density evolution of the biological oscillator considered. The response of the oscillator to both periodic and time-varying impulses can be analyzed using the operator's transient and stationary properties. Furthermore, an unreported stochastic dynamic bifurcation for the biological oscillator is obtained by using the eigenvalues of the product of the Markov operators.