Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

Abstract

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, 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 stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies 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.