  We analyze the spectral properties of a resonator coupled to a
superconducting single electron transistor (SSET) close to the Josephson
quasiparticle resonance. Focussing on the regime where the resonator is driven
into a limit-cycle state by the SSET, we investigate the behavior of the
resonator linewidth and the energy relaxation rate which control the widths of
the main features in the resonator spectra. We find that the linewidth becomes
very narrow in the limit-cycle regime, where it is dominated by a slow phase
diffusion process, as in a laser. The overall phase diffusion rate is
determined by a combination of direct phase diffusion and the effect of
amplitude fluctuations which affect the phase because the resonator frequency
is amplitude dependent. For sufficiently strong couplings we find that a regime
emerges where the phase diffusion is no longer minimized when the average
resonator energy is maximized. Finally we show that the current noise of the
SSET provides a way of measuring both the linewidth and energy relaxation rate.
