## 2 Two Dimensional Generalized Discrete Memristor Coupling Model
#### Memristor Coupled Low-Dimensional Discrete Chaos Model  
Due to its simple algebraic structure, one-dimensional discrete chaotic systems such as logistic map, sine map, and tent map have defects such as short period phenomenon, uneven distribution of output sequence, and predictable motion trajectory. However, these defects will limit the application of one-dimensional chaotic systems in some security fields. In order to enhance the chaotic dynamics characteristics of one-dimensional chaotic maps, a generalized two-dimensional discrete memristor model (2D-DMM) is obtained by coupling one-dimensional chaotic systems with the discrete memristor model constructed above. The structural block diagram of 2D-DMM is shown in Figure 2, in which \(F(\mu,x_{n})\) represents an existing 1D discrete map, and \(\mu\) is the control parameter of it. The parameter \(k\) is the coupling coefficient between the chaotic map and the discrete memristor model. Some 2D-DMM can be yielded by using such a system structure. The mathematical expression of 2D-DMM can be obtained  
\[\left\{\begin{array}{c}x_{n+1}=F(\mu,x_{n})+k(R_{1}q_{n}+R_{2}q_{n}{}^{2})x_ {n}\\ q_{n+1}=x_{n}+q_{n}\end{array}\right. \tag{5}\]  
According to Equation (5), based on different one-dimensional discrete chaotic systems, corresponding two-dimensional discrete memristor models can be obtained by using this system framework. This model combines the outputs of two nonlinear systems linearly to improve the dynamics of one system. This method is simple and effective to enhance the dynamic characteristics of chaotic systems.  
Figure 1: The \(i_{n}-v_{n}\) amplitude curves of the memristor: (**a**) \(i_{n}\) and \(v_{n}\) iterative sequences; (**b**) The pinched hysteresis loop of the memeristor at different frequencies \(\omega\); (**c**) The pinched hysteresis loop of the memeristor at different amplitudes \(A\); (**d**) The pinched hysteresis loop of the memeristor at different initial charges \(q\).  
Figure 2: Structure block diagram of the two-dimensional discrete memristor model.  
The stability of discrete chaotic system is described by its fixed point. A fixed point of a map is the value of iteration at the next moment, which is the same as the value at this moment. The fixed point of Equation (5) is denoted as \((\tilde{x},\tilde{q})\), and one can obtain the following expression.  
\[\begin{cases}&\tilde{x}_{n}=F(\mu,\tilde{x}_{n})+k(R_{1}\tilde{q}_{n}+R_{2} \tilde{q}_{n}^{2})\tilde{x}_{n}\\ &\tilde{q}_{n}=\tilde{x}_{n}+\tilde{q}_{n}\end{cases} \tag{6}\]  
Obviously, when \(F(\mu,x)=0\), the system has infinite fixed points, which are expressed as  
\[S=(\tilde{x},\tilde{q})=(0,\tilde{\xi}) \tag{7}\]  
where \(\tilde{\xi}\) is an arbitrary real number. While \(F(\mu,x)\
eq 0\), since Equation (6) has no real root, the system has no fixed point. The Jacobian matrix of Equation (5) is shown as follows  
\[J=\begin{bmatrix}k(R_{1}q_{n}+R_{2}{q_{n}}^{2})+\phi(\mu),&k(R_{1}q_{n}+R_{2}{q _{n}}^{2})x_{n}\\ 1,&1\end{bmatrix}, \tag{8}\]  
where \(\phi(\mu)=dF(\mu,x_{n})/dx_{n}\). The eigenvalue polynomial at the fixed point \(S\) of Equation (5) can be obtained  
\[P(\lambda)=(\lambda-1)(\lambda-k(R_{1}q_{n}+R_{2}{q_{n}}^{2})-\phi(\mu)). \tag{9}\]  
Therefore, the two eigenvalues at fixed points \(S\) of Equation (5) are calculated as  
\[\lambda_{1}=1,\lambda_{2}=k(R_{1}q_{n}+R_{2}{q_{n}}^{2})-\phi(\mu). \tag{10}\]  
According to the stability criterion of fixed points, when the modules of the two eigenvalues are in the unit circle, the fixed point is stable, otherwise it is unstable. Since \(\lambda_{1}\) is always on the unit circle, whether \(\lambda_{2}\) is inside or outside the unit circle depends on \(\phi(\mu)\), the coupling coefficient \(k\), and the initial state \(\tilde{\zeta}\) of the memristor. Therefore, the fixed points \(S\) of the model may be stable or unstable. While \(F(\mu,x)\
eq 0\), the Equation (6) has no fixed point. In this case, the trajectory of the map is periodic, chaotic, or even hyperchaotic.