Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

This paper investigates the stochastic P-bifurcation behavior of bistability in a generalized Van der Pol system with fractional time-delay feedback under additive and multiplicative Gaussian white noise excitations. First, using the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is transformed into an equivalent integer-order system. Second, the stationary probability density function of system amplitude is obtained by stochastic averaging, and according to singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the different types of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and the Monte Carlo simulation results verifies the theoretical analysis in this paper. The method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.