# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 1 Introduction  
In structural mechanics as well as in practically all fields of engineering, the periodicity characterizes the structuring of many systems such as layered composites, crystal lattices, bladed disks, turbines, multi-cylinder engines, ship hulls, aircraft fuselages, micro and nanoelectromechanical systems, etc. Periodicity implies an infinite or finite geometrical repetition of a unit cell in one, two or three dimensions and requires appropriate approaches to investigate it. Under the hypothesis of perfect periodicity, many works provided interesting insights in the behavior of these structures. In the context of wave propagation in periodic structures, the basic works performed in linear case by [PERSON] [1] and [PERSON] [2] are based on the Floquet's principle or the transfer matrix in order to compute propagation constants. Based on the transfer matrix theory, a combination of wave and finite element approaches was proposed by [PERSON] et al. [3] and used later by [PERSON] et al. [4] to calculate forced responses of waveguide structures. [PERSON] et al. [5] developed analytical and numerical models based on the transfer matrix approach to investigate the dispersion properties and bandgaps of a beam with a periodic array of airfoil-shaped resonating units bonded along its length. Using the Floquet-Bloch' theorem, [PERSON] et al. [6] completely described the behaviorof a heat exchanger periodic structure only from the vibroacoustic knowledge of the basic unit [PERSON] et al. [7] extended the analysis to two-dimensional periodic structures with complex damping configurations and underlined the reduced computational costs allowed by the Floquet-Bloch' theorem when representing whole structures by unit cell modeling. Recently, [PERSON] et al. [8] combined the Wave Finite Element Method (WFEM) with Component Mode Synthesis (CMS) to evaluate the dispersion characteristics of two-dimensional periodic waveguides. On the other hand, the wave propagation becomes considerably complicated when the governing wave equation contains nonlinear terms (i.e., contact, material or geometric nonlinearity). In this case, complex phenomena such as localization, solitons and breathers arise and traditional Floquet-Bloch and transfer matrix wave analyses are no longer applicable. In literature, other methodologies are developed to deal with nonlinear periodic structures such as perturbation approaches. For instance, [PERSON] and [PERSON] [9] investigated the harmonic wave propagation in one-dimensional periodic chain consisting of identical masses and weakly non-linear springs through single-frequency harmonic balance. They used a perturbation approach to calculate the propagation and attenuation constants. A straightforward perturbation analysis is applied by [PERSON] et al. [10] to investigate amplitude-dependent dispersion of a discrete one-dimensional nonlinear periodic chain with Hertzian contact. Otherwise, as an alternative to perturbation approach for strongly nonlinear systems, [PERSON] et al. [11] proposed a combination of shooting and pseudo-arc-length continuation to examine nonlinear normal modes and their bifurcations in cyclic periodic structures. Moreover, [PERSON] et al. used a secular perturbation theory to calculate the response of a coupled array of nonlinear oscillators under parametric excitation in [12] and of N nonlinearly coupled micro-beams in [13] using discrete models. The method of multiple scales is used by [PERSON] [14] to construct a first-order uniform expansion in the presence of internal resonance for the governing equations of parametrically excited multi-degree-of-freedom systems with quadratic nonlinearities. Using the same methodology, [PERSON] et al. [15] investigated the collective dynamics of a periodic structure of coupled nonlinear Duffing-Van Der Pol oscillators under simultaneous parametric and external excitations. An analytico computational model was used to compute the frequency responses and the basins of attraction of two and three coupled oscillators.