Bifurcation control and parametric influence analysis in a fractional bi-rhythm Van der pol-Rayleigh system driven by Gaussian colored noises

The stochastic P-bifurcation characteristics of a bistable Van der Pol-Rayleigh system incorporating fractional-order inertia and damping are examined under the joint effects of additive and multiplicative Gaussian colored noises. By invoking the minimum mean square error criterion, the fractional-order inertial and damping components are reformulated as the combinations of integer-order stiffness, damping, and inertial terms, allowing the original fractional system to be transformed into an isovalent integer-order dynamical system.

On this basis, the stationary probability density function of the system amplitude is obtained through the stochastic averaging technique. The parametric conditions governing the emergence of stochastic P-bifurcation are then identified using singularity theory. Furthermore, the qualitative topology evolution of the steady-state probability density is investigated by analyzing representative parameters located in distinct regions delineated by the transition curves.

To verify the analytical process, the numerical results obtained from Monte Carlo simulations and a Radial Basis Function Neural Network (RBFNN) are compared with the theoretical solutions. The high agreement among these results confirms the validity of the analytical framework and the accuracy of the transition set acquired. The findings provide theoretical insights into the control of bifurcation and vibration suppression in nonlinear fractional systems under random excitation, with potential applications in engineering design, aeroelastic stability, and vibration control of structures with memory-dependent properties.