# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 3 Numerical Examples
### Example 2: Three Coupled Pendulums
#### 3.2.2 Stochastic Study  
The greatest impact of uncertainty is obtained on the first pendulum responses since it contains the uncertain parameter. The second pendulum responses are less affected while the behavior of the third pendulum remains deterministic, because no coupling terms relate its equation with the one associated to the stochastic pendulum. Hence, the third pendulum responses are not presented hereafter. Figures 13-15 illustrate the envelopes of the responses amplitudes in generalized and physical coordinates, for \(\delta_{l}=5\%\) and \(\delta_{l}=2\%\). For \(\delta_{l}=5\%\), all branches are larger in terms of amplitude and frequency ranges. Consequently, the multimode solutions are enhanced and the multistability domain is wider. Nevertheless, the overlap of envelopes makes detection of extreme statistics of adjacent branches more difficult. Modal localization is here generated on the response of the first pendulum containing uncertainty. Figure 12: Basins of attraction of the deterministic amplitudes (**a**) \(\left|A_{1}\right|^{2}\), (**b**) \(\left|A_{2}\right|^{2}\) and (**c**) \(\left|A_{3}\right|^{2}\) in the Nyquist plane \(((a_{1})_{0},(b_{1})_{0})\) for \(\sigma=-1\), \((a_{2})_{0}=(b_{2})_{0}=\ 0.2\) and \((a_{3})_{0}=(b_{3})_{0}=\ 0.3\). Figure 13: _Cont._  
Figure 14: Envelopes of the modal intensity amplitude responses: (**a**) \(\left|\widetilde{A}_{1}\right|^{2}\), \(\left|\widetilde{A}_{2}\right|^{2}\) and \(\left|\widetilde{A}_{3}\right|^{2}\) of the first pendulum and (**b**) \(\left|A_{1}\right|^{2}\), \(\left|A_{2}\right|^{2}\) and \(\left|A_{3}\right|^{2}\) of the second pendulum, computed using the gPCE method for \(\delta_{l}=5\%\). Figure 15: _Cont._  
Figure 16 and Table 4 illustrate the uncertainty effects through the intervals \(\Delta\phi\) and \(\Delta f\) of the multistability domain. More important variation is detected on \(\Delta f\) than on \(\Delta\phi\). The ratios \(\Delta f/\Delta\phi\) show that the nonlinear aspect of the structure is enhanced. Indeed, the nonlinearity of the first pendulum response is more strengthened than the one of second pendulum response. Comparing uncertainty propagation methods, the gPCE allows generally an accurate approximation of the responses dispersion. Nevertheless, less accuracy is obtained for \(\delta_{l}=5\%\). Note that errors result also from lack of initial conditions and make detection of bifurcation points difficult. Furthermore, only 5 simulations are generated when applying the gPCE, compared to 200 samples used by the LHS method. A discontinuity of the envelopes' curves is thus detected, especially for \(\delta_{l}=5\%\). Figure 16: _Cont._  
Figure 15: Envelopes of the physical amplitude responses \(\tilde{\phi}_{1}\), \(\phi_{2}\) and \(\phi_{3}\), computed using the gPCE method, for (**a**) \(\delta_{l}=2\%\) and (**b**) \(\delta_{l}=5\%\). Regarding the dispersion of the basins of attraction (Figure 17 and Table 5), the envelopes of the attractors of \(\left|\widetilde{A}_{1}\right|^{2}\), computed using the LHS method, cover 69.27% of the basins, for \(\delta_{l}=2\%\). The gPCE approximation gives 66.85%. For \(\delta_{l}=5\%\), the basins are fully overlapped. However, the envelopes cover only 7.58% of the basins of \(\left|A_{1}\right|^{2}\), computed using the LHS method, for \(\delta_{l}=2\%\) and 16.40% for \(\delta_{l}=5\%\). The gPCE approximations are, respectively, 6.91% and 15.02%. Less accurate results are thus obtained by the gPCE for \(\delta_{l}=5\%\). To overcome this issue, one can elevate the polynomial order of the gPCE or use one of its recently developed variants [65; 66; 67]. As shown in Figure 17 and Table 5, a largest dispersion is detected on the DM-RB and DM-NRB attractors of \(\left|\widetilde{A}_{1}\right|^{2}\). Their contributions in the basins, approximated in deterministic case by \(1.69\%\) and 8.71% respectively, increase up to 14.5% and 47.5%, for \(\delta_{l}=2\%\). For \(\delta_{l}=5\%\), the attractors become totally overlapped (100%).