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. Keywords Generalized Hopfield neural network (GHNN) • AC equilibrium points • Remerging Feigenbaum trees • Coexistence of multiple stable states • Bursting oscillations • Pspice simulations B Z. Tabekoueng Njitacke
In this contribution, a new elegant hyperjerk system with three equilibria and hyperbolic sine nonlinearity is investigated. In contrast to other models of hyperjerk systems where either hidden or self-excited attractors are obtained, the case reported in this work represents a unique one which displays the coexistence of self-excited chaotic attractors and stable fixed points. The dynamic properties of the new system are explored in terms of equilibrium point analyses, symmetry and dissipation, and existence of attractors as well. Common analysis tools (i.e., bifurcation diagram, Lyapunov exponents, and phase portraits) are used to highlight some important phenomena such as period-doubling bifurcation, chaos, periodic windows, and symmetric restoring crises. More interestingly, the system under consideration shows the coexistence of several types of stable states, including the coexistence of two, three, four, six, eight, and ten coexisting attractors. In addition, the system is shown to display antimonotonicity and offset boosting. Laboratory experimental measurements show a very good coherence with the theoretical predictions.
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