Bifurcations for Hamiltonian systems via dual variational principle *

Guangcun Lu

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 the Rabinowitz-type alternative (Theorem 3.4) by a dual variational approach and abstract bifurcation theorems, carefully verifying the Morse index jump equals the Maslov-type index jump and obtaining the even-symmetry multiplicity bound; see the statement of Theorem 3.4 and its proof (; ). By contrast, the model’s argument incorrectly asserts compactness of the inverse of the first-order boundary operator and applies Leray–Schauder degree to I−Kλ without establishing Kλ’s compactness or properness. The paper’s assumptions and constructions (Assumption 1.1 and the dual functional set-up) and index relations are explicit and correct (; ; ), whereas the model conflates the Fredholm-degree orientation theory with Leray–Schauder degree and omits key technical conditions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript successfully derives a Rabinowitz-type bifurcation alternative for Hamiltonian systems with M-boundary via a dual variational principle and Maslov-type indices. It carefully verifies index relations and applies abstract splitting/bifurcation theorems to obtain both existence and multiplicity (even case) results. The exposition could be slightly streamlined with added pointers and a comparison to spectral-flow methods, but the results are correct and useful to specialists in Hamiltonian dynamics and variational methods.