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.

翻译：本文提出了一种新的边值问题公式，用于量化由小布朗噪声在确定性动力系统（无噪声条件下）的横向稳定周期轨道（极限环）与拟周期不变环面附近引起的不确定性。该公式利用伴随矩阵构建一组连续的横向超平面，这些超平面在极限环或拟周期不变环面附近的线性化确定性流下保持不变。对于在中间时间尺度内仍靠近确定性周期轨道或环面的随机轨迹，每个超平面与其的交点可近似为高斯分布，其协方差矩阵可通过相应边值问题的解获得。在极限环情形下，本文通过显式使用状态空间投影、横向约束和对称性破缺参数改进了现有文献结果，这些参数确保了在沿极限环缺乏双曲性情况下解的唯一性。随后将这些创新推广至任意维度的拟周期不变环面。在每个情形中，协方差边值问题的闭式解均以收敛级数形式给出。该方法通过两个随机扰动极限环示例和一个随机扰动二维拟周期不变环面示例的数值积分结果进行了验证。最后，将协方差边值问题实现于数值延拓软件包coco中，应用于$\mathbb{R}^4$中一个无法进行闭式分析的非线性确定性动力系统的二维拟周期不变环面附近的小噪声极限分析。