<p>This study proposes a novel multi-scroll realization method and its coexistence patterns for the memristive Hopfield neural network (MHNN). Specifically, a flux-controlled piecewise nonlinear memristor incorporated with step functions is designed to construct asymmetrically distributed multi-scroll attractors. Regulating system parameters overcomes the limitations of traditional methods that rely on generating uniform piecewise linear functions, allowing for the stable coexistence of multi-scroll attractors with arbitrary quantities and distributions. Adjusting the initial conditions of the system and memristor parameters can further expand the boundaries of the system’s dynamic behaviors, thereby revealing the coexistence phenomenon between periodic states and multi-scroll chaotic states. To verify the effectiveness of the proposed model, circuit simulation experiments on the MHNN are conducted, and the results demonstrate that the constructed model possesses good chaotic characteristics and stability. The multi-scroll generation and coexistence mechanism proposed in this paper provides a new idea for the flexible design and engineering implementation of multi-scroll chaotic systems.</p>