Stochastic bifurcation phenomenon and multistable behaviors in a fractional Rayleigh–Duffing oscillator under recycling noise

This study examines the stochastic bifurcation phenomenon in a fractional and multistable Rayleigh–Duffing oscillator subjected to recycling noise excitation. First, using the harmonic balance method and minimizing the mean-square error, an approximate integerorder equivalent system was derived for the original fractional-order system. Subsequently, the steady-state probability density function (sPDF) of the system amplitude was obtained via stochastic averaging. The critical conditions for stochastic P-bifurcation (SPB) were then determined using the singularity theory. The stationary PDF curves of the system amplitude were qualitatively analyzed across regions delineated by transition set curves. Finally, Monte Carlo simulations confirmed the analytical findings, validating the theoretical framework. These results provide insights for improving system response control through fractional-order controller design.