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

• Published in: Journal of systems science and complexity Vol. 35; no. 1; pp. 159 - 178
• Main Authors: Xing, Ruitao, Xiao, Min, Zhang, Yuezhong, Qiu, Jianlong
• Format: Journal Article
• Language: English
• Published: Beijing Academy of Mathematics and Systems Science, Chinese Academy of Sciences 01.02.2022; Springer Nature B.V
• Subjects: Complex Systems; Control; Eigenvalues; Eigenvectors; Flow stability; Hopf bifurcation; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Neural networks; Neurons; Operations Research/Decision Theory; Stability analysis; Statistics; Systems Theory; high-dimensional; Coates’ flow graph; Hopf bifurcation; stability
• ISSN: 1009-6124; 1559-7067
• DOI: 10.1007/s11424-021-0108-2

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.