# A Novel Intermittent Jumping Coupled Map Lattice Based on Multiple Chaotic Maps
## 3 Analysis of Dynamic Behaviors
### Computational Complexity Analysis  
Computational complexity reflects the time and space consumptions while executing a target algorithm so as to provide a guidance for developing practical software.  
First, we theoretically analyze the time complexity. Like the baseline systems [1, 7, 13, 14], the proposed IJCML system also has two nested \"for\" loops in its calculation procedure. The outer \"for\" loop runs the dynamical system by \(N\) iterations, while the inner one  
Figure 13: Space-amplitude plots for spatiotemporal behavior analysis. The setting of \(\mu\), from left to right, is 3.15, 3.57, and 3.99, respectively.  
traverses the \(L\) lattices. Hence, all the five CML-based systems investigated in this paper have the quadratic time complexity, namely \(\mathcal{O}(NL)\).  
Second, we theoretically analyze the space complexity. Clearly, it is necessary to save the resulting pseudo-random matrix of size \(N\ \times\ L\). This requires quadratic space complexity, namely \(\mathcal{O}(NL)\). Compared with the original CML system [1], the variants, i.e., the NCML system [7], the DCML system [13], the NDCML system [14], and the proposed IJCML system need to occupy additional space to save the intermediate variables obtained from the auxiliary chaotic maps. See Equations (3), (5), and (8) for details. Fortunately, since these auxiliary chaotic maps can be separately executed out of the two nested \"for\" loops, the final space complexity remains the same, namely \(\mathcal{O}(NL)\).  
However, it is undeniable that the variants may require more time and space consumptions in practice. To make an intuitive comparison, we experimentally count the average time for running 1000 iterations over 100 lattices, and inspect the size of the additional space. The parameters \(\mu\) and \(e\) are set to \([3,\ 3.99]\) and \([0,\ 0.99]\), respectively, and both of them take 0.01 as step size. Our computing device is a desktop computer with a 2.90 GHz Intel i7-10700 central processing unit, 16.00 GB memory. Our programming environment is Matlab R2017a installed on the Window 10 operation system. The intermediate variables are saved as \".mat\" format.  
Table 1 lists the simulation results. The IJCML system takes 0.5400 s, on average, to generate a pseudo-random matrix of size \(1000\times 100\), which is faster than the NCML system [7], the DCML system [13], and the NDCML system [14], and is comparable to the original CML system [1]. High efficiency of the IJCML system is due to the intermittent jumping mechanism because the coupling operation, as shown in Equation (9), will be skipped with a probability of about 50%. On the other hand, the NCML system [7] only requires 3.42 KB space to additionally save the time-invariant spatial positions. In contrast, the remaining three variants prepare about 746.00 KB space to accommodate the dynamical coupling coefficients [13; 14] or the intermediate variables \(w_{n}(l)\) obtained from the Chebyshev map. Fortunately, a space of size 746.00 KB is almost negligible for modern hardware devices. Consequently, the computational complexity of the proposed IJCML is better than or comparable to the baseline systems [1; 7; 13; 14].