# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 3 Numerical Examples
### Example 1: Two Coupled Pendulums
#### 3.1.1 Deterministic Study  
If uncertainty is not taken into account, \(p=0\), \(q=0\) and \(d=1\). Consequently, 4 algebraic equations are generated. Deterministic responses in both modal and physical coordinates are illustrated in Figure 2.  
A multiplicity of stable solutions is generated, in the multistability domain, by modal interactions [58] between the responses which are driven by the collective dynamics. Three stable solutions could thus be obtained at several frequencies in the multistability domain [59; 60; 61]. They are classified as Single (SM) and Double mode (DM) solutions. The only SM branch, presented by red curve, corresponds to the null trivial solution of the second amplitude. Two SM branches are identified for the first amplitude: resonant branch (SM-RB) and non-resonant branch (SM-NRB) (Figure 2a). The DM branches, which are presented by blue curves, result from coupling between the pendulums. A correspondence in term of bifurcation points between the amplitudes with respect to each branch type is observed. It is generated by the bifurcation topology transfer. Identical physical responses reflect the symmetry between the pendulums. More detailed illustration of the bifurcation topology transfer between the amplitudes \(|A_{1}|^{2}\) and \(|A_{2}|^{2}\) is allowed by the basins of attraction. The robustness of the attractors, corresponding to the multimode branches, and their contributions in the basins are investigated. In literature, [PERSON] et al. [62] illustrated numerically the basins of attraction of a plane pendulum under parametric excitation. An experimental investigation of the basins of attraction of two fixed points of a nonlinear mechanical nano-resonator was illustrated by [PERSON] et al. [63]. [PERSON] et al. [64] studied the basins of attraction of two coupled Kerr oscillators. [PERSON] et al. [15] studied the basins of attraction of two and three coupled oscillators under both external and parametric excitations. The basins of attraction are generally plotted in the phase plan \((\phi,\phi)\). In this work, the basins of attraction are  
Figure 2: Deterministic (**a**) modal amplitudes \(|A_{1}|^{2}\) and \(|A_{2}|^{2}\) and (**b**) physical amplitudes \(\phi_{1}\) and \(\phi_{2}\) of the pendulums responses. plotted in the Nyquist plane \((a,b)\) of real and imaginary parts of the responses amplitudes \(A_{m}\)[15; 29] regarding the proposed solving methodology. Fixing the detuning parameter on \(\sigma=-1\), in the multistability domain, two and three stable solutions are obtained for \(\left|A_{2}\right|^{2}\) and \(\left|A_{1}\right|^{2}\), respectively. Two and three attractors correspond to these stable solutions, respectively, in the basins of attraction. Figure 3 illustrates the basins of attraction plotted in the Nyquist plane \(\left(\left(a_{1}\right)_{0},\left(b_{1}\right)_{0}\right)\) for \(\left(a_{2}\right)_{0}=\left(b_{2}\right)_{0}=\ 0.25\). When \(\left|A_{1}\right|^{2}\) jumps between SM-RB and SM-NRB, \(\left|A_{2}\right|^{2}\) is stabilized on SM. Similarly, a correspondence exists between the DM attractors. Subsequently, one can restrict the analysis to the basins of attraction of \(\left|A_{1}\right|^{2}\). Varying the detuning parameter (\(\sigma=-1.2\), \(\sigma=-1\) and \(\sigma=-0.8\)), the contribution of each attractor is illustrated quantitatively through the ratio of its size with respect to the global size of the basins of attraction. For \(\sigma=-1.2\), the most robust attractors correspond to the SM-NRB; they dominate the basins with 52.5% of their total size, compared to 45.9% and only 1.6% for the SM-RB and DM attractors, respectively. Nevertheless, the SM-NRB attractors vanish for \(\sigma=-0.8\) and the DM attractors become the most robust; their contribution is quantified by 64.5%, compared to a contribution of 35.5% of the SM-RB. For \(\sigma=-1\), Figure 3, the DM attractors also dominate the basins of attraction; 60.9%. However, the contributions of the attractors of SM-RB and DM are quantified by 15.7% and 23.4%, respectively.