Stochastic transition behaviors in a tri-stable van der Pol oscillator with fractional delayed element subject to Gaussian white noise

Abstract

The stochastic P-bifurcation behavior of tri stability in a generalized Van der Pol system with fractional derivative under additive Gaussian white noise excitation is investigated. Firstly, based on the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the stationary probability density function of the system amplitude is obtained by stochastic averaging, and according to the singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the nature of stationary probability density function curves of the system amplitude is qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and Monte-Carlo simulation results verifies the theoretical results in this paper.