# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 3 Numerical Examples
### Example 1: Two Coupled Pendulums
#### 3.1.2 Stochastic Study
Indeed, the basins of attraction of \(\left|\widetilde{A}_{1}\right|^{2}\) are fully overlapped (100%). The size of the envelopes of the attractors of \(\left|A_{1}\right|^{2}\), computed by the LHS method, increases to 19.04%. The gPCE approximate it by 17.55%. The gPCE gives a satisfying approximation of the basins' envelopes, although errors proportionally increase with dispersion level. The comparison of the envelopes of the basins of \(\left|\widetilde{A}_{1}\right|^{2}\) to those of \(\left|A_{1}\right|^{2}\) highlights the modal localization generated on the first pendulum response. Moreover, 96.4% of CPU time reduction is achieved when applying the gPCE, with respect to the LHS method. In fact, nearly 15.8 h are needed to compute the frequency responses using the LHS method. However, less than 0.6 hours is required for the gPCE implementation. More important time reduction ratio is achieved when computing the basins of attraction: 97.6%. Indeed, the 350.4 h required by the LHS method are reduced to only 8.4 h needed for the gPCE implementation. Note that all computations are made using Matlab on 8-Core Dual Processor with 64 GB of RAM (Random Access Memory). For more detailed dispersion analysis, the attractors of SM-RB, SM-NRB and DM are plotted separately. The variability of the basins of attraction of \(\left|\widetilde{A}_{1}\right|^{2}\) is more important than the variability of the basins of \(\left|A_{1}\right|^{2}\), which illustrates modal localization on first pendulum response. The dispersion of each attractor is illustrated by the variability of its contribution in the basins of attraction from a minimal size (color defined for the considered attractor in the deterministic case illustrated by Figure 3) to a maximal size bounded by the limit (white area) corresponding to other attractors' contributions. The envelopes are represented by black areas. Figure 9 shows the envelopes of the attractors of \(\left|\widetilde{A}_{1}\right|^{2}\), for \(\delta_{l}=2\%\), computed using the gPCE. Since, the basins of \(\left|\widetilde{A}_{1}\right|^{2}\) are fully overlapped for \(\delta_{l}=5\%\), Figure 10 illustrates the envelopes of the basins of \(\left|A_{1}\right|^{2}\). As shown in Figure 9, the DM attractors are the most robust. Table 3 compares the contributions of the attractors of SM-RB, SM-NRB and DM in the basins of attraction in deterministic and stochastic case (for \(\delta_{l}=2\%\) and \(\delta_{l}=5\%\)). Table 3 lists both minimal sizes and envelopes of the attractors. Summing these two areas gives the global contribution of each attractor in the basins. Superposing the envelopes of the attractors of SM-RB, SM-NRB and DM gives the global envelopes. In deterministic case, the DM attractors of the first pendulum response are the most robust. However, the contributions of the attractors of SM-RB and SM-NRB in the basins become more important for \(\delta_{l}=2\%\) and even more for \(\delta_{l}=5\%\). The SM-NRB attractors are in fact the most robust. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline  & \multirow{2}{*}{**Dof**} & \multicolumn{6}{c}{**SM**} & \multicolumn{6}{c}{**DM**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**LHS**} & \multicolumn{2}{c}{**gPCE**} & \multicolumn{2}{c}{**LHS**} & \multicolumn{2}{c}{**gPCE**} \\ \cline{3-10}  & & **2\%** & **5\%** & **2\%** & **5\%** & **2\%** & **5\%** & **2\%** & **5\%** \\ \hline \multirow{2}{*}{\(\Delta f\) (\%)} & 1 & 204.17 & 356.25 & 204.17 & 343.75 & 182.50 & 267.50 & 167.50 & 253.75 \\  & 2 & 106.25 & 97.92 & 106.25 & 97.92 & 103.75 & 107.50 & 103.75 & 107.75 \\ \hline \multirow{2}{*}{\(\Delta\phi\) (\%)} & 1 & 120.86 & 133.14 & 118.49 & 132.69 & 113.57 & 123.13 & 110.11 & 123.55 \\  & 2 & 120.86 & 118.34 & 116.12 & 118.05 & 103.19 & 99.72 & 103.05 & 99.31 \\ \hline \multirow{2}{*}{\(\frac{\Delta f}{\Delta\phi}\) (\%)} & 1 & 204.23 & 267.61 & 171.83 & 259.15 & 160.36 & 217.12 & 152.25 & 205.41 \\  & 2 & 87.32 & 83.10 & 91.55 & 830 & 100.90 & 107.21 & 100.90 & 108.11 \\ \hline \hline \end{tabular}
\end{table}