Application of Complex Fractional Moment in nonlinear system with Gaussian colored noise

This paper studies the transient response of the nonlinear system with Gaussian colored noise. The one-dimensional Itô stochastic differential equation with respect to amplitude response is obtained by the stochastic averaging technique, and the Mellin transformation is applied to the related Fokker–Planck–Kolmogorov (FPK) equation to obtain a set of Complex Fractional Moment (CFM) differential equations. The semi-analytical solution of the FPK equation can be reconstructed by solving the CFM equations. Numerical method is adopted to verify the validity of the CFM method, and the dynamical temporal behaviors of the transient response Probability Density Function (PDF) are analyzed. In addition, the transient bifurcation phenomenon of the system is discussed, and the 3-D distribution diagram of peak value of transient PDF is produced.