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Stability and bifurcation analysis in tri-neuron model with time delay

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Abstract

A simple delayed neural network model with three neurons is considered. By constructing suitable Lyapunov functions, we obtain sufficient delay-dependent criteria to ensure global asymptotical stability of the equilibrium of a tri-neuron network with single time delay. Local stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the time delay varies and passes a sequence of critical values. The stability and direction of bifurcating periodic solution are determined by applying the normal form theory and the center manifold theorem. If the associated characteristic equation of linearized system evaluated at a critical point involves a repeated pair of pure imaginary eigenvalues, then the double Hopf bifurcation is also found to occur in this model. Our main attention will be paid to the double Hopf bifurcation associated with resonance. Some Numerical examples are finally given for justifying the theoretical results.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Chongqing University, Chongqing, 400030, P.R. China

    Xiaofeng Liao, Songtao Guo & Chuandong Li

  2. The Key Laboratory of Optoelectric Technology & Systems, Ministry of Education, Beijing, China

    Xiaofeng Liao, Songtao Guo & Chuandong Li

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  1. Xiaofeng Liao
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  2. Songtao Guo
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  3. Chuandong Li
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Correspondence to Xiaofeng Liao.

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Liao, X., Guo, S. & Li, C. Stability and bifurcation analysis in tri-neuron model with time delay. Nonlinear Dyn 49, 319–345 (2007). https://doi.org/10.1007/s11071-006-9137-6

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  • DOI: https://doi.org/10.1007/s11071-006-9137-6

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