Amplification in parametrically-driven resonators near instability based on Floquet theory and Green's functions

TL;DR

This paper employs Floquet theory and Green's functions to analyze the response of parametrically-driven resonators near instability, providing new estimates for response and spectral densities validated against numerical and analytical methods.

Contribution

It introduces novel theoretical expressions for response, power, and noise spectral densities in parametrically-driven systems near instability, validated through multiple approaches.

Findings

New estimates for system response in frequency domain

Expressions for power and noise spectral densities

Validation of theory with numerical and analytical results

Abstract

Here we use Floquet theory to calculate the response of parametrically-driven time-periodic systems near the onset of parametric instability to an added external ac signal or white noise. We provide new estimates, based on the Green's function method, for the response of the system in the frequency domain. Furthermore, we write novel expressions for the power and the noise spectral densities. We validate our theoretical results by comparing our predictions for the specific cases of a single degree of freedom parametric amplifier and of the parametric amplifier coupled to a harmonic oscillator with the numerical integration results and with analytical approximate results obtained via the averaging method up to second order.