  We theoretically investigate the dynamic properties of a Bose-Einstein
condensate in a toroidal trap. A periodic modulation of the transverse
confinement is shown to produce a density pattern due to parametric
amplification of phonon pairs. By imaging the density distribution after free
expansion one obtains i) a precise determination of the Bogoliubov spectrum and
ii) a sensitive detection of quantized circulation in the torus. The parametric
amplification is also sensitive to thermal and quantum fluctuations.
