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Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays

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Abstract

In this paper, a tri-neuron BAM neural network model with multiple delays is considered. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. There is a wide range of different dynamical behaviors which can be produced by varying the coupling strength. By choosing the connected weights c 21 and c 31 (the connection weights through the neurons from J-layer to I-layer) as bifurcation parameters, the critical values where a Bogdanov–Takens bifurcation occurs are derived. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, some interesting phenomena, such as saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation are found by choosing the different connection strengths. Some numerical simulations are given to support the analytic results.

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Acknowledgement

This work was supported in part by the National Natural Science Foundation of China under Grant 60973114 and 61170249, in part by the Research Fund of Preferential Development Domain for the Doctoral Program of Ministry of Education of China under Grant 20110191130005, in part by the Natural Science Foundation project of CQCSTC under Grant 2009BA2024, in part by Chongqing University of Posts and Telecommunications Social Science Foundation, under Grant k2011-110 and k2010-105, and in part by the program for Changjiang Scholars.

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Authors and Affiliations

  1. College of Electronics and Information Engineering, Southwest University, Chongqing, 400715, P.R. China

    Tao Dong & Xiaofeng Liao

  2. College of Software and Engineering, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China

    Tao Dong

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  1. Tao Dong
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  2. Xiaofeng Liao
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Correspondence to Tao Dong or Xiaofeng Liao.

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Dong, T., Liao, X. Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays. Nonlinear Dyn 71, 583–595 (2013). https://doi.org/10.1007/s11071-012-0683-9

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  • DOI: https://doi.org/10.1007/s11071-012-0683-9

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