# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 1 Introduction
The authors demonstrated the importance of the multimode solutions and the robustness of their attractors. The multiple scales method was also used by [PERSON] and [PERSON] [16] in a continuum-based model to investigate the dynamic behavior of an array of N nonlinearly coupled micro-beams. Furthermore, [PERSON] et al. [17] used the multiple scales method to investigate wave interactions in monoatomic mass-spring chain with a cubic nonlinearity. In [18], the method was combined with a finite-element discretization of a single unit cell, to study the wave propagation in continuous periodic structures subject to weak nonlinearities. The authors proposed later robust tools for wave interactions analysis in diatomic chain with two degrees of freedom per unit cell [19]. Recently, [PERSON] et al. [20] studied the wave interactions in a periodically perforated plate through the two-dimensional dispersion characteristics, group velocities and internal resonances investigation. [PERSON] and [PERSON] [21] identified the regions of existence of discrete breathers and guided their analysis using the nonlinear propagation region of chain of oscillators with cubic nonlinearity exhibiting periodic solutions. Furthermore, using the idea of harmonic balance in the periodic structures inspired from [9] the Harmonic Balance Method (HBM) was combined with multiple scales method in [22] in order to study the attenuation caused by weak damping of harmonic waves through a discrete periodic structure. The HBM was later used by [PERSON] et al. [23] to analyze the influence of nonlinearity and wave amplitude on the dispersion properties of plane waves in strongly nonlinear periodic uniform granular media. Particularly, periodic coupled pendulum structure has been the purpose of several researches in literature. [PERSON] [24], for instance, proved several theorems on the existence of oscillatory, rotary, and mixed periodic motions of \(N\) coupled simple pendulums. [PERSON] and [PERSON] [25] demonstrated numerically and experimentally the existence of three tristable stationary states. [PERSON] et al. [26] investigated the nonlinear dynamics of a two-dimensional array of coupled pendulums under parametric excitation and, recently [27],the energy localization phenomenon in an array of coupled pendulums under simultaneous external and parametric excitations by means of a nonlinear Schrodinger equation. The authors show that adding an external excitation increases the existence region of solitons. [PERSON] et al. [28; 29] investigated the collective nonlinear dynamics of perfectly periodic coupled pendulum structure under primary resonance using multiple scales and standing-wave decomposition. The authors studied the effects of modal interactions on the nonlinear dynamics. They highlighted the large number of multimode solutions and the bifurcation topology transfer between the modal intensities, in frequency domain. The analysis of the Basins of attraction illustrated the distribution of the multimodal solutions which increases by increasing the number of coupled pendulums. A detailed review was presented by [PERSON] et al. [30] dealing with the influence of modal interactions on the nonlinear dynamics of harmonically excited coupled systems. Besides, the study of collective nonlinear dynamics of coupled oscillators may serve to identify the Intrinsic Localized Modes (ILMs). ILMs are defined as localizations due to strong intrinsic nonlinearity within an array of perfectly periodic oscillators. Such localization phenomenon was studied by [PERSON] et al. [31] in the context of microcantilever and microresonator arrays. Authors used the multiple scales method and other methods to construct nonlinear normal modes and suggested to realize an ILM as a forced nonlinear vibration mode. It is important to note that the dynamic analysis of periodic structures is greatly simplified by assuming perfect periodicity.