Adjoint-Based Projections for Uncertainty Quantification near Stochastically Perturbed Limit Cycles and Tori

TL;DR

This work tackles uncertainty quantification for small Brownian perturbations near transversally stable limit cycles and quasiperiodic tori. It develops an adjoint-based construction of an invariant foliation of transversal hyperplanes and derives a covariance boundary-value problem that yields a Gaussian description of noise-induced fluctuations via a convergent series for the covariance $C$. The approach unifies periodic and quasiperiodic cases through adjoint BV problems and transverse projections, with explicit Lyapunov-type equations and integration into the Coco continuation framework. Validation against analytical solutions and numerical SDE simulations demonstrates accurate, coordinates-free predictions and supports simultaneous continuation of limit sets and covariance, enabling robust design under stochastic perturbations. The methods and open-source Coco implementation provide a practical, high-precision toolkit for uncertainty quantification near low-dimensional invariant sets in deterministic dynamics.

Abstract

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.

Figures (8)

• Figure 1: Comparison of the non-trivial eigenvalue of the covariance matrix function for the vector field in Eq. \ref{['eq: hopf normal form']} obtained from a discretization of the governing boundary-value problem in Proposition \ref{['prop: perorbbvp']} in the software package coco and from the analytical expression reported in \ref{['eq:eigenvalue']}, respectively.
• Figure 2: Comparison of leading-order theoretical predictions and a numerical time history describing the influence of small Brownian noise on the dynamics near a limit cycle of the deterministic vector field Eq. \ref{['eq: hopf normal form']} in cartesian (upper panel) and polar (lower panel) coordinates. Stochastic trajectories (red dots) were obtained using an Euler-Maruyama scheme with $\sigma=0.1$ and $\mathrm{d}t=10^{-4}$ for $500$ periods of the limit cycle. Dashed curves represent the deterministic limit cycle while solid curves represent predicted deviations from the limit cycle equal to one and two standard deviations, respectively, computed using \ref{['eq:eigenvalue']}. In the lower panel, the vertical axis equals the radial deviation from the limit cycle, since the normalized radial eigenvector $e(\tau(t))=\gamma(\tau(t))$ and $\lambda(\tau(t))^\mathsf{T}\left(x(t)-\gamma(\tau(t))\right)=0$. There, filled circles represent the statistical mean (blue) and standard deviation (black) of simulated data collected in bins of width $\Delta\tau=1/40$.
• Figure 3: Comparison of leading-order theoretical predictions and a numerical time history describing the influence of small Brownian noise on the dynamics near a limit cycle of the harmonically excited, damped, linear oscillator in Sec. \ref{['sec:numexpper']}. The top and bottom panels show samples (red dots) in sections of constant excitation phase ($t=0.1$ and $t=0.9$, respectively) obtained using an Euler-Maruyama scheme with $\sigma=0.1$ and $\mathrm{d}t=10^{-4}$ for $1,000$ periods of excitation. Solid (black) curves represent deviations from the intersection of the limit cycle (centered black dot) in terms of one and two standard deviations predicted using \ref{['eq:ex2pred']} and \ref{['eq:eigslinper']}. Dashed (blue) curves represent the one-standard deviation curve predicted from the statistical covariance matrix for the simulated data.
• Figure 4: Comparison of leading-order theoretical predictions and a numerical time history for the first example in Sec. \ref{['sec:numexpqua']} with $\Omega=\pi$ and $\omega=1$ in cartesian (upper panel) and polar (lower panel) coordinates. Stochastic trajectories sampled at integer values of the excitation phase (red dots) were obtained using an Euler-Maruyama scheme with $\sigma=0.1$ and $\mathrm{d}t=10^{-4}$ for $10,000$ periods of the excitation. Dashed curves represent the intersection of the deterministic quasiperiodic invariant torus while solid curves represent predicted deviations from this curve of intersection equal to one and two standard deviations, respectively, computed using \ref{['eq:quasiex1pred']}. In the lower panel, the vertical axis equals the radial deviation from the (circular) curve of intersection, since the normalized radial eigenvector $e(\psi,0)\parallel\gamma(\psi,0)$ and $w_\phi(\psi)^\mathsf{T}\left(x-\gamma(\psi,0)\right)=0$. There, filled circles represent the statistical mean (blue) and standard deviation (black) of simulated data collected in bins of width $\Delta\psi=1/40$.
• Figure 5: (Upper panel) Family of quasiperiodic invariant tori for the deterministic limit of the SDE \ref{['eq:tori_sde']} under simultaneous variations in $\beta$ and $\delta$ computed using coco. Here, $\epsilon=0.5$ and the rotation number $\rho$ is fixed at $140/62\sqrt{2}$. Each torus is represented by a finite collection of equal-duration trajectory segments coupled through all-to-all boundary conditions in terms of the rotation number. (Lower panel) The quasiperiodic invariant torus obtained for $\beta=0.5$ represented in terms of 29 trajectory segments, each of which is approximated by a continuous, piecewise-polynomial function on 20 mesh intervals.

Theorems & Definitions (59)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Lemma 2.5
• proof