Time delay feedbacks enhanced bifurcation in the birhythmic fractional self-sustained system subjected to correlated noise

The response of the birhythmic fractional Van der Pol (VdP) self-sustained oscillator driven both by time delay feedbacks and correlated noise is considered. We mainly focus on the effects of time-delay feedback parameter and fractional damping on stochastic properties of the birhytmic system, such as P-bifurcation. The minimum mean-square error is used to reduce the system to its equivalent integer-order nonlinear stochastic equation. An analytical approximation is adopted to obtain the system behavior and it is compared with numerical simulation. The delay self-control feedbacks are transformed into state variables without delay by means of the harmonic approximation. The maximum extension of the control parameters to have a birhythmic solution can be thus determined by an appropriated choice of the fractional order and delay time parameters. In particular, when the fractional damping is lowered below 2, the birhythmic region disappears. However, the influence of the fractional damping changes as a function of the correlation noise parameter. Instead, the delay feedback and the fractional derivative are not so effective on the transition. The input delay and fractional derivative can be greatly enhanced by the nonlinear interaction. Specifically, the amplitude of the induced oscillations can be much larger than that of the input fractional with the cooperation of an appropriated time delay and nonlinear terms. It is then shown that the time delay feedback parameters and the fractional damping also act as bifurcation parameters.