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Hopf bifurcation analysis of the Liu system

Author

Listed:
  • Zhou, Xiaobing
  • Wu, Yue
  • Li, Yi
  • Wei, Zhengxi

Abstract

In this paper, a three dimensional autonomous system which is similar to the Lorenz system is considered. By choosing an appropriate bifurcation parameter, we prove that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is presented by applying the normal form theory. Finally, an example is given and numerical simulations are performed to illustrate the obtained results.

Suggested Citation

  • Zhou, Xiaobing & Wu, Yue & Li, Yi & Wei, Zhengxi, 2008. "Hopf bifurcation analysis of the Liu system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1385-1391.
    • Handle: RePEc:eee:chsofr:v:36:y:2008:i:5:p:1385-1391
    DOI: 10.1016/j.chaos.2006.09.008
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    References listed on IDEAS

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    1. Čelikovský, Sergej & Chen, Guanrong, 2005. "On the generalized Lorenz canonical form," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1271-1276.
    2. Qi, Guoyuan & Du, Shengzhi & Chen, Guanrong & Chen, Zengqiang & yuan, Zhuzhi, 2005. "On a four-dimensional chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1671-1682.
    3. Qi, Guoyuan & Chen, Guanrong & Du, Shengzhi & Chen, Zengqiang & Yuan, Zhuzhi, 2005. "Analysis of a new chaotic system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 352(2), pages 295-308.
    4. Starkov, Konstantin E., 2005. "Localization of periodic orbits of polynomial vector fields of even degree by linear functions," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 621-627.
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    Citations

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    Cited by:

    1. Qinghui Liu & Xin Zhang, 2024. "Jacobi Stability Analysis of Liu System: Detecting Chaos," Mathematics, MDPI, vol. 12(13), pages 1-16, June.
    2. Wang, Xia, 2009. "Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2208-2217.
    3. Mello, L.F. & Messias, M. & Braga, D.C., 2008. "Bifurcation analysis of a new Lorenz-like chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1244-1255.
    4. Megam Ngouonkadi, E.B. & Fotsin, H.B. & Louodop Fotso, P. & Kamdoum Tamba, V. & Cerdeira, Hilda A., 2016. "Bifurcations and multistability in the extended Hindmarsh–Rose neuronal oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 85(C), pages 151-163.
    5. Wu, Ranchao & Fang, Tianbao, 2015. "Stability and Hopf bifurcation of a Lorenz-like system," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 335-343.
    6. Xiang Li & Ranchao Wu, 2013. "Dynamics of a New Hyperchaotic System with Only One Equilibrium Point," Journal of Mathematics, Hindawi, vol. 2013, pages 1-9, July.

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