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Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network

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  • Kaslik, E.
  • Balint, St.

Abstract

In this paper, a bifurcation analysis is undertaken for a discrete-time Hopfield neural network with a single delay. Conditions ensuring the asymptotic stability of the null solution are found, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, fold or Neimark–Sacker bifurcations occur, but codimension 2 (fold-Neimark–Sacker, double Neimark–Sacker and resonance 1:1) bifurcations may also be present. The direction and the stability of the Neimark–Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory.

Suggested Citation

  • Kaslik, E. & Balint, St., 2007. "Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network," Chaos, Solitons & Fractals, Elsevier, vol. 34(4), pages 1245-1253.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:4:p:1245-1253
    DOI: 10.1016/j.chaos.2006.03.107
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    1. Zhang, Chunrui & Zheng, Baodong, 2005. "Hopf bifurcation in numerical approximation of a n-dimension neural network model with multi-delays," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 129-146.
    2. Mohamad, S. & Gopalsamy, K., 2000. "Dynamics of a class of discrete-time neural networks and their continuous-time counterparts," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 53(1), pages 1-39.
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    1. Kaslik, E. & Balint, St., 2009. "Bifurcation analysis for a discrete-time Hopfield neural network of two neurons with two delays and self-connections," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 83-91.
    2. Yang, Degang & Hu, Chunyan & Chen, Yong & Wei, Pengcheng & Yang, Huaqian, 2009. "New delay-dependent global asymptotic stability criteria of delayed BAM neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 854-864.

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