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

MAKE IMPACT

Zaid Ahsan , Harry Dankowicz , Christian Kuehn · YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2404.13429

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.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/2404.13429

PDF

https://arxiv.org/pdf/2404.13429

OA Status

green

Cited By

2

Related Works

10

OpenAlex ID

https://openalex.org/W4395064984

Raw OpenAlex JSON

OpenAlex ID

https://openalex.org/W4395064984

Canonical identifier for this work in OpenAlex

DOI

https://doi.org/10.48550/arxiv.2404.13429

Digital Object Identifier

Title

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

Work title

Type

preprint

OpenAlex work type

Language

en

Primary language

Publication year

2024

Year of publication

Publication date

2024-04-20

Full publication date if available

Authors

Zaid Ahsan, Harry Dankowicz, Christian Kuehn

List of authors in order

Landing page

https://arxiv.org/abs/2404.13429

Publisher landing page

PDF URL

https://arxiv.org/pdf/2404.13429

Direct link to full text PDF

Open access

Yes

Whether a free full text is available

OA status

green

Open access status per OpenAlex

OA URL

https://arxiv.org/pdf/2404.13429

Direct OA link when available

Concepts

Limit (mathematics), Torus, Uncertainty quantification, Statistical physics, Physics, Mathematics, Applied mathematics, Mathematical analysis, Statistics, Geometry

Top concepts (fields/topics) attached by OpenAlex

Cited by

2

Total citation count in OpenAlex

Citations by year (recent)

2024: 2

Per-year citation counts (last 5 years)

Related works (count)

10

Other works algorithmically related by OpenAlex

Full payload