Stochastic Bifurcations in a Birhythmic Biological Model with Time-Delayed Feedbacks

Abstract

This paper presents a study on stochastic bifurcations in a time-delayed birhythmic oscillator possessing a bistability mode with coexistence of two stable limit cycles in the deterministic case. Relying on the approximate methods, the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity have been exhibited to characterize the qualitative properties of the system. The investigations indicate that the birhythmic region increases firstly and then decreases when time delay is monotonically varied. Further, system parameters and noise level can induce the appearance of stochastic P-bifurcation. Similar bifurcations can be induced by changing the strength of time delay and delayed feedbacks in displacement and velocity. Interestingly, joint PDF will reflect a more complex regime. And the role of the strength of the delayed velocity feedback on stochastic bifurcation is sensitive to the value of time delay. Numerical simulations are carried out for prototype models, which show basic agreement with our theoretical predictions.