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Stability and bifurcation analysis in an amplitude equation with delayed feedback

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  • Niu, Ben
  • Wei, Junjie

Abstract

An amplitude equation is considered. The linear stability of the equation with direct control is investigated, and hence a bifurcation set is provided in the appropriate parameter plane. It is found that there exist stability switches when delay varies, and the Hopf bifurcation occurs when delay passes through a sequence of critical values. Furthermore, the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical simulations are performed to illustrate the obtained results.

Suggested Citation

  • Niu, Ben & Wei, Junjie, 2008. "Stability and bifurcation analysis in an amplitude equation with delayed feedback," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1362-1371.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:5:p:1362-1371
    DOI: 10.1016/j.chaos.2006.10.034
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    References listed on IDEAS

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    1. Liu, Cheng-Lin & Tian, Yu-Ping, 2008. "Eliminating oscillations in the Internet by time-delayed feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 35(5), pages 878-887.
    2. Liao, Xiaofeng & Ran, Jiouhong, 2007. "Hopf bifurcation in love dynamical models with nonlinear couples and time delays," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 853-865.
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    Cited by:

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    2. Cai, Liming & Guo, Shumin & Li, XueZhi & Ghosh, Mini, 2009. "Global dynamics of a dengue epidemic mathematical model," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2297-2304.
    3. Li, Xue-Zhi & Li, Wen-Sheng & Ghosh, Mini, 2009. "Stability and bifurcation of an SIS epidemic model with treatment," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2822-2832.
    4. Li, Kai & Wei, Junjie, 2009. "Stability and Hopf bifurcation analysis of a prey–predator system with two delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2606-2613.
    5. Huang, Dongmei & Li, Wei & Yang, Guidong & He, Meijuan, 2018. "Analysis of limit cycles and stochastic responses of a real-power vibration isolation system under delayed feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 125-134.
    6. Yu, Chunbo & Wei, Junjie, 2009. "Stability and bifurcation analysis in a basic model of the immune response with delays," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1223-1234.
    7. Ji, J.C. & Zhang, N. & Gao, Wei, 2009. "Difference resonances in a controlled van der Pol-Duffing oscillator involving time delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 975-980.
    8. Zhang, Jianxiong & Tang, Wansheng, 2009. "Analysis and control for a new chaotic system via piecewise linear feedback," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2181-2190.

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