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Stability and Hopf bifurcation analysis of flux neuron model with double time delays

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Abstract

A flux neuron model with double time delays was developed by introducing a memristor. The first time delay is added to the memristor and another time delay is formed by the interaction of fast and slow variables in the HR model. The local stability and Hopf bifurcation of the equilibrium are obtained in four different combinations of time delays. The direction, stability, and periodic solution of the Hopf bifurcation are proved by the centre manifold theorem. Furthermore, using numerical simulations, the two-parameter bifurcation diagrams are obtained to give insight into the effect of time delays on chaos-mediated mixed-mode oscillations and non-chaos-mediated mixed-mode oscillations of the flux neuron model. Changing in time delays can lead to an increase or decrease in the number of periods, disappearance, chaos, and other rich phenomena in the system membrane voltage. These phenomena provide a satisfactory explanation for the abnormal brain discharges caused by magnetic field and time delays.

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Acknowledgements

This work was supported by the National Natural Science Foundation (No.61863022, 11962012), and the Innovation and Entrepreneurship of Lanzhou City (No.2018-RC-87).

Funding

National Natural Science Foundation of China, 61863022, Jiangang Zhang, 11962012, XinLei An.

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Authors and Affiliations

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, 730070, Gansu, People’s Republic of China

    Lixiang Wei, Jiangang Zhang, Xinlei An, Mengran Nan & Shuai Qiao

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  1. Lixiang Wei
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  2. Jiangang Zhang
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  3. Xinlei An
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  5. Shuai Qiao
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Correspondence to Jiangang Zhang.

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Wei, L., Zhang, J., An, X. et al. Stability and Hopf bifurcation analysis of flux neuron model with double time delays. J. Appl. Math. Comput. 68, 4017–4050 (2022). https://doi.org/10.1007/s12190-021-01682-y

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  • DOI: https://doi.org/10.1007/s12190-021-01682-y

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