Codimension one and two bifurcations of a discrete-time neural network model with ring structure
摘要
In this paper, a Hopfield neural network with ring structure has been considered. Analysis of codimension one bifurcations, such as flip, pitchfork and Neimark–Sacker, and codimension two bifurcations, such as generalized flip, Chenciner (generalized Neimark–Sacker) and 1:4 resonance, in the discrete-time neural network system has been investigated. The normal form coefficients for each bifurcation are determined through both analytical and numerical methods. In the subsequent analysis, we will assess the stability of the fixed point at the origin during pitchfork, flip and Neimark–Sacker bifurcations, taking into account the direction of these bifurcations. Finally, we will plot the bifurcations curve in parameter space and depict the Neimark–Sacker, flip, pitchfork, generalized flip, generalized Neimark–Sacker bifurcations and 1:4 resonance bifurcation in phase space.