Bifurcations and chaos in a discrete-time-delayed Hopfield neural network with ring structures and different internal decays

Abstract

In this paper, a discrete-time-delayed Hopfield type neural network model consisting of p neurons with ring architecture and different internal decays is considered. The stability domain of the null solution is found, the values of characteristic parameter for which bifurcations occur at the origin are identified and the existence of Fold/Cusp, Neimark–Sacker and Flip bifurcations are proved. These bifurcations are analyzed by applying the center manifold theorem and the normal form theory. Probability of resonant 1:3 and 1:4 bifurcations also are proved. It is shown that the dynamics in a neighborhood of the origin become more and more complicated as the characteristic parameter grows in magnitude and passes through the bifurcation values. The occurrence of chaos in the sense of Marotto is shown, if the magnitude of the interconnection coefficients are large enough and at least one of the activation functions has two simple real roots. Some numerical simulations are carried out to illustrate the analytical results.