Parametrically modulated oscillator dimer: an analytic solution

Abstract

A dimer consisting of two coupled oscillators undergoing periodic parametric modulations with a phase difference 0 reveals a rich panorama of stability-instability boundaries as a function of the system parameters. It was recently established that the instabilities observed in such dimers with a phase difference theta = 0 and with a phase difference theta = pi qualitatively and even quantitatively capture a great deal of the behavior of a mean field model of coupled parametric oscillators with random phases that undergo collective parametric instabilities. These similarities were established numerically for lack of an analytic solution for the 0 = pi dimer. Herein we present an analytic solution for such a parametrically modulated dimer. We present the exact associated instability boundaries and thus improve on earlier ones generated by numerical simulations. (C) 2002 Elsevier Science B.V. All rights reserved.