Transmission dynamics analysis of a bistable Van der Pol oscillator under Gaussian colored noise with fractional inertial element

The stochastic response and bifurcation phenomenon of system amplitude in a bi-stable Van der Pol system with fractional inertial element driven by the Gaussian colored noise is investigated. In the initial step of the process, the mean square error is minimized, and the harmonic balance technique is employed. This results in the fractional derivative being found to be isovalent to a linear combination of inertial and damping forces. Simultaneously, the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the steady-state probability density function of the system amplitude is acquired via stochastic averaging. According to the principles of singularity theory, the critical parametric conditions for the stochastic P-bifurcation of system amplitude can be deter-mined. A qualitative analysis of the stationary PDF curves of amplitude is finally conducted in each area, with the transition set curves serving as a dividing point. The congruence between the analytical outcomes and the numerical results obtained from Monte Carlo simulation and the radial basis function neural network testifies to the theoretical analysis. In this paper, the methodology employed and the results obtained can enhance the design of the fractional-order controller to control the response of these systems