Parametrically modulated oscillator dimer: an analytic solution

A dimer consisting of two coupled oscillators undergoing periodic parametric modulations with a phase difference θ 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 θ=0 and with a phase difference θ=π 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 θ=π 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.