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

Abstract

Abstract This work investigates the dynamics and implementation of a circulant inertia three Hopefield neuron model with each neuron activated by a non-monotonic Crespi function. Owing its source to the work previously done by Song and co-authors [Song et al (2019) Mathematical Biosciences and Engineering 16. 6406–6426], we propose a network made up of three neurons connected cyclically. We demonstrate that the model is capable of the coexistence of two, three, four, six, seven, eight and ten attractors basing on different initial states. The system is dissipative and presents fifteen unstable equilibrium points for a given rank of parameters. Accordingly, we demonstrate the Hopf bifurcation in the model when the bifurcation parameter is the first synaptic weight. Moreover, using bifurcation diagrams, Maximum Lyapunov Exponent diagram, phase portraits, two parameters Lyapunov diagrams, double-sided Poincaré section and basin of attraction, intriguing phenomena have been revealed such as hysteresis, coexistence of parallel branches of bifurcation, antimonotonicity and transient chaos to name a few. 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. To verify the feasibility of the model, a microcontroller-based implementation has been used to demonstrate the period-doubling route to chaos obtained numerically.