Transition behaviors of system energy in a bi-stable van Ver Pol oscillator with fractional derivative element driven by multiplicative Gaussian white noise

The stochastic P-bifurcation behavior of system energy in a bi-stable Van der Pol oscillator with fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging. Then, according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis in this paper, and the method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.