With the rapid development of artificial intelligence, it has important theoretical and practical significance to construct neural network models and study their dynamical behaviors. This article mainly focuses on the bionic model and chaotic dynamics of the asymmetric neural network as well as its engineering application. We first construct a memristor-coupled asymmetric neural network (MANN) utilizing two asymmetrical sub-neural networks and a coupled multipiecewise memristor synapse. Then, the chaotic dynamics of the proposed MANN is studied and analyzed by using basic dynamics methods like equilibrium stability, bifurcation diagrams, Lyapunov exponents, and Poincare mappings. Research results show that the proposed MANN exhibits multiple complex dynamical characteristics including infinitely wide hyperchaos with amplitude control, hyperchaotic initial-boosted behavior, and arbitrary number of hyperchaotic multi-structure attractors. More importantly, the phenomena of the infinitely wide hyperchaos and the hyperchaotic multi-structure attractors are observed in neural networks for the first time. Meanwhile, applying the hyperchaotic multi-structure attractors, a color image encryption scheme is designed based on the proposed MANN. Performance analyses show that the designed encryption scheme has some merits in correlation, information entropy, and key sensitivity. Finally, a physical circuit of the MANN is implemented and various typical dynamical behaviors are verified by hardware experiments.