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A parameterization method for quasi-periodic systems with noise: computation of random invariant tori

Pingyuan Wei, Lei Zhang

TL;DR

This work develops a rigorous, computation-ready framework for random invariant tori in quasi-periodically forced, noisy systems by extending the parameterization method to random settings. It first transforms SDEs to RDEs via an Ornstein–Uhlenbeck–based random change of variables, then defines random invariant manifolds and embeds tori with a random parameterization $K(\theta,\omega)$ satisfying a random invariance equation. A central existence-persistence theorem shows that for small noise $\varepsilon$ and adequate hyperbolicity, a nearby random torus $\mathcal{K}_\varepsilon$ exists, is uniquely determined in $\mathcal{L}^\infty(C^0)$, and remains normally hyperbolic; the framework also provides a perturbation theory to compute higher-order corrections $K_k$ and Lyapunov exponents. Numerically, the authors present an algorithm that uses adapted frames and reducibility to efficiently compute $K_k$ and the exponents, enabling direct estimation of NHIM statistics in random quasi-periodic settings. The approach yields a practical path to quantify transition states and reaction rates in noisy, time-dependent environments, with potential applications to Langevin-type models and other stochastic driven systems.

Abstract

This work is devoted to studying normally hyperbolic invariant manifolds (NHIMs) for a class of quasi-periodically forced systems subject to additional stochastic noise. These systems can be understood as skew-product systems. The existence of NHIMs is established by developing a parameterization method in random settings and applying the Implicit Function Theorem in appropriate Banach spaces. Based on this, we propose a numerical algorithm to compute the statistics of NHIMs and Lyapunov exponents.

A parameterization method for quasi-periodic systems with noise: computation of random invariant tori

TL;DR

This work develops a rigorous, computation-ready framework for random invariant tori in quasi-periodically forced, noisy systems by extending the parameterization method to random settings. It first transforms SDEs to RDEs via an Ornstein–Uhlenbeck–based random change of variables, then defines random invariant manifolds and embeds tori with a random parameterization satisfying a random invariance equation. A central existence-persistence theorem shows that for small noise and adequate hyperbolicity, a nearby random torus exists, is uniquely determined in , and remains normally hyperbolic; the framework also provides a perturbation theory to compute higher-order corrections and Lyapunov exponents. Numerically, the authors present an algorithm that uses adapted frames and reducibility to efficiently compute and the exponents, enabling direct estimation of NHIM statistics in random quasi-periodic settings. The approach yields a practical path to quantify transition states and reaction rates in noisy, time-dependent environments, with potential applications to Langevin-type models and other stochastic driven systems.

Abstract

This work is devoted to studying normally hyperbolic invariant manifolds (NHIMs) for a class of quasi-periodically forced systems subject to additional stochastic noise. These systems can be understood as skew-product systems. The existence of NHIMs is established by developing a parameterization method in random settings and applying the Implicit Function Theorem in appropriate Banach spaces. Based on this, we propose a numerical algorithm to compute the statistics of NHIMs and Lyapunov exponents.

Paper Structure

This paper contains 16 sections, 10 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A stochastic differential equation (SDE)
  4. Random transformations: Converting SDEs to RDEs
  5. Random invariant manifolds
  6. The parameterization method and random invariance equations
  7. Existence and persistence of random invariant tori
  8. Spaces of differentiable and measurable functions
  9. Transfer operators and the normal hyperbolicity
  10. The main theorem for random invariant tori
  11. Perturbation analysis of random invariant tori
  12. Expansion formulas for $F_\varepsilon$ and $K_\varepsilon$
  13. A perturbation theory for random invariant tori
  14. Numerical algorithms for computing random invariant tori
  15. Computing $K_k$
  16. ...and 1 more sections

Key Result

Lemma 2.1

(i) There exists a $\{\Phi_t \}_{t\in\mathbb{R}}$-invariant subset ${\Omega}^\ast\subset{\Omega}$ of full measure with sublinear growth, that is, $\lim_{t\to\pm\infty}{|\omega(t)|}/{|t|}=0$ for a.e. $\omega\in{\Omega}^\ast$. (ii) For $\omega\in{\Omega}^\ast$ the random variable exists and generates a unique solution of eq: linear given by (iii) For $\omega\in{\Omega}^\ast$ the solution $\zeta(\P

Theorems & Definitions (33)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • ...and 23 more