Bifurcation analysis of a tristable system with fractional derivative under colored noise excitation

Stochastic bifurcation has received much attention recently and is of great significance in the research on the dynamics of nonlinear systems. In this paper, we study the bifurcation of a self-sustained tristable system containing fractional derivative under the excitation of two colored noises, where the self-sustained tristable system consists of stable limit cycles and a stable state. The multiple scale method and generalized harmonic function are used to convert the original system into an equal system without obvious time delay and fractional derivative. Then the stationary probability density function (SPDF) of the system is obtained by using the stochastic averaging method to discuss stochastic bifurcation. Based on singularity theory, the equation satisfying amplitude is obtained to derive the corresponding bifurcation diagram. From the bifurcation analysis of the system, it is revealed that fractional order, fractional coefficient, and the intensity and correlation time of colored noise can be utilized as bifurcation parameters, causing peculiar stochastic bifurcation phenomena and regulating the output of the system, as well as the relatively big colored noise intensity and relatively small correlation time facilitate the realization of large magnitude limit cycle. Monte Carlo numerical results verify the validity of the theoretical approach. In addition, appropriately changing the delayed feedback parameter helps to govern the bifurcation range and causes richer bifurcation phenomena. The results of this paper contribute to a better understanding of self-sustained tristable systems with fractional derivative and may provide reference value in resolving the problems of chattering in high-speed aircraft and applications to viscoelastic materials.