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Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays

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Abstract

In this paper, we consider the effect of distributed delays in a three-neuron unidirectional ring. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. Taking the average delay as a bifurcation parameter, we find two critical values at which the system undergoes Hopf bifurcations. The orbital asymptotic stability of the Hopf bifurcating periodic solutions is investigated by using the method of multiple scales. The global Hopf bifurcation is also studied. Finally, the theoretical results are illustrated by some numerical simulations.

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, China

    Yanyan Han & Yongli Song

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  1. Yanyan Han
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  2. Yongli Song
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Correspondence to Yongli Song.

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Han, Y., Song, Y. Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays. Nonlinear Dyn 69, 357–370 (2012). https://doi.org/10.1007/s11071-011-0269-y

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  • DOI: https://doi.org/10.1007/s11071-011-0269-y

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