Stability and Hopf Bifurcation Analysis of a Continuous Neural Network With Mixed Delays

Abstract

A three-neuron network model with mixed delays involving multiple discrete and distributed delays is considered in this paper. Taking the discrete and distribute delays as bifurcation parameters respectively, we investigate the stability of the system structure and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the root of the characteristic equation of the linearized system at the equilibrium state of the nonlinear system. The intervals of parameters that make the system stable and unstable are also given. In addition, when the conditions of Hopf bifurcation theorem are satisfied, the calculation formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution are presented by means of the central manifold theorem and normal form method. Finally, numerical experiments are carried out to support the correctness of the theoretical results in different cases. It can be concluded that the oscillation and instability caused by delays obviously affect the stability of the network.