Stability and Hopf Bifurcation Analysis of an (n + m)-Neuron Double-Ring Neural Network Model with Multiple Time Delays

XING Ruitao, XIAO Min, ZHANG Yuezhong, QIU Jianlong

系统科学与复杂性(英文) ›› 2022, Vol. 35 ›› Issue (1) : 159-178.

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系统科学与复杂性(英文) ›› 2022, Vol. 35 ›› Issue (1) : 159-178. DOI: 10.1007/s11424-021-0108-2

Stability and Hopf Bifurcation Analysis of an (n + m)-Neuron Double-Ring Neural Network Model with Multiple Time Delays

    XING Ruitao1, XIAO Min1, ZHANG Yuezhong1, QIU Jianlong2
作者信息 +

Stability and Hopf Bifurcation Analysis of an (n + m)-Neuron Double-Ring Neural Network Model with Multiple Time Delays

    XING Ruitao1, XIAO Min1, ZHANG Yuezhong1, QIU Jianlong2
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文章历史 +

摘要

Up till the present moment, researchers have always featured the single-ring neural network. These investigations, however, disregard the link between rings in neural networks. This paper highlights a high-dimensional double-ring neural network model with multiple time delays. The neural network has two rings of a shared node, where one ring has n neurons and the other has m+1 neurons. By utilizing the sum of time delays as the bifurcation parameter, the method of Coates' flow graph is applied to obtain the relevant characteristic equation. The stability of the neural network model with bicyclic structure is discussed by dissecting the characteristic equation, and the critical value of Hopf bifurcation is derived. The effect of the sum of time delays and the number of neurons on the stability of the model is extrapolated. The validity of the theory can be verified by numerical simulations.

Abstract

Up till the present moment, researchers have always featured the single-ring neural network. These investigations, however, disregard the link between rings in neural networks. This paper highlights a high-dimensional double-ring neural network model with multiple time delays. The neural network has two rings of a shared node, where one ring has n neurons and the other has m+1 neurons. By utilizing the sum of time delays as the bifurcation parameter, the method of Coates' flow graph is applied to obtain the relevant characteristic equation. The stability of the neural network model with bicyclic structure is discussed by dissecting the characteristic equation, and the critical value of Hopf bifurcation is derived. The effect of the sum of time delays and the number of neurons on the stability of the model is extrapolated. The validity of the theory can be verified by numerical simulations.

关键词

Coates' flow graph / high-dimensional / Hopf bifurcation / stability

Key words

Coates' flow graph / high-dimensional / Hopf bifurcation / stability

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XING Ruitao , XIAO Min , ZHANG Yuezhong , QIU Jianlong. Stability and Hopf Bifurcation Analysis of an (n + m)-Neuron Double-Ring Neural Network Model with Multiple Time Delays. 系统科学与复杂性(英文), 2022, 35(1): 159-178 https://doi.org/10.1007/s11424-021-0108-2
XING Ruitao , XIAO Min , ZHANG Yuezhong , QIU Jianlong. Stability and Hopf Bifurcation Analysis of an (n + m)-Neuron Double-Ring Neural Network Model with Multiple Time Delays. Journal of Systems Science and Complexity, 2022, 35(1): 159-178 https://doi.org/10.1007/s11424-021-0108-2

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基金

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61573194, 62073172, 61877033, and the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20181389.
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