# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 2 Mechanical Model
### Multiple Scales Method Applied to Stochastic Model  
The multiple scales method [46; 47] consists on replacing the single time variable by an infinite sequence of independent time scales (\(T_{i}=\varepsilon^{i}\ t\)), where \(\varepsilon\) is a dimensionless parameter assumed to be small, and eliminating secular terms in the fast time variable \(T_{0}=t\).  
Limiting the study to a first order perturbation, (\(T=T_{1}=\varepsilon^{1}\ t\)), Equation (1) takes the form  
\[\ddot{\phi}_{n}+\varepsilon\ c_{n}\dot{\phi}_{n}+\omega_{0}^{2}\dot{\phi}_{n} +\varepsilon\ \mathcal{L}_{c}(\phi,\widetilde{\phi})+\varepsilon\,\alpha_{n}\dot{\phi}_{n} ^{3}=\varepsilon\,f_{n}\cos(\Omega t), \tag{6}\]  
where the excitation frequency \(\Omega\) is expressed as: \(\Omega=\omega_{0}+\varepsilon\ \sigma\), \(\sigma\) being the detuning parameter.  
The solution of Equation (6) can generally be given by a formal power series expansion: \(\phi_{n}=\sum_{i}\varepsilon^{i}\phi_{ni}\). Up to the order \(\varepsilon^{1}\), the solution is of the form  
\[\phi_{n}=\phi_{n0}+\varepsilon\ \phi_{n1}. \tag{7}\]  
Its derivatives are given by  
\[\dot{\phi}_{n}=\frac{d\phi_{n}}{dt}=\phi_{n0}^{(0,1)}+\varepsilon\ \Big{[}\phi_{n0}^{(1,0)}+\phi_{n1}^{(0,1)}\Big{]}, \tag{8}\]  
\[\ddot{\phi}_{n}=\frac{d^{2}\phi_{n}}{dt^{2}}=\phi_{n0}^{(0,2)}+2\ \varepsilon\ \phi_{n0}^{(1,1)}+\varepsilon\ \phi_{n1}^{(0,2)}, \tag{9}\]  
with \(\phi_{n0}^{(0,1)}=\frac{\partial\phi_{n0}}{\partial t}\), \(\phi_{n0}^{(0,2)}=\frac{\partial^{2}\phi_{n0}}{\partial t^{2}}\), \(\phi_{n0}^{(1,0)}=\frac{\partial\phi_{n0}}{\partial T}\) et \(\phi_{m0}^{(1,1)}=\frac{\partial^{2}\phi_{n0}}{\partial t\partial T}\).  
Substituting Equations (7)-(9) into Equation (6) and separating the terms with different orders of \(\varepsilon\), one obtains a hierarchical set of equations. For the order \(\varepsilon^{0}\), an unperturbed equation is obtained  
\[\ddot{\phi}_{n0}+\omega_{0}^{2}\dot{\phi}_{n0}=0. \tag{10}\]  
The solution of Equation (10), which appears in every order in the expansion of the approximate solution, is expressed as  
\[\phi_{n0}=A_{n}\ exp(i\omega_{0}t)+c.c. \tag{11}\]for the order \(\varepsilon^{1}\), one obtains an equation of the form  
\[\phi_{n1}^{(0,2)}+\omega_{0}^{2}\phi_{n1}+c_{n}\phi_{n0}^{(0,1)}+2\phi_{n0}^{(1,1)}+k_{n}(2\phi_{n0}-\phi_{n0-1}-\phi_{n0+1})+\alpha_{n}\phi_{n0}^{3}=\frac{f_{ n}}{2}exp[i(\omega_{0}t+\sigma T)]. \tag{12}\]  
Substituting Equations (4) and (5) into Equation (12) leads to \(N\) equations of the form  
\[\phi_{n1}^{(0,2)}+\omega_{0}^{2}\phi_{n1}=\sum_{m=1}^{N}\left(m^{th}secular\ terms\right)e^{i\omega_{0}t}+other\ terms, \tag{13}\]  
where the secular terms (coefficients of \(e^{i\omega_{0}t}\)) should be equated to zero to satisfy the condition of solvability of the multiple scales method. Projecting the response on the standing-wave modes implies to multiply all terms by \(\sin(nq_{m})\) and sum over \(n\). Consequently, a generic complex equation of the \(m^{th}\) amplitude \(A_{m}\) is obtained:  
\[\begin{array}{c}2i\omega_{0}A_{m}^{(1,0)}+i\omega_{0}c_{n}A_{m}+k_{n}(2A_{m }-cos[q_{m}]*G_{m})\\ +S*\frac{2k_{n}}{N+1}\sum_{n=1}^{N}sin[nq_{m}]\sum_{x=1}^{N}cos[nq_{x}]sin[q_{ x}](A_{x}-\widetilde{A}_{x})+\frac{3}{4}\alpha_{n}\sum_{j,k,l}A_{j}A_{k}A_{l}^{*} \Delta_{jkl,m}^{(1)}-\\ \frac{1}{(N+1)}f_{n}\ exp(i\sigma T)\sum_{n=1}^{N}sin[nq_{m}]=0,\end{array} \tag{14}\]  
where \(\Delta_{jkl,m}^{(1)}\) is the delta function [12; 15] defined in terms of the Kronecker deltas as  
\[\begin{array}{c}\Delta_{jkl,m}^{(1)}=\delta_{-j+k+l,m}-\delta_{-j+k+l,-m}- \delta_{-j+k+l,2(N+1)-m}\\ +\delta_{j-k+l,m}-\delta_{j-k+l,-m}-\delta_{-j-k+l,2(N+1)-m}\\ -\delta_{j+k+l,m}-\delta_{j+k+l,2(N+1)-m}-\delta_{j+k+l,2(N+1)-m},\end{array} \tag{15}\]  
with \(\delta_{p,p}\) is the Kronecker delta equal to 1 if \(v=w\) and to 0 otherwise. The functions \(G_{m}\) and \(S\) are defined in the Appendix A.