Simplified frequency method for stability and bifurcation of delayed neural networks in ring structure

In this paper, a novel procedure for the stability and Hopf bifurcation of delayed neural networks with ring topology and multiple time delays is proposed. This procedure mainly focuses on the distribution of roots of exponential polynomial which is necessary for analyzing the asymptotic properties of dynamic systems at an equilibrium. The purely imaginary roots (PIRs) of the exponential polynomial (which regards the delay ττ as its parameter) are exactly determined by an intuitive graphic method. Moreover, the critical delay associating with those PIRs can be computed according to its periodicity. The obtained results are applied to the analysis of a general delayed neural network model (which means no restriction of number of neurons is posed on this general model). Eventually, the complete picture of stable regions and the Hopf bifurcation point in ττ-parameter space is given. This work offers an exact, structured methodology for the local dynamical analysis of delayed neural networks. Some illustrative simulations based on a powerful continuation software are presented to prove the effectiveness of our theoretical analysis.