Grid-like multi-scroll dynamics of memristive autapse-driven Hopfield neural networks

This paper introduces a novel non-volatile memristor based on a saturated function series and incorporates it as an autapse into a four-neuron Hopfield Neural Network (HNN). By replacing neuronal self-connections with memristive autapses, we construct three distinct topologies of the Memristive Hopfield Neural Network (MHNN). These networks are capable of generating grid-like multi-scroll chaotic attractors in one- to three-dimensional spaces. The saturated-function-series-based memristor offers flexible tunability and scalability in the number of stable equilibrium points. A quantitative relationship is established between the memristor parameters and the number of stable equilibria (i.e., [Formula: see text]). Simulation results show that by modulating memristor parameters, the memristive autapses on neurons 1, 2, and 3 can precisely control both the quantity and spatial distribution of the scrolls. Specifically, the memristive autapses on neurons 1, 2, and 3 govern the number of scrolls along the [Formula: see text], [Formula: see text], and [Formula: see text]-axis directions, which are given by [Formula: see text], [Formula: see text], and [Formula: see text], respectively. Compared with existing MHNNs, the proposed memristor offers more flexible parameter tuning and more precise scroll control, enabling quantifiable enhancement of system complexity. Dynamical analysis further confirms that the multi-scroll chaotic attractors arise from the global dynamics of the system rather than from the system equilibria. Modulation of the memristive autapse parameters drives the system through transitions among chaotic states, periodic orbits, and point attractors via typical routes such as period-doubling bifurcations and intermittent periodic windows. Finally, an MHNN analog circuit has been implemented using discrete components, with its correctness and feasibility demonstrated through circuit simulations. These findings highlight the role of memristive autapses in regulating neural network dynamics and provide theoretical insights for neuromorphic computing and high-dimensional secure communication.