Response Control and Bifurcation Phenomenon of a Tristable Rayleigh–Duffing System with Fractional Inertial Force Under Recycling Noises

This study investigates stochastic bifurcation in a generalized tristable Rayleigh–Duffing oscillator with fractional inertial force under both additive and multiplicative recycling noises. The system’s dynamic behavior is influenced by its inherent spatial symmetry, represented by the potential function, as well as by temporal symmetry breaking caused by fractional memory effects and recycling noise. First, an approximate integer-order equivalent system is derived from the original fractional-order model using a harmonic balance method, with minimal mean square error (MSE). The steady-state probability density function (sPDF) of the amplitude is then obtained via stochastic averaging. Using singularity theory, the conditions for stochastic P bifurcation (SPB) are identified. For different fractional derivative’s orders, transition set curves are constructed, and the sPDF is qualitatively analyzed within the regions bounded by these curves—especially under tristable conditions. Theoretical results are validated through Monte Carlo simulations and the Radial Basis Function Neural Network (RBFNN) approach. The findings offer insights for designing fractional-order controllers to improve system response control.