  This paper presents a new boundary-value problem formulation for quantifying
uncertainty induced by the presence of small Brownian noise near transversally
stable periodic orbits (limit cycles) and quasiperiodic invariant tori of the
deterministic dynamical systems obtained in the absence of noise. The
formulation uses adjoints to construct a continuous family of transversal
hyperplanes that are invariant under the linearized deterministic flow near the
limit cycle or quasiperiodic invariant torus. The intersections with each
hyperplane of stochastic trajectories that remain near the deterministic cycle
or torus over intermediate times may be approximated by a Gaussian distribution
whose covariance matrix can be obtained from the solution to the corresponding
boundary-value problem. In the case of limit cycles, the analysis improves upon
results in the literature through the explicit use of state-space projections,
transversality constraints, and symmetry-breaking parameters that ensure
uniqueness of the solution despite the lack of hyperbolicity along the limit
cycle. These same innovations are then generalized to the case of a
quasiperiodic invariant torus of arbitrary dimension. In each case, a
closed-form solution to the covariance boundary-value problem is found in terms
of a convergent series. The methodology is validated against the results of
numerical integration for two examples of stochastically perturbed limit cycles
and one example of a stochastically perturbed two-dimensional quasiperiodic
invariant torus. Finally, an implementation of the covariance boundary-value
problem in the numerical continuation package coco is applied to analyze the
small-noise limit near a two-dimensional quasiperiodic invariant torus in a
nonlinear deterministic dynamical system in $\mathbb{R}^4$ that does not
support closed-form analysis.
