## 4 Analysis on the Features of 2D-DPRCML
### Analysis on Coupling Scheme  
The conventional 2D-CML system's coupling method is adjacent and fixed. The coupled lattices of \(x_{n}\)\((i,j)\) in 2D-CML are always \(x_{n}\)\((l+1,j)\), \(x_{n}\)\((l-1,j)\), \(x_{n}\)\((i,j+1)\) and \(x_{n}\)\((i,j-1)\) at each iteration. Figure 3a illustrates 2D-CML's coupling method.  
The indexes of the coupling lattices for 2D-NLCML are calculated from the Arnold cat map:  
\[\left[\begin{array}{c}a\\ b\end{array}\right]=\begin{bmatrix}1&p\\ q&pq+1\end{bmatrix}\left[\begin{array}{c}i+1\\ i-1\end{array}\right](\text{mod}R), \tag{15}\]  
\[\left[\begin{array}{c}c\\ d\end{array}\right]=\begin{bmatrix}1&p\\ q&pq+1\end{bmatrix}\left[\begin{array}{c}j+1\\ j-1\end{array}\right](\text{mod}L) \tag{16}\]  
in which \(p\) and \(q\) are control parameters. The number of the lattices is \(R\ \times\ L\). The coupled lattices are denoted by \(x_{n}\)\((a,j)\), \(x_{n}\)\((b,j)\), \(x_{n}\)\((i,c)\) and \(x_{n}\)\((i,d)\). Even though Equations (15) and (16) are nonlinear and some values of \(p\) and \(q\) can lead to chaos, \(p\) and \(q\) are fixed at each iteration. Moreover, Equations (15) and (16) only iterate one time in the system. Thus, the coupling lattices of \(x_{n}\)\((i,j)\) are invariant in any iteration. It means that this system's coupling method is non-adjacent and fixed, as shown in Figure 3b.  
The coupling method is improved in 2D-MCPML and is not fixed as in the above two systems. Besides the two adjoining lattices \(x_{n}\)\((i+1,\ j)\) and \(x_{n}\)\((i,\ j-1)\), a non-adjacent lattice \(x_{n}\)\((a,\ b)\) with a random location is also coupled with a lattice of \(x_{n}\)\((i,\ j)\). Furthermore, \(a\) and \(b\) are decided by the state value of \(x_{n}\)\((a,\ b)\). For instance, we assign the size of the lattices as \(R\ \times\ L\) if \(x_{n}\)\((i,\ j)=0.40967142\), \(a=(40\text{ mod }R)+1\) and \(b=(96\text{ mod }L)+1\). Therefore, \(a\) and \(b\) are changed at each iteration, increasing the instability of potential periodic orbits, and enhancing the system's complexity. The coupling of a 2D-MCPML system is illustrated in Figure 3c.  
Figure 3: \(Cont.\)  
Figure 3: The coupling schemes of (**a**) 2D-CML, (**b**) 2D-NLCML, (**c**) 2D-MCPML and (**d**) 2D-DPRCML. The dark blue lattice denotes the current lattice for calculation, and the light blue lattices represent the lattices for coupling.  
In contrast, the coupling method in 2D-DPRCML presented in Figure 3d varies considerably from that of the above three systems. This coupling scheme is somewhat similar to [PERSON]'s method of selecting neighboring cells in cellular neural networks [40]. However, in the proposed scheme, the coupling lattices are not centered on the current lattice, and only four lattices are involved in the coupling rather than all lattices in a range. Its coupling is pseudo-random and dynamic rather than fixed or adjacent. The four coupled lattices break the constraints of row \(i\) and column \(j\), and they can be lattices at any position in the system. Moreover, the four coupled lattices are all changeable at each iteration. Thus, the complexity of the system is enhanced efficiently.