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Stochastic Stability And Parametric Control In A Generalized And Tri-Stable Van Der Pol System With Fractional Element Driven By Multiplicative Noise

Author

Listed:
  • YA-JIE LI

    (Henan Engineering Research Centre of Building-Photovoltaics, School of Mathematics and Physics, Henan University of Urban, Construction, Pingdingshan, P. R. China)

  • ZHI-QIANG WU

    (��Tianjin Key Laboratory of Nonlinear Dynamics and Control, School of Mechanical Engineering, Tianjin, University, Tianjin, P. R. China)

  • YONG-TAO SUN

    (��Tianjin Key Laboratory of Nonlinear Dynamics and Control, School of Mechanical Engineering, Tianjin, University, Tianjin, P. R. China)

  • YING HAO

    (��School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao, P. R. China)

  • XIANG-YUN ZHANG

    (�Basic Course Department, Tianjin Sino-German, University of Applied Sciences, Tianjin, P. R. China)

  • FENG WANG

    (�School of Automobile and Transportation, Tianjin University of Technology and Education, Tianjin, P. R. China)

  • HE-PING SHI

    (�School of Automobile and Transportation, Tianjin University of Technology and Education, Tianjin, P. R. China)

Abstract

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.

Suggested Citation

  • Ya-Jie Li & Zhi-Qiang Wu & Yong-Tao Sun & Ying Hao & Xiang-Yun Zhang & Feng Wang & He-Ping Shi, 2023. "Stochastic Stability And Parametric Control In A Generalized And Tri-Stable Van Der Pol System With Fractional Element Driven By Multiplicative Noise," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(07), pages 1-13.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:07:n:s0218348x23500834
    DOI: 10.1142/S0218348X23500834
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