A 5D Fractional-Order Dual-Memristor Hopfield Neural Network: Hidden Multi-Scroll Attractors, FPGA Implementation, and Image Encryption

Abstract

Unlike conventional models that typically rely on a single memristive synapse, this study uniquely proposes a novel 5D fractional-order memristive Hopfield neural network (FOMHNN) modulated by dual memristors to simultaneously emulate internal synaptic plasticity and external electromagnetic radiation effects in brain-like computing. Analytically, the FOMHNN features multiple parallel lines of equilibria with double-zero eigenvalues, rigorously proving the generation of hidden attractors. The continuous dynamical behaviors are systematically evaluated using the Adomian Decomposition Method (ADM), revealing rich phenomena including transient chaos, grid multi-scroll hidden attractors, and frequency-controllable extreme multistability with fractal-like basin boundaries. The theoretical model is physically validated on a Field Programmable Gate Array (FPGA) platform, demonstrating high precision and ultra-low power consumption. To bridge theoretical dynamics with cryptographic applications, a novel pseudo-random number generator is designed. By incorporating a chaotic derivative extractor, the generated sequences significantly reduce topological periodicity, successfully passing all rigorous NIST SP 800-22 statistical tests. Furthermore, an adaptive color image encryption scheme is developed, utilizing bidirectional feedback diffusion and least significant bit (LSB) key embedding. Security analyses confirm that the cipher, under the fractional order q=0.95, achieves near-ideal information entropy, optimal resistance against differential attacks, with NPCR and UACI values reaching 99.6114% and 33.4910%, both extremely close to their theoretical ideals (99.6094% and 33.4635%), and robust resilience against noise. Ultimately, the FOMHNN provides a highly secure and physically realizable chaotic source for advanced secure communications.