Building artificial neural network models and studying their dynamic behaviors is extremely important from both a theoretical and practical standpoint due to the rapid advancement of artificial intelligence . In addition to its engineering applications, this article concentrates primarily on the memristor model and chaotic dynamics of the asymmetric memristive neural network. First, we develop a novel memristor model, which is multistable and highly tunable. Using this memristor model to build an asymmetric memristive Hopfield neural network (AMHNN), the chaotic dynamics of the proposed AMHNN are investigated and analyzed using fundamental dynamics techniques such as equilibrium stability, bifurcation diagrams, and Lyapunov exponents. According to the findings of this study, the proposed AMHNN possesses a number of complex dynamic properties, including scaling amplitude chaos with coupling strength control, and coexisting uncommon chaotic attractors with initial control and coupling strength control. Significantly, the proposed AMHNN has been observed to exhibit the phenomenon of infinitely persisting uncommon chaotic attractors. In the interim, a system for image encryption based on the proposed AMHNN is constructed. By analyzing correlation, information entropy, and key sensitivity, the devised encryption method reveals a number of benefits. The feasibility of the encryption method is validated through field-programmable gate arrays hardware experiments, and the proposed memristor and AMHNN models have been translated into a Simulink model.