TY - JOUR
TI - Transition behaviors of system energy in a bi-stable van Ver Pol oscillator with fractional derivative element driven by multiplicative Gaussian white noise
AU - Li Yajie
AU - Wu Zhiqiang
AU - Lan Qixun
AU - Cai Yujie
AU - Xu Huafeng
AU - Sun Yongtao
JN - Thermal Science
PY - 2022
VL - 26
IS - 3
SP - 2727
EP - 2736
PT - Article
AB - The stochastic P-bifurcation behavior of system energy in a bi-stable Van der  Pol oscillator with fractional damping under multiplicative Gaussian white  noise excitation is investigated. Firstly, using the principle of minimal  mean square error, the non-linear stiffness terms can be equivalent to a  linear stiffness which is a function of the system amplitude, and the  original system is simplified to an equivalent integer order Van der Pol  system. Secondly, the system amplitude's stationary probability density  function is obtained by stochastic averaging. Then, according to the  singularity theory, the critical parametric conditions for the system  amplitude's stochastic P-bifurcation are found. Finally, the types of the  system's stationary probability density function curves of amplitude are  qualitatively analyzed by choosing the corresponding parameters in each area  divided by the transition set curves. The consistency between the analytical  results and the numerical results obtained from Monte-Carlo simulation  verifies the theoretical analysis in this paper, and the method used in this  paper can directly guide the design of the fractional-order controller to  adjust the response of the system.