Non-linear stochastic response and bifurcation analysis of a multistable Rayleigh system with a fractional element subjected to noise excitation

The study examines the stochastic bifurcation phenomenon of a generalized and multistable Rayleigh system subjected to fractional damping driven by Gaussian white noise. First, the harmonic balance technique is employed to minimize the error in terms of mean square, thereby deriving the approximate equal integer-order system from the original system with fractional-order elements. Subsequently, the stationary probability density function of the system is determined using the stochastic averaging method. Subsequently, employing singularity theory, the critical conditions of system parameters for stochastic P-bifurcation of the original system are identified. Finally, a qualitative analysis of the stationary probability density function curves of the system amplitude is conducted in each region delineated by the boundary set curves. The analytical solutions were found to align with the numerical findings obtained from Monte-Carlo simulation, thereby corroborating the theoretical deductions. The methodology and findings presented in this study have the potential to enhance system response control through the design of fractional-order controllers.