Parametric influences of a modified fractional Van der Pol oscillator under Gaussian colored noise

We investigate the bistable dynamic behaviors of a fractional and modified Van der Pol system subjected to Gaussian colored noise, especially focusing on the control of bifurcation parameters. Firstly, according to the minimal mean square error concept and the generalized harmonic balance technique, the fractional derivative term can be expressed as a linear combination of the damping force and restoring force. Consequently, the initial system can be simplified as an equivalent integer-order Van der Pol system. And the Probability Density Function (PDF) of the system’s stationary amplitude can be obtained by utilizing the stochastic averaging method. Subsequently, the critical condition for stochastic P-bifurcation of system amplitude is determined by using the singularity theory. Finally, the qualitative analysis of the stationary PDF curves of amplitude is discussed in each region divided by the transition set curves. The agreement between the analytical solutions and the numerical outcomes obtained through Monte Carlo simulation confirms the theoretical analysis presented in this paper. Additionally, the results attained in this paper have a potential guidance to control the system’s response by optimizing the parametric design of the fractional-order controller.