# A Novel Intermittent Jumping Coupled Map Lattice Based on Multiple Chaotic Maps
## 2 Intermittent Jumping Coupled Map Lattice
In this paper, we propose a novel Intermittent Jumping Coupled Map Lattice (IJCML), in which the pseudo-random information generated from the Logistic map, Sine map, and Chebyshev map will be integrated together in a dynamical manner. The definitions of the Sine map and the Chebyshev map can be described by the following equations:  
\[y_{k+1}=g(y_{k})=\gamma\cdot\sin(\pi\,\cdot\,y_{k}), \tag{7}\]  
\[z_{k+1}=h(z_{\,k})=\cos(\phi\cdot\arccos(z_{\,k})), \tag{8}\]  
where \(\gamma\) and \(\phi\) are the systems' parameters. As shown in Figure 0(b), the Sine map shares similar chaotic behaviors with the Logistic map. As shown in Figure 0(c), \(z_{k}\) fills in the range \([-1,\,1]\), and the Lyapunov exponent increases monotonically with \(\phi\). The Chebyshev map starts to enter the chaotic regime when \(\phi\geq 1\). With these preparations, we formulate the IJCML system as:  
\[x_{n+1}(l)=\left\{ \begin{array}{ll}(1-e)\cdot f(x_{\,n}(l))\,+\,(e/2)\cdot[g (x_{\,n}(u_{\,n}(l)))\,+\,g(x_{\,n}(v_{\,n}(l)))],&\text{if }w_{n}(l)\geq 0.5,\\ f(x_{\,n}(l)),&\text{otherwise},\end{array} \right. \tag{9}\]  
where \(w_{n}(l)\) determines the couple state of the \(l\)th lattice at the \(n\)th time step. In this paper, the Chebyshev map is used to generate \(w_{n}(l)\), taking the form \(w_{n}(l)=(z_{\,n}(l)+1)/2\), where \(z_{n}(l)\) is the reshaped version of \(z_{k}\). The parameter \(\phi\) is set to 3.999 so that \(w_{n}(l)\) varies with the spatiotemporal indices dynamically. The intermittence is reflected in the phenomenon that a lattice sometimes interacts with other ones (when \(w_{n}(l)\geq 0.5\)), and other times updates alone (when \(w_{n}(l)<0.5\)). Clearly, the coupling states are mutually different between lattices. When \(w_{n}(l)\geq 0.5\), the spatial positions of coupled lattices are \(u_{n}(l)\) and \(v_{n}(l)\), respectively. In this paper, we abandon the Arnold cat map, and resort to a chaotic map to determine \(u_{n}(l)\) and \(v_{n}(l)\). Specifically, the pseudo-random information \(w_{n}(l)\) is reused here in the following form:  
\[\left\{ \begin{array}{l}u_{n}(l)=mod(\left\lfloor w_{n}(l)\cdot 10^{\mathcal{T}} \right\rfloor,\;L)+1,\\ v_{n}(l)=mod(\,\left\lfloor w_{n}(l)\cdot 10^{\mathcal{T}}\right\rfloor,\;L)+1, \end{array} \right. \tag{10}\]  
where the floor sign \(\left\lfloor\cdot\right\rfloor\) rounds down to the nearest integer of the number enclosed within the sign. The sign \(\left\lvert\right\rangle\) taking its right operand, namely \(\left\lfloor w_{n}(l)\cdot 10^{\mathcal{T}}\right\rfloor\) in Equation (10), as input represents a compound S-box substitution operation. Specifically, the right operand is first converted to a six-digit hexadecimal integer. For example, if \(\left\lfloor w_{n}(l)\cdot 10^{\mathcal{T}}\right\rfloor=1234567\), its hexadecimal representation equals '12D687'. Second, the six-digit hexadecimal integer is divided into three groups, each of which contains two digits. The three groups of '12D687' are '12', 'D6', and '87', respectively. Third, apply substitution operation to each group in turn, where we use the S-box recommended in AES [30]. See Figure 2 for details. By doing so, we obtain 'C9', 'F6', and '17', respectively. Forth, concatenate the substitution values, yielding 'C9F617'. Convert this six-digit hexadecimal integer into decimal representation, yielding 13235735. As shown in Equation (10), the compound S-box substitution operation is only used in the course of calculating \(v_{n}(l)\), and the nonlinearity of the S-box ensures that \(u_{n}(l)\
eq v_{n}(l)\) holds. This is a straightforward way to avoid sampling the same spatial position for a specific spatiotemporal index. The pseudo-randomness of \(u_{n}(l)\) and \(v_{n}(l)\) inherits from \(w_{n}(l)\), so that they vary with the spatiotemporal indices as well. The new non-adjacent coupling relations are time-varying, compared with the one in NCML [13] or NDCML [14]. In this paper, we use the term \"jumping\" to characterize the pseudo-randomness and the spatiotemporal variability of the coupling relation.