Collective dynamics in arrays of coupled nonlinear resonators

The study of collective nonlinear dynamics of coupled mechanical resonators has been regaining attention in recent years thanks to rapid developments in the fields of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). This chapter reviews a wide range of collective dynamical phenomena, while highlighting the common concepts and theoretical tools that have been developed for treating them. It provides detailed derivations of amplitude equations, which allow for the obtaining of reduced descriptions for the relevant dynamics of these complex systems. These amplitude equations are then applied to the study of resonant response to parametric excitation; pattern selection, or the nonlinear competition between extended modes in situations of multistability; the formation and dynamics of intrinsically localized modes (ILM); and spontaneous synchronization of oscillators with differing frequencies. All the predictions obtained from analyzing the different amplitude equations are in excellent agreement with numerical solutions of the underlying equations of motion, suggesting that the predicted effects can be observed in arrays of real micromechanical or nanomechanical resonators, thus motivating new experiments in these systems.