Remerging Feigenbaum Trees, Coexisting Behaviors and Bursting Oscillations in a Novel 3D Generalized Hopfield Neural Network

Abstract

This paper, a novel 3D generalized Hopfield neural network is proposed and investigated in order to generate and highlight some unknown behaviors related to such type of neural network. This generalized model is constructed by exploiting the effect of an external stimulus on the dynamics of a simplest 3D autonomous Hopfield neural network reported to date. The stability of the model around its ac-equilibrium point is studied. We note that the model has three types of stability depending on the value of the external stimulus. The stable node-focus, the unstable saddle focus, and the stable node characterize the stability of the equilibrium point of the model. Traditional nonlinear analyses tools are used to highlight and support several complex phenomena such as remerging Feigenbaum trees, the coexistence of up to two, four and six disconnected stable states when the amplitude of the external stimulus is set to zero. Furthermore, for some values of the frequency and amplitudes of the external stimulus, bursting oscillations occur. This latter behavior is characterized by the fact that oscillations switch between quiescent states and spiking states, repetitively. The transformed phase portraits are used to support the bursting mode of oscillations found. Finally, PSpice simulations enable to support the results of the theoretical studies.