Fluctuations in a parametrically excited subharmonic oscillator

The behavior of the classical degenerate parametric oscillator ( ) with small linear dissipation is considered and an expression for the steady-state probability distribution for the subharmonic amplitude is obtained. The treatment is limited to the case where Qpand Qsare the factors at respectively the pump and signal frequencies. The behavior is analogous to that of the Brownian motion of a particle in a bistable potential well. This leads to a tractable equation for the relaxation towards the steady-state distribution by thermally activated jumps over the barrier. Near threshold, the behavior is similar to that of a system undergoing a second order phase transition in the mean field approximation. Analogies between first-order phase transitions and transitions in oscillating systems are also pointed out.