## 1 Introduction  
In 1971, [PERSON] et al. theoretically predicted the existence of a nonlinear passive electronic component between charge and magnetic flux according to the theory of electronics, which they named the memristor, and proposed the concept of the generalized memristor [1, 2]. For a long time, due to the lack of suitable materials, the research of memristors has been in the theoretical stage. Until 2008, HP Laboratory in the United States manufactured a solid memristor by using double-layer _TiO\({}_{2}\)_ thin film, which made the memristor model physically realized [3]. In 2013, [PERSON] and [PERSON] [4, 5] proposed three fingerprints of the memristor, showing that a memristor can exhibit a pinched hysteresis loop when applying a bipolar sinusoidal current. When the frequency exceeds the critical value, the area of hysteresis sidelobe decreases monotonically with the increase of the frequency. As the frequency increases to infinity, the tight hysteresis loop of the device degenerates into a single-valued function. As the memristor is a nonlinear component and has memory capability, memristors are widely used in neural networks [6, 7, 8], nonlinear circuit systems [9, 10], nanotechnology [11], electrical engineering, [12, 13] and other fields. Chaos is an inherent randomness of deterministic systems and a special motion of nonlinear dynamic systems widely existing in nature [14]. Due to the defects of the classical chaotic systems, such as simple algebraic structure and few control parameters, the system output has a short period and the value distribution of the sequence is not uniform [15, 16]. Many scholars are devoted to improving the complexity of classical chaotic systems. [PERSON] et al. [17] proposed a parametric switching chaotic system and applied it to image encryption schemes. [PERSON] et al. [18, 19] took the output of a low-dimensional system as the input of a nonlinear function to improve the dynamic characteristics of low-dimensional chaotic system. All these works were based on mathematical concepts without physical background. In 2008, [PERSON] [20] replaced the original Chua diode with an active memristor in a chaotic circuit, and the results show that the circuit can also generate chaotic behavior. Later, scholars have done a lot of research on the memristor chaotic system, such as multi-stable states [21; 22], co-existence attractor [23; 24], and hidden attractor [25]. However, all these work are based on continuous-time memristor models (CMM). Compared with CMM, discrete-time memristor models (DMM) have not received much attention. In recent years, [PERSON] et al. [26] proposed a discrete-time ideal charge-controlled memristor model. The output of the model was used as a control parameter of the two-dimensional Henon mapping, and the output of the Henon mapping was disturbed to enhance the dynamic characteristics of the Henon mapping. [PERSON] et al. [27] proposed a discrete time ideal charge-controlled memristor model, and linearly coupled the model with a logistic map to obtain a new two-dimensional memristor model. [PERSON] et al. [28] also proposed a two-dimensional hyperchaotic system based on a discrete memristor and analyzed the nonparametric bifurcation mechanism of the system. However, the above works just replace the output of the discrete memristor with the parameters of the chaotic system, or use the output of the discrete memristor as the input of the chaotic system. This method can improve the complexity of the chaotic system, but the effect is not obvious enough. In this paper, a generalized two-dimensional hyperchaotic system based on a discrete memristor is proposed. The discrete memristor is coupled with a classical logistic map, a sine map, and a tent map, respectively, to generate a two-dimensional memristor chaotic map. These three models are chaotic in a wide range of parameters, and even hyperchaotic in a suitable range of parameters. This method is simple and effective to enhance the dynamic characteristics of chaotic systems.