Infinitely many coexisting hidden attractors in a new hyperbolic-type memristor-based HNN

In this article, a new model of Hopfield Neural Network (HNN) with two neurons considering a synaptic weight with a hyperbolic-type memristor is studied. Equilibrium points analysis shows that the system has an unstable line of equilibrium in the absence of the external stimuli (i.e. I-1 = 0) and presents no equilibrium point in the presence of the external stimuli (i.e. I-1 not equal 0); hence the model admits hidden attractors. Analyses are carried out for both cases I-1 = 0 and I-1 not equal 0 using appropriate tools (bifurcation diagrams and the Lyapunov exponents, phase portraits, etc.). For both modes of operations, the system exhibits complex homogeneous and heterogeneous bifurcations, respectively marked by a large number of coexisting attractors. The roads to chaos unfold in the same scenario of period doubling. The Hamiltonian plot for the case I-1 = 0 allows us to observe an increase in the energy of the neuronal structure when it migrates from regular oscillations to irregular ones. Moreover, the existence of infinitely many coexisting homogeneous solutions (chaotic or periodic) is revealed for case I-1 = 0. In contrast, for I-1 not equal 0 (i.e I-1 = 0.1) the new model presents infinitely many coexisting hidden heterogeneous attractors (periodic and chaotic). An electronic circuit design of the new hyperbolic memristor enables the analog computer of the whole system to be designed for future engineering applications. Simulation results based on this analog computer in PSpice confirm those of the numerical investigations.

Automated summary

This article studies a new model of Hopfield Neural Network (HNN) with two neurons and a hyperbolic-type memristor. The equilibrium points analysis shows that the system has an unstable line of equilibrium in the absence of external stimuli and no equilibrium point in the presence of external stimuli, indicating hidden attractors. Analyses for both cases reveal complex homogeneous and heterogeneous bifurcations with multiple coexisting attractors.