  The response of a large array of coupled nonlinear oscillators to parametric
excitation is studied, 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 contains uncommon nonlinear gradient terms which yield a unique
wave-number dependent bifurcation similar in character to the behavior known to
exist in fluids undergoing the Faraday wave instability. We suggest a number of
experiments with nanomechanical or micromechanical resonators to test the
predictions of our theory, in particular the strong hysteretic dependence on
the drive amplitude.
