Dynamic analysis of multi-spiral chaotic inertia model with a cyclic configuration involving four homogeneous HNN cells: stability analysis, analog and digital verifications.

This paper investigates the behavior of a Hopfield neural network consisting of four interconnected inertial neurons arranged in a loop configuration. The mathematical equation that governs the overall dynamic of the model is consists of a set of eight first-order ordinary differential equations (ODEs) with odd symmetry. The system has 81 equilibrium points, some of which undergo multiple Hopf bifurcations as a control parameter is varied. The maximum number of coexisting states is related to the maximum number of active equilibrium points. Through numerical investigations, intriguing nonlinear properties are discovered, including both homogeneous and heterogeneous multistability and the coexistence of up to sixteen bifurcation branches, the presence of multi-spiral chaos, crisis phenomenon, period splitting and the oscillation death phenomenon. In order to obtain a comprehensive understanding of the dynamics, various tools are used, such as phase portraits, bifurcation diagrams, Poincare maps, frequency spectra, Lyapunov exponent spectra, and attraction basins. A Significant achievement of this study is the demonstration that coupling inertial neurons can be an effective method to generate multi-spiral chaotic signals. The overall dynamics is non-hidden and meticulous adjustment of the gradient connected to the fourth neuron allows to complete annihilate oscillations (no motion) in the neural network in a particular interval. Finally, an electronic circuit inspired by the coupled inertial neuron system is designed using Orcad-PSpice software and implemented using an Arduino-based microcontroller. The simulation results from PSpice and microcontroller confirm the findings from the theoretical analysis.