# A Novel Intermittent Jumping Coupled Map Lattice Based on Multiple Chaotic Maps
## 2 Intermittent Jumping Coupled Map Lattice  
There are three essential elements in the CML system [1]: coupling mechanism, nonlinear mapping function, and multiple processing units (lattices), which forms a reaction-diffusion architecture. In each time step, the reaction procedure updates the state variable of a lattice via the nonlinear mapping function. At the same time, the diffusion procedure establishes a coupling relation between the current lattice and a set of other lattices. The traditional CML system can be expressed as:  
\[x_{n+1}(l)=(1-e)\cdot f(x_{n}(l))+(e/2)\cdot[f(x_{n}(l-1))+f(x_{n}(l+1))], \tag{1}\]  
where \(n=1\), \(2,\cdots\), \(N\) represent the time steps. The indices \(l=1\), \(2,\cdots\), \(L\) stand for the spatial positions of lattices with a periodic boundary condition, and \(L\) is the size of the CML system. The symbol \(e\) denotes the coupling coefficient, whose value lies in the interval [0, 1]. In general, the Logistic map \(x_{k+1}\!=\!f(x_{k})\!=\!\mu\cdot x_{k}\cdot(1-x_{k})\) serves as the nonlinear mapping function, where the parameter \(\mu\in(0,\ 4]\), and \(x_{k}\) represents the state variable after \(k\) iterations. Remarkably, when \(\mu\in(3.57,\ 4]\), the Logistic map exhibits chaotic behaviors, including ergodicity and high sensitivity to initial conditions. See its bifurcation diagram and Lyapunov exponent in Figure 1a. Figure 1: The bifurcation diagrams and Lyapunov exponents for (**a**) Logistic map, (**b**) Sine map, and (**c**) Chebyshev map, respectively. Clearly, the coupling mechanism in Equation (1) is static, due to the constant coupling coefficient and the regularity of the coupling relations. To alleviate this problem, the NCML system [7] defines a non-adjacent coupling mechanism as follows:  
\[x_{n+1}(l)=(1\,-\,e)\cdot f(x_{\,n}(l))+(e\,/\,2)\cdot[f(x_{\,n}(u(l)))+f(x_{\,n }(v(l)))], \tag{2}\]  
where the two spatial positions \(u(l)\) and \(v(l)\) are obtained from the Arnold cat map. That is:  
\[\left[ \begin{array}{c}u(l)\\ v(l)\end{array} \right]=\left[ \begin{array}{cc}1&s\\ t&st+1\end{array} \right]\left[ \begin{array}{c}l\\ l\end{array} \right]\text{mod}(L)+\left[ \begin{array}{c}1\\ 1\end{array} \right], \tag{3}\]  
where \(s\) and \(t\) are the parameters of the Arnold cat map. However, as discussed before, the Arnold cat map used in Equation (3) belongs to a time-invariant transformation, and tends to sample the same spatial position repeatedly within a single time step. The DCML system [13] provides another way to break the static defect, which can be formulated as:  
\[x_{n+1}(l)=(1\,-\,\varepsilon_{n}(l))\cdot f(x_{\,n}(l))+(\varepsilon_{\,n}(l )\,/\,2)\cdot[f(x_{\,n}(l-1))+f(x_{\,n}(l+1))], \tag{4}\]  
where the coupling coefficient is no longer a constant, but varies with the spatiotemporal indices \(l\) and \(n\). To be specific, the DCML system [13] leverages an auxiliary Logistic map of the form:  
\[\varepsilon_{k+1}=f(\varepsilon_{k})=\mu^{aux}\cdot\varepsilon_{k}\cdot(1\,- \,\varepsilon_{k}), \tag{5}\]  
where \(\,k=1,\,2,\cdots,\,LN\), to generate the sequence of dynamical coupling coefficients. In practical use, this sequence is reshaped into a matrix of size \(\,L\times N\,\) so as to gear towards the spatiotemporal indices in Equation (4). The auxiliary parameter \(\mu^{aux}\) is set to 3.99 for achieving outstanding dynamics, and the initial value of \(\varepsilon_{k}\,\) is set to \(e\). The NDCML system [14], which combines the non-adjacent coupling mechanism and the dynamical coupling coefficient together, can be viewed as a natural extension of the NCML [7] and DCML [13] systems. That is:  
\[x_{n+1}(l)=(1\,-\,\varepsilon_{n}(l))\cdot f(x_{\,n}(l))+(\varepsilon_{\,n}(l )\,/\,2)\cdot[f(x_{\,n}(u(l)))+f(x_{\,n}(v(l)))], \tag{6}\]  
where \(u(l)\) and \(v(l)\) are determined by Equation (3), and \(\varepsilon_{n}(l)\) is obtained from the auxiliary Logistic map defined in Equation (5).