2007.04675
Weak tracking in nonautonomous chaotic systems
Hassan Alkhayuon, Peter Ashwin
incompletemedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper convincingly formulates weak tracking and provides a clear piecewise-linear reduction and numerical shooting evidence, but it does not actually prove density or zero-measure of the critical-rate set for the Rössler example; the crucial step from preimages on a Poincaré section to dense, measure-zero parameter values is only sketched (or claimed) without a rigorous argument. The model supplies a plausible rigorous route (via Poincaré return maps, λ-lemma, and transversality) but relies on strong hyperbolicity/mixing assumptions (e.g., a basic set around Γ+ with transverse homoclinics) not established for the Rössler attractor at the quoted parameters. Hence, the paper’s argument is incomplete as written, and the model’s proof is conditional on unverified hypotheses.
Referee report (LaTeX)
\textbf{Recommendation:} major revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The study articulates a fresh viewpoint (weak tracking) on rate-induced behavior in systems with chaotic future limits and demonstrates compelling numerics on the Rössler system. However, the theoretical claims about dense and zero-measure sets of critical rates are either conjectural or insufficiently justified in the current text. The piecewise-linear surrogate provides a clean setup but still lacks the rigorous bridge from a preimage criterion to dense, measure-zero parameter sets. The paper would substantially benefit from a precise theorem (with explicit hyperbolicity/transversality hypotheses) and proof for the piecewise-linear case, or a clear repositioning of these statements as conjectures supported by numerics.