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Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller

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  • Li, Wei
  • Huang, Dongmei
  • Zhang, Meiting
  • Trisovic, Natasa
  • Zhao, Junfeng

Abstract

Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker–Planck–Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.

Suggested Citation

  • Li, Wei & Huang, Dongmei & Zhang, Meiting & Trisovic, Natasa & Zhao, Junfeng, 2019. "Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller," Chaos, Solitons & Fractals, Elsevier, vol. 121(C), pages 30-38.
    • Handle: RePEc:eee:chsofr:v:121:y:2019:i:c:p:30-38
    DOI: 10.1016/j.chaos.2019.01.026
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    References listed on IDEAS

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    1. Huang, Chengdai & Cao, Jinde & Xiao, Min & Alsaedi, Ahmed & Alsaadi, Fuad E., 2017. "Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 293-310.
    2. Li, Wei & Xu, Wei & Zhao, Junfeng & Jin, Yanfei, 2007. "Stochastic stability and bifurcation in a macroeconomic model," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 702-711.
    3. Lokenath Debnath, 2003. "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-30, January.
    4. Kavyanpoor, Mobin & Shokrollahi, Saeed, 2017. "Challenge on solutions of fractional Van Der Pol oscillator by using the differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 44-45.
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    Cited by:

    1. Yong-Ge Yang & Ya-Hui Sun & Wei Xu, 2019. "Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises," Complexity, Hindawi, vol. 2019, pages 1-10, October.

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