Chaotic Dynamics of a Delayed Discrete-Time Hopfield Network of Two Nonidentical Neurons with no Self-Connections

A bifurcation analysis is undertaken for a discrete-time Hopfield neural network of two neurons with two delays, two internal decays and no self-connections, choosing the product of the interconnection coefficients as the characteristic parameter for the system. The stability domain of the null solution is found, the values of the characteristic parameter for which bifurcations occur at the origin are identified, and the existence of Fold/Cusp, Neimark–Sacker and Flip bifurcations is proved. All these bifurcations are analyzed by applying the center manifold theorem and the normal form theory. It is shown that the dynamics in a neighborhood of the null solution become more and more complex as the characteristic parameter grows in magnitude and passes through the bifurcation values. Under certain conditions, it is proved that if the magnitudes of the interconnection coefficients are large enough, the neural network exhibits Marotto’s chaotic behavior.