Nonlinear saturation of the parametric instability for three coupled oscillators

The case of three parametrically coupled and damped oscillators is considered. One (with natural frequency ω0) is externally driven at a frequency Ω close to the sum (ω1 + ω2) of the other two. In the vicinity of the parametric instability threshold, we determine the time-asymptotic oscillator amplitudes and nonlinear frequency shifts by using secular-free perturbation analysis. The resulting amplitudes are sensitive to the mismatch delta (= Ω - ω1 - ω2), if Ω and ω0 are dissimilar, and rise sharply as delta → 0. If Ω is close to ω0, the behavior is quite insensitive to delta, and models results obtained for parametric instabilities in plasma. The single parametrically-driven oscillator threshold result is also obtained and compared to a general small amplitude result of Landau. This Landau result is then generalized to two coupled oscillators and compared to a result obtained by perturbation analysis. These two cases all behave as the three-oscillator does when ω0 and Ω are dissimilar. All solutions are shown to be stable