Fractional-order-induced symmetric multi-scroll chaotic attractors and double bubble bifurcations in a memristive coupled Hopfield neural networks

The multi-state memristor is a type of memristor capable of memorizing multiple conductance states. This special feature is successfully used to mimic the behavior of neural synapses and simulate the influences of electromagnetic induction and radiation in artificial neural networks such as Hopfield neural networks. This paper introduces and examines a novel fractional-order Hopfield neural network model consisting of two non-identical sub-Hopfield neural networks coupled by a multi-stable flux controlled memristor (MCFHNN). The pinched hysteresis loops of the multi-stable flux controlled memristor are analyzed via numerical simulations. The equilibrium points of MCFHNN model and their stability are investigated. The introduced MCFHNN model is solved by using Adomian Decomposition Method (ADM). By using some examination methods such as spectral entropy complexity, bifurcation diagrams, phase portraits, maximum Lyapunov exponent and basins of attraction, we analyze the dynamic behaviors of MCFHNN model associated with coupling strength, fractional-order and initial states. Numerical results demonstrate that the proposed MCFHNN model is capable to develop abundant and complicated dynamics including, chaos, multiple transient transition behaviors, double bubble bifurcations, initial-boosted behavior and symmetric multi-scroll chaotic attractors. It is found that the complexity results are in agreement with those obtained from the bifurcation diagrams. This proves that the complexity can well reflect the dynamic behaviors of MCFHNN model. In order to support theoretical and numerical results, the designed MCFHNN model is implemented with the help of an Arduino-Due microcontroller board based on the Atmel SAM3X8E ARM Cortex-M3 processor. The results are in very good agreement with those obtained from computational simulations.