Stochastic response analysis of a birhythmic oscillator under Poisson white noise excitation via path integration method

This paper investigates the stochastic response of a classical model of the birhythmic Van der Pol oscillator under Poisson white noise excitation. The improved path integration (PI) method is comprehensively derived in this paper and the probability density of the system is calculated using this method. Monte Carlo simulations are employed to validate the accuracy of the improved PI method. The findings reveal that the nonlinear parameters of the system significantly influence the mechanism by which Poisson white noise impacts the birhythmic oscillator. For the stationary response, we compare the effects of Gaussian white noise and Poisson white noise at equivalent intensities, and specifically analyze the mechanisms of two key parameters of Poisson white noise. This analysis offers general insights into the mechanisms by which Poisson noise impacts the birhythmic oscillator. For transient response, the occurrence time of birhythmicity, along with the system's probability distribution at different moments, are influenced by random perturbations, yet the nonlinear parameters of the system remain pivotal factors. This study provides a perspective for investigating the dynamic behavior of birhythmic oscillators under discontinuous stochastic disturbances.