Nonlinear Saturation of the Parametric Instability for Three Coupled Oscillators.

摘要

The case of three parametrically coupled and damped oscillators is considered. One (with natural frequency omega sub 0 ) is externally driven at a frequency omega close to the sum ( omega sub 1 + omega sub 2 ) 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 (= omega - omega sub 1 - omega sub 2 ), if omega and omega sub 0 are dissimilar, and rise sharply as delta implies 0. If omega is close to omega sub 0 , 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 omega sub 0 and omega are dissimilar. All solutions are shown to be stable. (ERA citation 10:052998)