中国物理B ›› 2010, Vol. 19 ›› Issue (3): 30511-030511.doi: 10.1088/1674-1056/19/3/030511

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Hopf bifurcation analysis of Chen circuit with direct time delay feedback

任海鹏, 李文超, 刘 丁   

  1. School of Automation and Information Engineering, Xi'an University of Technology, Xi'an 710048, China
  • 收稿日期:2008-09-02 修回日期:2009-08-30 出版日期:2010-03-15 发布日期:2010-03-15
  • 基金资助:
    Project supported in part by the National Natural Science Foundation of China (Grant No.~60804040) and Fok Ying-Tong Education Foundation for Young Teacher (Grant No.~111065).

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

Ren Hai-Peng(任海鹏), Li Wen-Chao(李文超), and Liu Ding(刘丁)   

  1. School of Automation and Information Engineering, Xi'an University of Technology, Xi'an 710048, China
  • Received:2008-09-02 Revised:2009-08-30 Online:2010-03-15 Published:2010-03-15
  • Supported by:
    Project supported in part by the National Natural Science Foundation of China (Grant No.~60804040) and Fok Ying-Tong Education Foundation for Young Teacher (Grant No.~111065).

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摘要： Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit (oscillation) in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.

Abstract: Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit (oscillation) in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.

Key words: direct time delay feedback, bifurcation diagram, Hopf bifurcation, bifurcation boundary

中图分类号:  (Numerical simulations of chaotic systems)

  • 05.45.Pq
02.30.Oz (Bifurcation theory) 84.30.Bv (Circuit theory) 02.30.Hq (Ordinary differential equations)