## 3 Analysis and Results
### Analysis of Coupled Dynamic Model
The coupling strength between \(x1\) and \(x2\) was set to 0.1, 0.3, and 0.5, respectively, and the results are shown in Figure 2. Figure 2a shows the results of the CBTN method and Figure 2b shows the results of the CPE method. It can be found that the two methods can make a good distinction between different coupling strengths. The CPE method can reach a stable state when the window width is less than 1000 samples, and the CBTN method can reach a stable state when the window width is more than 2000 samples. It was speculated that the CPE was appropriate for the analysis of short time series when it was proposed. Therefore, the appropriate window width for the CBTN is 2000 samples. It should be noted that the differences \(H_{CPE}^{x1-x2^{Surrogate}}-H_{CPE}^{x1-x2}\) had a negative value with the CPE method when the coupling strength was 0.1. This is inconsistent with the theory and indicates that the CPE was not capable of differentiating the weak coupling state. With the determined window width of 2000 samples, the impact of the embedding dimension \(d\) on the estimation of the coupling strength was further investigated using the same method as above. The dimension \(d\) varied from 3 to 15 with a step of 1. Figure 3 compares the results between the CBTN method and the CPE method. It can be seen that the results of CBTN method tended to be stable with the increase in the embedding dimension \(d\). When the embedding dimension was greater than 10, the measured difference under three coupling strength levels basically reached a stable state. In contrast, the measured difference using the CPE method was greatly affected by the embedded dimension. Within the varying range of the embedded dimension, the measured difference under three coupling strength levels could reach a stable state. When the embedding dimension was greater than 11, the measured difference under the stronger coupling strength (C = 0.3) was even smaller than that under the weaker coupling strength (C = 0.1 and C = 0.2). These results demonstrated that the CBTN method can be less affected by the embedding dimension \(d\) compared with the CPE method. More specifically, the embedding dimension would have little influence when it is greater than a certain value. Accordingly, for the unidirectional coupled Henon model, the recommended embedding dimension \(d\) for the CBTN method was set to 10. After the embedding dimension and the sliding window width were determined, the CBTN was used to analyze the unidirectional coupled Honen model under different coupling strengths \(C\). Specifically, the coupling strength \(C\) increased from 0 to 0.9 with a step of 0.05. For a given coupling strength, the data of \(x1\), \(x2\) and \(x2^{Surrogate}\) within individual sliding windows were analyzed using the CBTN and CPE methods, respectively, to obtain the \(H_{NCP}^{\text{A1-A2}}\), \(H_{CPE}^{\text{A1-A2}}\), \(H_{NCP}^{\text{A1-A2}}\) and \(H_{CPE}^{\text{A1-A2}^{Surrogate}}\). The procedure was also repeated 30 times under each coupling strength, and the mean value and the standard deviation across 30 repetitions were calculated. Figure 4 illustrates the average value of \(H_{NCPE}^{\text{A1-A2}}\), \(H_{CPE}^{\text{A1-A2}}\), \(H_{NCPE}^{\text{A1-A2}^{Surrogate}}\) and \(H_{CPE}^{\text{A1-A2}^{Surrogate}}\) across all repetitions under different  
Figure 3: At different coupling strengths \(C=0.1,0.3,0.5\), the sliding time window width is fixed at 2000 samples, and the embedding dimension \(d\) is taken from 3 to 15; the results of unidirectional coupled Henon model using CBTN and CPE for coupling analysis, respectively. The values of ordinate are \(H_{1}^{\text{A1-A2}^{surrogate}}-H_{1}^{\text{A1-A2}}\). \(x_{2}^{surrogate}\) can be obtained by surrogating \(x_{2}\) using the iAAFT method. **(a)** The results of coupling analysis using CBTN (30 repeated calculations); **(b)** The results of coupling analysis using CPE, \(\tau=2\) (30 repeated calculations).