We introduce a variational method for analyzing limit cycle oscillators in R
d driven by Gaussian
noise. This allows us to derive exact stochastic differential equations for the amplitude and phase of
the solution, which are accurate over times of order (Cb −1
), where is the amplitude of the noise
and b the magnitude of decay of transverse fluctuations. Within the variational framework, different
choices of the amplitude-phase decomposition correspond to different choices of the inner product
space R
d
. For concreteness, we take a weighted Euclidean norm, so that the minimization scheme
determines the phase by projecting the full solution onto the limit cycle using Floquet vectors. Since
there is coupling between the amplitude and phase equations, even in the weak noise limit, there
is a small but nonzero probability of a rare event in which the stochastic trajectory makes a large
excursion away from a neighborhood of the limit cycle. We use the amplitude and phase equations
to bound the probability of it doing this: finding that the typical time the system takes to leave
a neighborhood of the oscillator scales as exp(Cb −1
). We also show how the variational method
provides a numerically tractable framework for calculating a stochastic phase, which we illustrate
using a modified version of the Morris–Lecar model of a neuron.