# 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  
Impact of uncertainty is illustrated by the envelopes of the frequency responses amplitudes computed using the LHS method and the gPCE, in generalized and physical coordinates, for two dispersion levels: \(\delta_{l}=2\%\) and \(\delta_{l}=5\%\) (Figures 4-6). In order to enable comparative study with deterministic responses, we use the initial conditions fixed in deterministic case, in the multistability domain. The discontinuity of the envelopes is due to the lack of initial conditions. Note that varying more the initial conditions makes computation prohibitive and it still difficult to cover all possible solutions and so obtain smooth curves. Increasing the dispersion level affects more the responses variability. All multimode branches are larger in terms of amplitude and frequency. Consequently, the multimode solutions are enhanced and the multistability domain is wider. The uncertainty effect on each pendulum response depends on its position with respect to the stochastic one. The greatest impact is on the first pendulum responses (larger envelopes) since it contains the uncertain parameter. The second one is less affected by uncertainty. Modal localization is consequently generated on the first pendulum response. Figure 4: Envelopes of the amplitudes (**a,b**) \(\left|\widetilde{A}_{1}\right|^{2}\) and \(\left|\widetilde{A}_{2}\right|^{2}\) of the first pendulum modal response and (**c,d**) \(\left|A_{1}\right|^{2}\) and \(\left|A_{2}\right|^{2}\) of the second pendulum modal response computed using the LHS and gPCE methods, respectively, for \(\delta_{l}=2\%\). Figure 5: Envelopes of the amplitudes (**a,b**) \(\left|\widetilde{A}_{1}\right|^{2}\) and \(\left|\widetilde{A}_{2}\right|^{2}\) of the first pendulum modal response and (**c,d**) \(\left|A_{1}\right|^{2}\) and \(\left|A_{2}\right|^{2}\) of the second pendulum modal response computed using the LHS and gPCE methods, respectively, for \(\delta_{l}=5\%\). Table 2 and Figures 7 and 8 illustrate the uncertainty effects through the intervals of variation of the amplitude (\(\Delta\phi\)) and frequency (\(\Delta f\)) ranges of the multistability domain. These intervals are computed between extreme bifurcation points, according to the example shown in Figure 7. \(\widetilde{\phi}_{1}\) is more affected by uncertainty than \(\phi_{2}\) since the first pendulum length is stochastic. More important variation is detected on \(\Delta f\) than on \(\Delta\phi\). The ratio \(\Delta f/\Delta\phi\) shows that the nonlinearity of the first pendulum response is strongly strengthened. Moreover, dispersion and nonlinearity are proportional. Satisfying approximations are allowed by the gPCE with respect to the LHS method, considered as reference. Good agreement between modal and physical responses is proved by small accuracy errors. For \(\delta_{l}=5\%\), the gPCE errors increase, revealing the limits of the method against high level of uncertainty. It is important to note that some errors result from the luck of initial conditions making the detection of bifurcation points more difficult. Figure 7: Example of measured amplitude and frequency ranges of the multistability domain according to the SM and DM. Figure 8: Evolution of (**a**) the frequency ranges and (**b**) the amplitude ranges of the multistability domain with respect to deterministic ones according to the SM and DM. In presence of uncertainty, a set of solutions is generated by the LHS method or the gPCE. Several basins of attraction are thus superposed. The envelopes of the attractors are quantified by the overlapped areas. For \(\delta_{l}=2\%\), the envelopes of the attractors of \(\left|\widetilde{A}_{1}\right|^{2}\) (first pendulum), computed using the LHS method, represent 64.57% of the basins. The gPCE approximation gives 62.34%. The envelopes of the attractors of \(\left|A_{1}\right|^{2}\) (second pendulum) represent 9.67% of the basins (gPCE: 8.97%). For \(\delta_{l}=5\%\), the impact of uncertainty is more important.