In this paper we report a theoretical model based on Green functions, Floquet theory and averaging techniques up to second order that describes the dynamics of parametrically-driven oscillators with added thermal noise. Quantitative estimates for heating and quadrature thermal noise squeezing near and below the transition line of the first parametric instability zone of the oscillator are given. Furthermore, we give an intuitive explanation as to why heating and thermal squeezing occur. For small amplitudes of the parametric pump the Floquet multipliers are complex conjugate of each other with a constant magnitude. As the pump amplitude is increased past a threshold value in the stable zone near the first parametric instability, the two Floquet multipliers become real and have different magnitudes. This creates two different effective dissipation rates (one smaller and the other larger than the real dissipation rate) along the stable manifolds of the first-return Poincare map. We also show that the statistical average of the input power due to thermal noise is constant and independent of the pump amplitude and frequency. The combination of these effects cause most of heating and thermal squeezing. Very good agreement between analytical and numerical estimates of the thermal fluctuations is achieved.
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