At present, discrete memristor is a focus of chaotic mapping. However, the study of discrete memcapacitor and discrete meminductor has not begun. In order to study the dynamics of discrete meminductor and the dynamics of chaotic map containing discrete meminductor, in this article, a 5-D Hyperchaotic mapping is constructed by combining discrete memristor and discrete meminductor. In addition, the dynamical behavior of chaotic mapping is analyzed by means of the phase diagram of a chaotic attractor, lyapunov exponents (LEs), bifurcation diagram, and spectral entropy complexity. Theoretical analysis and numerical simulation show that the 5-D chaotic map not only has hyperchaotic phenomenon, but also can generate different types of chaotic attractors with the change of parameters. Especially, the special phenomenon of coexistence of attractors is discovered. This makes this 5-D hyperchaotic map more suitable for many chaotic-based engineering applications. So, an application of discrete chaotic system in chaotic image encryption is proposed. The security and reliability of the encryption scheme are analyzed by national institute of standards and technology (NIST) test, entropy, correlation, and histogram. Finally, the results of the numerical simulation are verified by a digital circuit. Due to its complex dynamical characteristics, the 5-D hyperchaotic map has great value in engineering applications.