# A Novel Intermittent Jumping Coupled Map Lattice Based on Multiple Chaotic Maps
## 3 Analysis of Dynamic Behaviors
### Diffusion Energy Analysis  
The goal of this analysis is to check whether the distribution of diffusion energies between lattices is uniform or not. In this paper, the diffusion energy of the \(l\)th lattice is defined as:  
\[E(l)=\sum_{l^{\prime}=1}^{L}\sum_{n^{\prime}=1}^{N}\frac{\delta(l-u_{n^{\prime }}(l^{\prime}))+\delta(l-v_{n^{\prime}}(l^{\prime}))}{2}\Big{[}\varepsilon_{n^ {\prime}}(l^{\prime})\Big{]}^{2}, \tag{11}\]  
where \(\delta(\cdot)\) denotes the Dirac delta function. Equation (11) reveals that there are two factors determining the diffusion energy. One is the total number of times that the \(l\)th lattice is sampled after \(N\) iterations. Another is the square of the coupling coefficient. Note that, for a specific CML-based system, the spatiotemporal variability of \(u_{n^{\prime}}(l^{\prime})\), \(v_{n^{\prime}}(l^{\prime})\), and \(\varepsilon_{n^{\prime}}(l^{\prime})\) in Equation (11) may be absent. For example, for the original CML system, we shall set that \(u_{n^{\prime}}(l^{\prime})=l^{\prime}-1\), \(v_{n^{\prime}}(l^{\prime})=l^{\prime}+1\), and \(\varepsilon_{n^{\prime}}(l^{\prime})=e\), respectively. Since this analysis is primarily concerned with whether the distribution is uniform or not, rather than the magnitude of \(E(l)\), we normalize the diffusion energy as follows:  
\[\overline{E}(l)=E(l)/\underset{l}{max}E(l), \tag{12}\]  
where \(l=1\), \(2,\cdots\), \(L\). It is desirable to equalize this distribution so that each lattice contributes equally to the diffusion of spatiotemporal dynamics.  
As demonstrated in [7; 8], the non-adjacent coupling mechanism can strengthen the diffusion of spatiotemporal dynamics to some extent. Hence, in this paper, we define inter-lattice diffusion distances as follows:  
\[D(l)=\frac{\sum_{r=\{u,v\}}\sum_{l^{\prime}=1}^{L}\sum_{n^{\prime}=1}^{N} \delta(l-r_{n^{\prime}}(l^{\prime}))\cdot min(\,|\,l-r_{n^{\prime}}(l^{\prime} )\,|\,L-|\,l-r_{n^{\prime}}(l^{\prime})\,|\,)}{\sum_{r=\{u,v\}}\sum_{l^{ \prime}=1}^{L}\sum_{n^{\prime}=1}^{N}\delta(l-r_{distances. In addition, we print the standard deviation (\(std^{-}\)) of the blue histogram, and print the average (\(avg^{+}\)) and the standard deviation (\(std^{-}\)) of the orange one, where the superscript \"\(+\)\" (or \"\(-\)\") is intended to indicate that a higher (or lower) value is better.  
We find that although the CML system achieves the best uniformity with zero-valued standard deviations, \(D(l)\) equals 1 for all \(l\), reflecting that each lattice only interacts with its adjacent counterpart in a regular way. As shown in Figure 3b, the inter-lattice diffusion distances reach larger values, which demonstrates that the NCML system indeed enlarges the range of coupling. However, there are several peaks that appear periodically in the histograms. This is due to the use of the Arnold cat map, which repeatedly samples the same spatial position at a regular period. Such non-uniform distribution implies that only several lattices may dominate the diffusion of spatiotemporal dynamics, leading to the local chaotic behaviors. As shown in Figure 3c, the dynamical coupling coefficient can slightly modulate the diffusion energies. However, it fails to equalize NDCML's histograms. In contrast, the proposed IJCML system equalizes the distribution of diffusion energies, and achieves larger inter-lattice diffusion distances at the same time. This validates that the IJCML system can uniformly spread the diffusion energies to non-adjacent lattices, so as to encourage the diffusion of spatiotemporal dynamics.