Nonlinear Dynamics of Three-Neurons-Based Hopfield Neural Networks (HNNs): Remerging Feigenbaum Trees, Coexisting Bifurcations and Multiple Attractors

Abstract

In this work, the dynamics of a simplified model of three-neurons-based Hopfield neural networks (HNNs) is investigated. The simplified model is obtained by removing the synaptic weight connection of the third and second neuron in the original Hopfield networks introduced in Ref. 11 . The investigations have shown that the simplified model possesses three equilibrium points among which origin of the systems coordinates. It is found that the origin is always unstable while the symmetric pair of fixed points with conditional stability has values depending on synaptic weight between the second and the first neuron that is used as bifurcation control parameter. Numerical simulations, carried out in terms of bifurcation diagrams, graph of Lyapunov exponents, phase portraits, Poincaré section, time series and frequency spectra are employed to highlight the complex dynamical behaviors exhibited by the model. The results indicate that the modified model of HNNs exhibits rich nonlinear dynamical behaviors including symmetry breaking, chaos, periodic window, antimonotonicity (i.e., concurrent creation and annihilation of periodic orbits) and coexisting self-excited attractors (e.g., coexistence of two, four and six disconnected periodic and chaotic attractors) which have not been reported in previous works focused on the dynamics of HNNs. Finally, PSpice simulations verify the results of theoretical analyses of the simplified model of three-neurons-based HNNs.