A Second-Order Memristor Method to Construct Memristive Neural Networks With Multi-Butterfly and Multi-Scroll Dynamics

Memristor-based Hopfield neural networks (MHNNs) exhibit rich chaotic dynamics and bear closer hardware resemblance to the biological brain, making them well suited for emulating neural dynamical behaviors. However, most existing MHNNs are constructed with first-order memristors. This paper proposes a novel second-order memristor (SOM) approach for constructing MHNNs with enriched chaotic dynamics. Specifically, a second-order memristor is incorporated into a three-neuron Hopfield neural network to emulate the magnetic coupling mechanism between neurons, thereby forming a second-order memristor-based neural network (SOM-HNN). Comprehensive dynamical analyses, including bifurcation diagrams, Lyapunov exponent spectra, and numerical simulations, confirm that the proposed SOM-HNN exhibits richer and more intricate chaos behaviors than its first-order counterparts. Remarkably, the proposed SOM-HNN can simultaneously generate butterfly and scroll attractors, multi-butterfly and multi-scroll attractors, as well as initial-boosed coexisting multi-butterfly and multi-scroll attractors, thereby substantially enhancing its dynamical diversity. To the best of our knowledge, this is the first report of both multi-butterfly and multi-scroll dynamics in a neural network. Furthermore, the SOM-HNN is implemented in hardware using analog circuits and a digital field-programmable gate array (FPGA) platform. Experimental results demonstrate the network’s abundant dynamical features and its feasibility for efficient hardware realization in neuromorphic engineering applications.