Adjoint-Based Projections for Quantifying Statistical Covariance near Stochastically Perturbed Limit Cycles and Tori

Abstract

This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near normally hyperbolic attracting 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 in $\mathbb{R}^2$, $\mathbb{R}^2\times S^1$, and $\mathbb{R}^2\times S^1$, respectively, for which explicit expressions may be found for the associated covariance functions using the proposed series solutions. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package \textsc{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. Excellent agreement with numerical evidence from stochastic time integration shows the potential for using deterministic continuation techniques to study the influence of stochastic perturbations for both autonomous and periodically excited deterministic vector fields.