Dynamical Bifurcation for a Class of Large-Scale Fractional Delayed Neural Networks With Complex Ring-Hub Structure and Hybrid Coupling

Real neural networks are characterized by large-scale and complex topology. However, the current dynamical analysis is limited to low-dimensional models with simplified topology. Therefore, there is still a huge gap between neural network theory and its application. This article proposes a class of large-scale neural networks with a ring-hub structure, where a hub node is connected to n peripheral nodes and these peripheral nodes are linked by a ring. In particular, there exists a hybrid coupling mode in the network topology. The mathematical model of such systems is described by fractional-order delayed differential equations. The aim of this article is to investigate the local stability and Hopf bifurcation of this high-dimensional neural network. First, the Coates flow graph is employed to obtain the characteristic equation of the linearized high-dimensional neural network model, which is a transcendental equation including multiple exponential items. Then, the sufficient conditions ensuring the stability of equilibrium and the existence of Hopf bifurcation are achieved by taking time delay as a bifurcation parameter. Finally, some numerical examples are given to support the theoretical results. It is revealed that the increasing time delay can effectively induce the occurrence of periodic oscillation. Moreover, the fractional order, the self-feedback coefficient, and the number of neurons also have effects on the onset of Hopf bifurcation.

Automated summary

This article investigates a large-scale neural network model with a ring-hub structure using fractional-order delayed differential equations. The stability and Hopf bifurcation of the model are analyzed by obtaining the characteristic equation of the linearized model and providing numerical examples to support the theoretical results.