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Analysis of stability and Hopf bifurcation for a delayed logistic equation

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  • Sun, Chengjun
  • Han, Maoan
  • Lin, Yiping

Abstract

The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799–38.]

Suggested Citation

  • Sun, Chengjun & Han, Maoan & Lin, Yiping, 2007. "Analysis of stability and Hopf bifurcation for a delayed logistic equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 672-682.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:3:p:672-682
    DOI: 10.1016/j.chaos.2005.10.019
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    Cited by:

    1. Gan, Qintao & Xu, Rui & Yang, Pinghua, 2009. "Bifurcation and chaos in a ratio-dependent predator–prey system with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1883-1895.
    2. Xu, Rui & Ma, Zhien, 2008. "Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 669-684.
    3. Sen, Ayan & Mukherjee, Debasis, 2009. "Chaos in the delay logistic equation with discontinuous delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2126-2132.
    4. Xu, Beibei & Chen, Diyi & Zhang, Hao & Wang, Feifei, 2015. "Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 50-61.

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