**ABSTRACT**

The stochastic P-bifurcation behavior of bi-stability in a Duffing oscillator with fractional damping under multiplicative noise excitation is investigated. Firstly, in order to consider the influence of Duffing term, the nonlinear stiffness can be equivalent to a linear stiffness which is a function of the system amplitude, and then, using the principle of minimal mean square error, the fractional derivative term can be equivalent to a linear combination of damping and restoring forces, thus, the original system is simplified to an equivalent integer order Duffing system. Secondly, the system amplitude's stationary Probability Density Function (PDF) is obtained by stochastic averaging, and 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 PDF 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, and the method used in this paper can directly guide the design of the fractional order controller to adjust the behaviors of the system.

**KEYWORDS**

PAPER SUBMITTED: 2020-02-10

PAPER REVISED: 2020-06-20

PAPER ACCEPTED: 2020-06-20

PUBLISHED ONLINE: 2021-01-31

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