Probabilistic evolution analysis and first passage analysis of a class of stochastic dynamic systems with fractional derivative based on Complex Fractional Moment method

In this paper, the application of complex fractional moment(CFM) method in the stochastic dynamic systems with Caputo-type fractional derivative under the excitation of Gaussian white noise are investigated. By the approximation method, the FPK equation governing the amplitude is derived, and the semi-analytic solution of FPK equation is obtained by CFM method. The results are verified by numerical simulation. Meanwhile, the probability evolution of transient Probability Density Function(PDF) is analyzed when the stochastic bifurcation induced by fractional coefficient for the first time. In addition, the relation of equivalence between first passage time(FPT) and CFM is established by a novel method, the accuracy of this method is verified, the influences of system parameters change on reliability function and the FPT are discussed by the novel method.