Phase-Induced (In)-Stability in Coupled Parametric Oscillators

We report results on a model of two coupled oscillators that undergo periodic parametric modulations with a phase difference $\theta$. Being to a large extent analytically solvable, the model reveals a rich $\theta$ dependence of the regions of parametric resonance. In particular, the intuitive notion that anti-phase modulations are less prone to parametric resonance is confirmed for sufficiently large coupling and damping. We also compare our results to a recently reported mean field model of collective parametric instability, showing that the two-oscillator model can capture much of the qualitative behavior of the infinite system.