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Stability and Hopf bifurcation of a Lorenz-like system

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  • Wu, Ranchao
  • Fang, Tianbao

Abstract

Hopf bifurcation is one of the important dynamical behaviors. It could often cause some phenomena, such as quasiperiodicity and intermittency. Consequently, chaos will happen due to such dynamical behaviors. Since chaos appears in the Lorenz-like system, to understand the dynamics of such system, Hopf bifurcation will be explored in this paper. First, the stability of equilibrium points is presented. Then Hopf bifurcation of the Lorenz-like system is investigated. By applying the normal form theory, the conditions guaranteeing the Hopf bifurcation are derived. Further, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also presented. Finally, numerical simulations are given to verify the theoretical analysis. It is found that Hopf bifurcation could happen when conditions are satisfied. The stable bifurcating periodic orbit is displayed. Chaos will also happen when parameter further increases.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:262:y:2015:i:c:p:335-343
    DOI: 10.1016/j.amc.2015.04.072
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    1. Tigan, Gheorghe & Opriş, Dumitru, 2008. "Analysis of a 3D chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1315-1319.
    2. 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.
    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.
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    1. Zarei, Amin & Tavakoli, Saeed, 2016. "Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 323-339.

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