2501.01279
On the dynamics of contact Hamiltonian systems II: Variational construction of asymptotic orbits
Liang Jin, Jun Yan, Kai Zhao
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves Theorem A and Theorem B under (H1)–(H3) with pointwise convergence, using a global characteristics/variational construction. The candidate solution captures the right ideas for (A1) and (B1) but leaves two essential gaps: (i) for (A2) it only obtains a limsup-with-closure and does not actually prove the exact hitting J^1 u^- ⊂ ⋂_{T>0} ⋃_{t≥T} Φ_H^t(J^1 φ); (ii) for (B2) it invokes Conley’s attractor–repeller theorem without establishing that Ñ_{u^-} and Ñ_{v^+} form an attractor–repeller pair in a compact isolating neighborhood. The paper, by contrast, supplies a concrete variational proof of (A2) and (B2).
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The manuscript develops a clear variational/characteristics framework for contact Hamiltonian systems under (H1)–(H3), proving existence of semi-infinite orbits asymptotic to Mañé slices and heteroclinic connections between distinct slices under a strict order on weak KAM limits. The arguments are careful, technically competent, and self-contained relative to the authors' prior works. Minor clarifications (e.g., the exact-hitting step in (A2) and a brief roadmap in Section 3.3) would enhance readability.