# A Novel Intermittent Jumping Coupled Map Lattice Based on Multiple Chaotic Maps
## 3 Analysis of Dynamic Behaviors
### Lyapunov Exponent Analysis  
Lyapunov exponent [31] is a canonical metric in describing a dynamical system. It quantifies the average exponential rate of separation (or convergence) between two initially close trajectories along each direction of phase space. Mathematically, the definition of Lyapunov exponent is formulated as:  
\[\lambda=\lim_{n\to 0}\frac{1}{n}\sum_{i=1}^{n}\ln\left|\frac{\mathrm{d}F(x )}{\mathrm{d}x}\right|_{x=x_{i}}, \tag{19}\]  
Figure 8: Average values of MIED for the inter-lattice independence analysis. (**a**) Result for the CML system. (**b**) Result for the NCML system. (**c**) Result for the DCML system. (**d**) Result for the NDCML system. (**e**) Result for the proposed IJCML system. The symbol ‘max’ on the vertical axis signifies the maximum value of the average information entropy, namely 3.3219.  
where \(\lambda\) denotes the Lyapunov exponent of \(F(x)\), while \(i\) stands for the time step. Typically, a positive Lyapunov exponent indicates that \(F(x)\) has chaotic behaviors with the trajectories diverging exponentially. Kolmogorov-Sinai Entropy Density (KSED), which incorporates the Lyapunov exponents of all lattices, is devoted to measuring the overall dynamics of a CML-based system. Formally, its definition is written as:  
\[KSED=\sum_{l=1}^{L}\lambda^{+}(l)/L, \tag{20}\]  
where \(\lambda^{+}(l)\) represents the positive Lyapunov exponent of the \(l\)th lattice. In other words, Equation (20) overlooks all negative Lyapunov exponents during its calculation. In addition, Kolmogorov-Sinai Entropy Breadth (KSEB) [32] is used, in this paper, to count the proportion of chaotic lattices. We calculate its value as follows:  
\[KSEB=L^{+}/L, \tag{21}\]  
where \(L^{+}\) denotes the number of lattices whose Lyapunov exponents are positive. Clearly, the higher the values of KSED and KSEB, the better chaotic behaviors of a dynamical system. Figure 9 exhibits the resulting values of KSED under the setting \(3\leq\mu<4\) and \(0\leq e\leq 1\). Starting from \(\mu=3.57\), the DCML system possesses a smooth surface with positive curvatures, validating that increasing \(\mu\) can constantly strengthen the chaotic behaviors. For the NDCML system, however, there exist two objectionable ravines appearing at \(\mu=3.74\) and \(3.84\), and straightly passing through the surface. The CML and NCML systems behave unstably when \(\mu>3.57\), as the values of KSED are varying sharply and a distinct valley appears at the range \(3.57<\mu<3.99\) and \(0.11<e<0.16\). The IJCML system achieves numerous high values of KSED that form a flat plateau. Noticeably, even when \(\mu<3.57\), setting \(e\) to a higher value can still lead the system into the chaotic regime. Thus, the proposed system has a larger parameter space of spatiotemporal chaos, which benefits from the intermittent jumping mechanism. Figure 9: Resulting values of KSED for the Lyapunov exponent analysis. (**a**) Result for the CML system. (**b**) Result for the NCML system. (**c**) Result for the DCML system. (**d**) Result for the NDCML system. (**e**) Result for the proposed IJCML system. In Figure 10, we display the resulting values of KSEB. The maximum value of KSEB, being equal to 1.0, means that all the lattices in the system are activated into spatiotemporal chaos. At first glance, the IJCML system's flat plateau, as shown in Figure 10e, is of the largest area. We note that, among the baseline systems, only the DCML system has a stable flat plateau when \(\mu>3.57\). For the other baseline systems, there exist deep valleys or straight ravines embedded into the flat plateau. In other words, even though \(\mu\) is set to a higher value, some lattices are still far from spatiotemporal chaos. Therefore, one shall spend time inspecting the dynamics carefully for each lattice before deploying them in an image cryptosystem. Further, we design a statistical tool called Cumulative Percentage Function (CPF) for analyzing _KSEB_.