On a discrete-time-delayed Hopfield neural network with ring structures and different internal decays: Bifurcations analysis and chaotic behavior

This paper concerns to a discrete-time-delayed Hopfield multi-dimensional neural network model with ring architecture and different internal decays. Stability of the origin equilibrium is studied and the region of this stability is obtained. The existence of some codimension one and codimension two bifurcations at the origin are identified and necessary circumstances for fold/cusp, flip, Neimark–Sacker bifurcations and also existence of some resonances are proved. With the purpose of doing these, we apply the center manifold theorem and the normal form theory. In fact we compute the critical coefficients of normal forms in order to find out the bifurcations properties. Finally occurrence of chaos in the sense of Marotto is showed, if the magnitude of the interconnection coefficients are large enough and at least one of the activation functions has two simple real roots. In a separate section, some numerical simulations are carried out to illustrate the analytical results.