Collective Dynamics in Arrays of Coupled Nonlinear Resonators

TL;DR

This paper reviews the collective nonlinear dynamics of coupled mechanical resonators, emphasizing theoretical tools like amplitude equations to predict phenomena such as pattern formation and synchronization in MEMS and NEMS arrays.

Contribution

It provides a comprehensive theoretical framework with detailed derivations of amplitude equations for analyzing complex coupled resonator systems, connecting theory with potential experiments.

Findings

Amplitude equations accurately predict resonant responses and pattern selection.

Theoretical predictions match numerical simulations, confirming their validity.

Spontaneous synchronization and localized modes can be observed in real resonator arrays.

Abstract

The study of collective nonlinear dynamics of coupled mechanical resonators is regaining attention in recent years thanks to rapid developments in the fields of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). We review a wide range of collective dynamical phenomena, while highlighting the common concepts and theoretical tools that we have developed for treating them. We provide detailed derivations of amplitude equations, which allow us to obtain reduced descriptions for the relevant dynamics of our complex systems. We apply these amplitude equations to study (a) resonant response to parametric excitation; (b) pattern selection, or the nonlinear competition between extended modes in situations of multistability; (c) formation and dynamics of intrinsically localized modes (ILM); and (d) 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.