Hopf bifurcation of a fractional-order double-ring structured neural network model with multiple communication delays

In this paper, we delve into the Hopf bifurcation of a novel high-dimensional delayed fractional neural network model with bicyclic structure sharing a node. Under some mild assumptions, the developed model is equivalently transformed into a system with two delays representing, respectively, the sum of delays in each ring. By applying the formula of Coates's flow graph, we first derive the associated characteristic equation of the transformed system. Then, picking one delay as a control parameter and regarding the other one as a constant in its stable interval, we establish the respective critical value of each delay for the occurrence of Hopf bifurcation. It is shown that the stability behavior remains unchanged for the small control delay and will be lost once the delay passes through the critical value. Numerical simulations and sensitivity analysis are performed to validate the theoretical results and identify the most sensitive parameter to the Hopf bifurcation point.