Stochastic stability of the fractional and tri-stable Van der vol oscillator with time-delay feedback driven by Gaussian white noise

Abstract

The stochastic P-bifurcation behavior of tri-stability in a fractional-order van der Pol system with time-delay feedback under additive Gaussian white noise excitation is investigated. Firstly, according to the equivalent principle, the fractional derivative and the time-delay term can be equivalent to a linear combination of damping and restoring forces, so the original system can be simplified into an equivalent integer-order system. Secondly, the stationary probability density function of the system amplitude is obtained by the stochastic averaging, and based on the singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the properties of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing corresponding parameters in each sub-region divided by the transition set curves. The consistence between numerical results obtained by Monte-Carlo simulation and analytical solutions has verified the accuracy of the theoretical analysis. The method used in this paper has a direct guidance in the design of fractional-order controller to adjust the dynamic behavior of the system.