A new general-purpose voltage-controlled memristor with a fourth power term is designed in this paper. Based on this memristor, a new four-dimensional chaotic system, the memristor cell neural network (CNN) chaotic system, is constructed with complex dynamic characteristics, high spectral entropy complexity (SE), and wavelet entropy complexity (C0). Combination with analysis including phase portraits, equilibrium points, dissipativity, Lyapunov exponent spectrum, bifurcation diagrams, transient phenomena, spectral entropy complexity (SE), and wavelet entropy complexity (C0), the dynamic characteristics of the new system are thoroughly explored. Constructed analog circuits for the new memristor and new memristor CNN chaotic system, and a digital circuit is implemented using a field-programmable gate array. Combining the new memristor CNN chaotic system with DNA encoding operations for image encryption applications, its excellent encryption effectiveness and high security are discovered through analysis of key sensitivity, histogram, information entropy, correlation, keyspace, and robustness.