## 4 Analysis on the Features of 2D-DPRCML
### Analysis on the Lyapunov Exponent and Kolmogorov-Sinai Entropy  
The Lyapunov exponent (LE) [4] is a vital measure for evaluating the separation rate of infinitely approaching trajectories in phase space. It is a universal methodology for characterizing the predictability of dynamic systems. It is defined as:  
\[\lambda=\lim_{n\ \rightarrow\ \infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln\left|\frac{ dF(x)}{dx}\right|_{x=x_{i}}, \tag{17}\]  
where \(\lambda\) stands for the LE of a dynamic system \(F(x)\) and \(i\) stands for the time dimension. The system requires at least one LE greater than zero to be in chaos. In addition, the higher the \(\lambda\), the better the system exhibits chaotic properties. Without loss of generality, we employ the Wolf method [41] utilized in the literature [6; 26; 27] to calculate LE. On this basis, Kolmogorov-Sinai entropy density (KED) is the mean value of LEs greater than zero in all lattices. It is introduced to characterize the chaotic properties of the system [1; 4; 27]. The KED is calculated as follows:  
\[h=\frac{\sum\limits_{i=1}^{R}\sum\limits_{j=1}^{L}\lambda^{+}(i,j)}{R\times L}, \tag{18}\]  
in which \(h\) denotes the KED and \(\lambda^{+}\) is the positive LE of \(R\ \times\ L\) lattices. Here, \(i\) and \(j\) are space dimensions. An \(h\) value higher than zero means that the system contains lattices existing in chaos. Moreover, the magnitude of \(h\) determines the strength of the system's chaotic properties. The performance of the KEDs of the four systems is displayed in Figure 4.  
In Figure 4, the X-axis, Y-axis and \(Z\)-axis denote coupling coefficient \(\varepsilon\), control parameter \(\mu\) and KED \(h\), respectively. In Figure 4a,b, the KEDs of these two systems are positive only when \(\mu>3.6\), which means that chaotic lattices only exist in interval \(\mu\in\) (3.6, 4]. Moreover, the chaos is weak around \(\varepsilon=0.2\) in the 2D-CML system shown in Figure 4a, while this drawback is ameliorated in 2D-NLCML shown in Figure 4b. In the first two systems, only 2.24% and 1.12% of the parameter pairs, respectively, make the system's KED higher than 0.5. By contrast, as shown in Figure 4c,d, 89.76% and 97.76% of the control parameters in 2D-MCPML and 2D-DPRCML, respectively, make the KED of the system higher than 0.5, and some KEDs of these two systems even reach 0.8. The KEDs of these two systems are positive in the entire interval of \(\varepsilon\) and \(\mu\). All the above analyses indicate that the chaotic behaviors of the latter two CMLs are stronger than those of the former two systems. Additionally, more parameters are available to be used as keys when the latter two systems are employed in cryptosystems. In other words, using 2D-MCPML and 2D-DPRCML systems can efficiently broaden the key space of cryptography. Furthermore, the minimum and average values of KED in Figure 4d are 0.4266 and 0.7034, respectively, which are higher than 0.1994 and 0.6487 in Figure 4c. Therefore, the chaotic properties of 2D-DPRCML are superior to those of 2D-MCPML. In addition to the KED, [PERSON] [22; 42] proposed the Kolmogorov-Sinai entropy breadth (KEB) to indicate the proportion of chaotic lattices in the CML systems. The KEB is calculated as follows:  
\[hu=\frac{L^{+}}{L}, \tag{19}\]  
in which \(hu\) is the KEB. \(L^{+}\) refers to the number of lattices where LE is greater than zero. The system's lattice count is indicated by \(L\). KEB which is employed to evaluate the chaotic performance of the spatiotemporal chaotic system from the spatial perspective. A larger \(hu\) value implies that more lattices are in chaos, and the system possesses stronger chaotic properties. The KEBs of the four systems mentioned above are shown in Figure 5.  
In Figure 5a,b, the KEBs reach '1' only when \(\mu>3.6\), which corresponds to the KEDs in Figure 4a,b, which means that all lattices are in chaos only when \(\mu>3.6\) in 2D-CML and 2D-NLCML systems.