Response of discrete nonlinear systems with many degrees of freedom

TL;DR

This paper investigates the collective response of large arrays of coupled nonlinear oscillators to parametric excitation, deriving an amplitude equation that reveals wavenumber-dependent bifurcations similar to Faraday wave phenomena, with implications for MEMS and NEMS devices.

Contribution

The study derives a first-principles amplitude equation for coupled nonlinear oscillators, revealing wavenumber-dependent bifurcations akin to Faraday instability, and confirms these behaviors numerically.

Findings

Wavenumber-dependent bifurcation behavior identified.

Amplitude equation derived from first principles.

Numerical confirmation of bifurcation phenomena.

Abstract

We study the response of a large array of coupled nonlinear oscillators to parametric excitation, motivated by the growing interest in the nonlinear dynamics of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). Using a multiscale analysis, we derive an amplitude equation that captures the slow dynamics of the coupled oscillators just above the onset of parametric oscillations. The amplitude equation that we derive here from first principles exhibits a wavenumber dependent bifurcation similar in character to the behavior known to exist in fluids undergoing the Faraday wave instability. We confirm this behavior numerically and make suggestions for testing it experimentally with MEMS and NEMS resonators.