2404.10986
Melnikov Method for Perturbed Completely Integrable Systems
F. Crespo, M. Uribe, E. Martínez
incompletemedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper derives the same first-order nT-time displacement expansion and Melnikov vector as the model and states an implicit-function-theorem (IFT) criterion to obtain subharmonic orbits. However, in the key IFT step it defines a “modified displacement” whose last component is Ω(I0) nT + O(ε) and then asserts zeros coincide with those of the true displacement, but it omits the necessary resonance shift by mT. As written, ∆̃(z*0,0) = 0 cannot hold unless that mT subtraction is included; the model explicitly fixes this by solving ∆I = 0 and ∆θ = mT and by desingularizing with ε (and, in the degenerate case, scaling by Ω(I)), yielding a correct IFT application. The expansion and Melnikov integrals match the paper’s formulas, but the paper’s proof of the IFT step is incomplete without the mT shift and precise scaling. See the paper’s standard form and displacement definition (13), (20), its Melnikov vector and expansion (25)–(27), and its Theorem 1 conditions (28), (29); compare the “modified displacement” step, where the mT shift is missing in the displayed formula for ∆̃ and equivalence-of-zeros is stated without justification.
Referee report (LaTeX)
\textbf{Recommendation:} major revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} The manuscript develops a standard Melnikov framework for completely integrable systems with a global action–angle-type chart and provides clean existence conditions for subharmonic periodic orbits, including a degenerate-frequency case. The main derivations are sound and the applications are informative. However, in the central IFT step, the modified displacement omits the necessary mT shift in the angular component and claims an equivalence of zeros without proof. These are focal to the correctness of Theorem 1 and should be fixed explicitly. With these revisions, the paper would be a solid contribution.