Extremely rich dynamics from hyperchaotic Hopfield neural network: Hysteretic dynamics, parallel bifurcation branches, coexistence of multiple stable states and its analog circuit implementation

In this work, we investigate the dynamics of a model of 4-neurons based hyperchaotic Hopfield neural network (HHNN) with a unique unstable node as a fixed point. The basic properties of the model including symmetry, dissipation, and condition of the existence of an attractor are explored. Our numerical simulations highlight several complex phenomena such as periodic orbits, quasi-periodic orbits, and chaotic and hyperchaotic orbits. More interestingly, it has been revealed several sets of synaptic weights matrix for which the HHNN studied display multiple coexisting attractors including two, three and four symmetric and disconnected attractors. Both hysteretic dynamics and parallel bifurcation branches justify the presence of these various coexisting attractors. Basins of attraction with the riddle structure of some of the coexisting attractors have been computed showing different regions in which each solution can be captured. Finally, PSpice simulations are used to further support the results of our previous analyses.