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Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms

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  • Yang, Xiaofan
  • Yang, Maobin
  • Liu, Huaiyi
  • Liao, Xiaofeng

Abstract

This paper addresses the dynamics of a class of two-neuron networks with resonant bilinear terms, which are of the formx˙1=(α1+a)f(x1)+(α2+b)f(x2)+cx1x2,x˙2=(α2-b)f(x1)+(α1-a)f(x2)+dx1x2,where α1 and α2 are two independent parameters. Our main contribution is to present a sufficient condition for the occurrence of a Bautin bifurcation in such a network by using the standard normal form theory and with the Maple software. Two numerical examples are given to demonstrate the utility of our result. To our knowledge, this is the first time to study Bautin bifurcation for recurrent neural networks.

Suggested Citation

  • Yang, Xiaofan & Yang, Maobin & Liu, Huaiyi & Liao, Xiaofeng, 2008. "Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 575-589.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:2:p:575-589
    DOI: 10.1016/j.chaos.2007.01.001
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    References listed on IDEAS

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    1. Huang, Chuangxia & Huang, Lihong & Feng, Jianfeng & Nai, Mingyong & He, Yigang, 2007. "Hopf bifurcation analysis for a two-neuron network with four delays," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 795-812.
    2. 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.
    3. Chunguang Li & Guangrong Chen & Xiaofeng Liao & Juebang Yu, 2004. "Hopf bifurcation and chaos in a single inertial neuron model with time delay," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 41(3), pages 337-343, October.
    4. Yuan, Yuan, 2007. "Dynamics in a delayed-neural network," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 443-454.
    5. Zhang, Chunrui & Zheng, Baodong, 2007. "Stability and bifurcation of a two-dimension discrete neural network model with multi-delays," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1232-1242.
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    2. Yang, Yu & Ye, Jin, 2009. "Stability and bifurcation in a simplified five-neuron BAM neural network with delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2357-2363.

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