Bifurcation Analysis of a Self-Sustained Birhythmic Oscillator Under Two Delays and Colored Noises

In this manuscript, an investigation on bifurcations induced by two delays and additive and multiplicative colored noises in a self-sustained birhythmic oscillator is presented, both theoretically and numerically, which serves for the purpose of unveiling extremely complicated nonlinear dynamics in various spheres, especially in biology. By utilizing the multiple scale expansion approach and stochastic averaging technique, the stationary probability density function (SPDF) of the amplitude is obtained for discussing stochastic bifurcations. With time delays, intensities and correlation time of noises regarded as bifurcation parameters, rich bifurcation arises. In the case of additive noise, it is identified that the bifurcations induced by the two delays are entirely distinct and longer velocity delay can accelerate the conversion rate of excited enzyme molecules. A novel type of P-bifurcation emerges from the process in the case of multiplicative colored noise, with the SPDF qualitatively changing between crater-like and bimodal distributions, while it cannot be generated when the multiplicative colored noise is coupled with additive noise. The feasibility and effectiveness of analytical methods are confirmed by the good consistency between theoretical and numerical solutions. This investigation may have practical applications in governing dynamical behaviors of birhythmic systems.