TY - JOUR
TI - Stochastic stability of the fractional and tri-stable Van der vol oscillator with time-delay feedback driven by Gaussian white noise
AU - Li Yajie
AU - Sun Yongtao
AU - Hao Ying
AU - Zhang Xiangyun
AU - Wang Feng
AU - Shi Heping
AU - Wang Bin
JN - Thermal Science
PY - 2023
VL - 27
IS - 3
SP - 2155
EP - 2164
PT - Article
AB - 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.