Local stability and bifurcation of multi-delay fractional-order bidirectional ring neural networks with reaction-diffusion

This paper investigated the Hopf bifurcation in fractional-order ring neural networks incorporating multiple time delays (leakage, transmission, and distributed delays) and reaction-diffusion effects. By introducing virtual neurons into the original system, a new model capable of equivalently describing the influence of distributed time delays was constructed. The critical conditions for the emergence of pure imaginary roots in the characteristic equation were analytically determined using the Coates flow graph and the holistic element method. Furthermore, by selecting the leakage and transmission delays as bifurcation parameters, explicit criteria for local stability and the existence of Hopf bifurcation were established. The theoretical findings were substantiated through numerical simulations.