Fractional Hopfield Neural Networks: Fractional Dynamic Associative Recurrent Neural Networks

This paper mainly discusses a novel conceptual framework: fractional Hopfield neural networks (FHNN). As is commonly known, fractional calculus has been incorporated into artificial neural networks, mainly because of its long-term memory and nonlocality. Some researchers have made interesting attempts at fractional neural networks and gained competitive advantages over integer-order neural networks. Therefore, it is naturally makes one ponder how to generalize the first-order Hopfield neural networks to the fractional-order ones, and how to implement FHNN by means of fractional calculus. We propose to introduce a novel mathematical method: fractional calculus to implement FHNN. First, we implement fractor in the form of an analog circuit. Second, we implement FHNN by utilizing fractor and the fractional steepest descent approach, construct its Lyapunov function, and further analyze its attractors. Third, we perform experiments to analyze the stability and convergence of FHNN, and further discuss its applications to the defense against chip cloning attacks for anticounterfeiting. The main contribution of our work is to propose FHNN in the form of an analog circuit by utilizing a fractor and the fractional steepest descent approach, construct its Lyapunov function, prove its Lyapunov stability, analyze its attractors, and apply FHNN to the defense against chip cloning attacks for anticounterfeiting. A significant advantage of FHNN is that its attractors essentially relate to the neuron's fractional order. FHNN possesses the fractionalorder-stability and fractional-order-sensitivity characteristics.