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Controlling Hopf bifurcations of discrete-time systems in resonance

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  • Wen, Guilin
  • Xu, Daolin
  • Xie, Jianhua

Abstract

Resonance in Hopf bifurcation causes complicated bifurcation behaviors. To design with certain desired Hopf bifurcation characteristics in the resonance cases of discrete-time systems, a feedback control method is developed. The controller is designed with the aid of discrete-time washout filters. The control law is constructed according to the criticality and stability conditions of Hopf bifurcations as well as resonance constraints. The control gains associated with linear control terms insure the creation of a Hopf bifurcation in resonance cases and the control gains associated with nonlinear control terms determine the type and stability of bifurcated solutions. To derive the former, we propose the implicit criteria of eigenvalue assignment and transversality condition for creating the bifurcation in a desired parameter location. To derive the latter, the technique of the center manifold reduction, Iooss’s Hopf bifurcation theory and Wan’s Hopf bifurcation theory for resonance cases are employed. In numerical experiments, we show the Hopf circles and fixed points from the created Hopf bifurcations in the strong and weak resonance cases for a four-dimensional control system.

Suggested Citation

  • Wen, Guilin & Xu, Daolin & Xie, Jianhua, 2005. "Controlling Hopf bifurcations of discrete-time systems in resonance," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1865-1877.
    • Handle: RePEc:eee:chsofr:v:23:y:2005:i:5:p:1865-1877
    DOI: 10.1016/j.chaos.2004.07.002
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    References listed on IDEAS

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    1. Guanrong Chen & Jin-quing Fang & Yiguang Hong & Huashu Qin, 2000. "Controlling hopf bifurcations: Discrete-time systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 5, pages 1-5, January.
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    Cited by:

    1. Cui, Yan & Liu, Suhua & Tang, Jiashi & Meng, Yimin, 2009. "Amplitude control of limit cycles in Langford system," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 335-340.
    2. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Bifurcation control of the Hodgkin–Huxley equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 217-224.
    3. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Analysis and control of the bifurcation of Hodgkin–Huxley model," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 247-256.

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