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Zero-Hopf bifurcation and multistability coexistence on a four-neuron network model with multiple delays

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Abstract

In this paper, a four-neuron neural system with four delays is investigated to exhibit the effects of multiple delays and coupled weights on system dynamics. Zero-Hopf bifurcation is obtained, where the system characteristic equation has a simple zero and a simple pair of pure imaginary eigenvalues. The coupled weight and time delay are considered as bifurcation parameters to study dynamical behaviors derived from zero-Hopf bifurcation. Various dynamical behaviors are analyzed near the bifurcation singularity qualitatively and quantitatively in detail by using perturbation- incremental method, and bifurcation diagrams are obtained. Numerical simulations and theoretical results are given to display a stable resting state, multistability coexistence of two resting states and a pair of periodic activities in the neighbor of the zero-Hopf bifurcation point.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11202068 and 11572224), University Key Teacher Foundation for Youths of Henan Province (Grant No. 2014GGJS-076), and Key Technological Research Project of Henan Province (Grant Nos. 152102210089 and 152102310320).

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Authors and Affiliations

  1. School of Mathematics and Information Science, He’nan University of Economics and Law, Zhengzhou, 450046, China

    Juhong Ge & ZhiQiang Li

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Jian Xu

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  1. Juhong Ge
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  2. Jian Xu
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  3. ZhiQiang Li
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Correspondence to Juhong Ge.

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Ge, J., Xu, J. & Li, Z. Zero-Hopf bifurcation and multistability coexistence on a four-neuron network model with multiple delays. Nonlinear Dyn 87, 2357–2366 (2017). https://doi.org/10.1007/s11071-016-3195-1

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  • DOI: https://doi.org/10.1007/s11071-016-3195-1

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