A circulant inertia three Hopfield neuron system: dynamics, offset boosting, multistability and simple microcontroller- based practical implementation

Abstract

This work investigates the dynamics and implementation of a three inertia circulant neurons model with each neuron activated by a non-monotonic Crespi function. Owing its source to the work previously done by Z. Song and co-authors[1], we proposes a network made up of three neurons connected cyclically. Z. Song and collaborators studied the dynamics of an inertia neural system and demonstrated the mixed coexistence of periodic orbits and chaotic attractors but they show the coexistence of only two and four attractors. Recently, B.F.Boya and co-workers[2] break the symmetry of the previous system then, demonstrated the coexistence of two, three, four and six attractors. Here, is analyzed a model capable of the coexisting of two, three, four, six, eight and ten attractors basing on different starting points. Analytically, the system is dissipative and presents fifteen equilibrium unstable for a given rank of parameters. This is the reason of the demonstration of Hopf bifurcation in the model when the bifurcation parameter is the first synaptic weight. Also, using bifurcation diagrams, Maximum Lyapunov Exponent diagram, phase portraits, two parameters Lyapunov diagrams, double-side Poincaré section and basin of attraction, intriguing phenomena have been demonstrated such as hysteresis, coexistence of parallel branches of bifurcation, antimonotonicity to name these few. Moreover, offset boosting is exhibiting associated once with transient oscillation leading from one chaotic state to another which is an input to the dynamics of inertia neural systems particularly and chaotic system in general. A number of coexisting attractors have been developed by the new network which can be used to build sophisticated cryptosystem or to explain the possible tasks of a brain in normal or abnormal cases. In other to emphasize the feasibility of the model, a microcontroller-based implementation has been applied to demonstrate the period-doubling route to chaos obtained numerically.