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Stability and Bifurcation Analysis in a Class of Two‐Neuron Networks with Resonant Bilinear Terms

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  • Changjin Xu
  • Xiaofei He

Abstract

A class of two‐neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.

Suggested Citation

  • Changjin Xu & Xiaofei He, 2011. "Stability and Bifurcation Analysis in a Class of Two‐Neuron Networks with Resonant Bilinear Terms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    • Handle: RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:697630
    DOI: 10.1155/2011/697630
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Zizhen Zhang & Dianjie Bi, 2016. "Dynamical Analysis of a Computer Virus Propagation Model with Delay and Infectivity in Latent Period," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    2. Zizhen Zhang & Huizhong Yang, 2013. "Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    3. Juan Liu, 2014. "Hopf Bifurcation in a Delayed SEIQRS Model for the Transmission of Malicious Objects in Computer Network," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    4. Zizhen Zhang & Huizhong Yang, 2013. "Hopf Bifurcation Analysis for a Four‐Dimensional Recurrent Neural Network with Two Delays," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    5. Kaiyu Liu & Zhengqiu Zhang & Liping Wang, 2012. "Existence and Global Exponential Stability of Periodic Solution to Cohen‐Grossberg BAM Neural Networks with Time‐Varying Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).

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