Investigation on the regular and chaotic dynamics of a ring network of five inertial Hopfield neural network: theoretical, analog and microcontroller simulation.

The studies conducted in this contribution are based on the analysis of the dynamics of a homogeneous network of five inertial neurons of the Hopfield type to which a unidirectional ring coupling topology is applied. The coupling is achieved by perturbing the next neuron's amplitude with a signal proportional to the previous one. The system consists of ten coupled ODEs, and the investigations carried out have allowed us to highlight several unusual and rarely related dynamics, hence the importance of emphasizing them. The main analysis tools that have helped in obtaining the results presented are phase portraits, bifurcation diagrams, and the Maximal Lyapunov exponent. In this system, we have observed phenomena such as the coexistence of homogeneous and heterogeneous attractors, period-doubling crisis, parallel branches, and the path leading to hyperchaotic multi-spiral. All attractors are non-hidden as they originate from well-known equilibrium points. The system has 254 equilibrium points, among which only 32 undergo a Hopf bifurcation followed by period-doubling, leading to a merging crisis phenomenon until the final hyperchaotic multi-spiral attractor. For the same parameter values (coupling or dissipation), a maximum of 30 attractors for the coupling coefficient and 32 attractors for dissipation coexist, and illustrated by the phase portraits. Virtual verification using Pspice and practical verification using an Arduino Mega 2580 microcontroller of the model have also been reported. They are in perfect agreement with the behaviors resulting from numerical investigations. The circuit energy and dimensionless energy has been estimated and the scale relation established. The results presented further enrich previous and recent work in the study of the nonlinear dynamics of Hopfield-type neural networks. Additionally, it is important to mention that cyclic coupling typology may be used as an alternative approach in generating multi-spiral signals in Hopfield oscillators.