A classification of global attractors for S^1-equivariant parabolic equations

Carlos Rocha, Bernold Fiedler, Alejandro López-Nieto

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that for scalar parabolic PDEs on S1 with f in the SturmP(u,ux) class under dissipativity and hyperbolicity, the connection graph CPf is isomorphic to a Neumann graph CNg modulo the hidden symmetry, via a freeze-and-symmetrize homotopy (Theorems 2.1–2.2 and Corollary 2.3: CPf ≅ CNg/∼) . In the reversible setting, full lap signatures of the ODE period map are in bijection with integrable Sturm involutions and determine the connection graph (Theorems 4.2–4.3 and Lemma 4.1) . Enumerating all full lap signatures with n+q ≤ 7 yields exactly 21 isomorphism classes (Theorem 1.2; Figure 1.1; Table 5.1) . The candidate solution follows precisely this pipeline—freeze, symmetrize, classify by lap signature, and enumerate—so it matches the paper’s argument and result .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript articulates a clean homotopy-based reduction from the periodic SO(2) setting to a reversible Neumann framework and then deploys full lap signatures of period maps to obtain a complete list of connection graphs for n+q ≤ 7. The structural pipeline is persuasive and well connected to established Sturm machinery. Some proofs (e.g., preservation of connections under homotopy) are summarized with references; adding a succinct lemma or roadmap would make the paper more self-contained.