  Many physical and biological systems exhibit intrinsic cyclic dynamics that
are altered by random external perturbations. We examine continuous-time
autonomous dynamical systems exhibiting a stable limit cycle, perturbed by
additive Gaussian white noise. We derive a formal approximation for the
dynamics of sample paths that stay close to the limit cycle, in terms of a
phase coordinate and a deviation perpendicular to the limit cycle. To leading
order in the deviation, the phase advances at the deterministic speed
superimposed by a Brownian-motion-like drift. The deviation itself takes the
form of an (n-1)-dimensional Ornstein-Uhlenbeck process. We apply these results
to the case of limit cycles emerging through a supercritical Hopf bifurcation,
which is widespread in ecological and epidemiological models. We derive
approximation formulas for the system's stationary autocovariance and power
spectral density. The latter two reflect the effects of perturbations on the
temporal coherence and spectral bandwidth of perturbed limit cycles. We verify
our results using numerical simulations and exemplify their application to the
El Ni\~no Southern Oscillation.
