On stochastic response of fractional-order generalized birhythmic van der Pol oscillator subjected to delayed feedback displacement and Gaussian white noise excitation

This paper addresses the occurrence of P-bifurcation in a fractional-delay modified birhythmic van der Pol (BVDP) oscillator, for enzyme-substrate reactions in brain waves, under Gaussian white noise excitation. The minimum mean-square error is used to reduce the system to its equivalent integer-order nonlinear stochastic equation. An averaged Itô equation is obtained via the stochastic averaging method, with the amplitude being the solution of the Fokker-Planck-Kolmogorov equation. From the latter, the stationary density functions are found analytically. This helps to predict the appearance of birhythmicity theoretically and shows itself to respond to parameter changes, namely, fractional-orders, fractional coefficient, and noise intensity. There is an agreement between the theoretical solutions and the numerical solution, which confirms the accuracy of our predictions. In general, bifurcation scenarios are dominated by the changes in fractional orders, as strongly supported by the behaviors of the calculated potential barriers.