In order to extend the research and application of discrete memristor, a discrete memristor model with square nonlinearity is constructed based on difference operator, and Simulink simulation is realized. The simulation results show that the designed memristor satisfies the definition of a generalized memristor. A new three-dimensional memristor chaotic mapping is designed by introducing the obtained discrete memristor into the three-dimensional chaotic map, and a Simulink model of this chaotic map is established. The complex dynamic characteristics of the system are analyzed by equilibrium point, bifurcation diagram, Lyapunov exponent spectra, sample entropy complexity and multi-steady state. From the perspective of system modeling, this paper constructs the chaotic mapping between discrete memristor and discrete memristor, further verifies the feasibility of discrete memristor, and lays the foundation for the application research of discrete memristor.