Bifurcations in a fractional birhythmic biological system with time delay

This paper presents a bifurcation analysis in a stochastic time-delayed birhythmic oscillator containing fractional derivative. Analytical criterion for birhythmic region is obtained by adopting approximate methods and verified by numerical results. The detailed parameter space study reveals that the rhythmic properties of the oscillator can be efficiently controlled by the fractional order damping. The maximum birhythmic region can be detected in parameter space (α β) by choosing an approximate fractional order and a larger absolute value of fractional coefficient is conducive to a smaller optimal fractional order. In the stochastic case, the smaller fractional order, the smaller the critical value of time delay and its feedback required for the occurrence of birhythmicity. Similarly, a smaller time delay admits to a smaller value of fractional order and absolute value of fractional coefficient. Astonishingly, fractional derivative and time delay have opposite effects on the direction of the transition. Our study sheds new insight lights on the understanding of nontrivial effects of fractional derivative on a birhythmic oscillator.