  Fluctuations and noise may alter the behavior of dynamical systems
considerably. For example, oscillations may be sustained by demographic
fluctuations in biological systems where a stable fixed point is found in the
absence of noise. We here extend the theoretical analysis of such stochastic
effects to models which have a limit cycle for some range of the model
parameters. We formulate a description of fluctuations about the periodic orbit
which allows the relation between the stochastic oscillations in the fixed
point phase and the oscillations in the limit cycle phase to be elucidated. In
the case of the limit cycle, a suitable transformation into a co-moving frame
allow fluctuations transverse and longitudinal with respect to the limit cycle
to be effectively decoupled. While longitudinal fluctuations are of a diffusive
nature, those in the transverse direction follow a stochastic path more akin to
an Ornstein-Uhlenbeck process. Their power spectrum is computed analytically
within a van Kampen expansion in the inverse system size. This is carried out
in two different ways, and the subsequent comparison with numerical simulations
illustrates the effects that can occur due to diffusion in the longitudinal
direction.
