Robust computation of higher-dimensional invariant tori from individual trajectories

Maximilian Ruth, Jackson Kulik, Joshua Burby

incompletemedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proposes a four-step algorithm (Birkhoff RRE → MAP labeling → KZ reduction → LS parameterization) and demonstrates strong empirical performance, but explicitly leaves key theoretical pieces open—notably, convergence of RRE roots to true frequencies and higher-dimensional residual rates—so its argument is incomplete by the authors’ own admission . The model’s solution mirrors the overall pipeline and gives ideal-case guarantees, but it replaces the paper’s eigenvalue-based frequency extraction with a periodogram step and adopts simplified priors/likelihoods in the MAP stage; its guarantees hinge on assuming exact frequency identification and thus also remain incomplete.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper introduces a practical, broadly applicable pipeline that computes higher-dimensional invariant tori from single trajectories, addressing a longstanding pain point in dynamical systems computations. The Birkhoff RRE + MAP + KZ + LS architecture is well-motivated, and the empirical evaluation across coupled standard maps and ER3BP is convincing. However, the theoretical underpinnings of the RRE-root frequency convergence and higher-dimensional residual decay are left open. Clarifying these limits and detailing sensitivity to hyperparameters would strengthen the contribution. Given the quality of results and clear exposition, I recommend minor revisions.