Multi-scroll hopfield neural network excited by memristive self-synapses and its application in image encryption

Abstract The functionality of the biological brain is closely related to the dynamic behavior generated by synapses in its complex neural system. The self-connection synapse, as a critical form of feedback synapse in Hopfield neurons, plays an essential role in understanding the dynamic behavior of the brain. Synaptic memristors can bring neural network models closer to the complexity of the brain’s neural networks. Inspired by this, this study incorporates the nonlinear memory characteristics of synapses into the Hopfield neural network (HNN) by replacing a single self-synapse in a four-dimensional HNN model with a novel cosine memristor model, aiming to more realistically reproduce the dynamical behavior of biological neurons in artificial systems. By performing a dynamical analysis of the system using numerical methods, we find that the model exhibits infinitely many equilibrium points and can induce the formation of rare transient attractors, as well as an arbitrary number of multi-scroll attractors. Additionally, the model demonstrates complex coexisting attractor dynamics, including transient chaos, periodicity, decaying periodicity, and coexisting chaos. Furthermore, the feasibility of the proposed HNN model is verified using a field-programmable gate array (FPGA). Finally, an electronic codebook (ECB)–mode block cipher encryption algorithm is proposed for image encryption. The encryption performance is evaluated, with an information entropy value of 7.9993, demonstrating the excellent randomness of the system-generated numbers.