2104.06043
Where the Liénard–Levinson–Smith (LLS) theorem cannot be applied for a generalised Liénard system
Sandip Saha, Gautam Gangopadhyay
correctmedium confidence
- Category
- Not specified
- Journal tier
- Note/Short/Other
- Processed
- Sep 28, 2025, 12:56 AM
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Audit review
For the case study ẍ + ε(x^2 − 1)ẋ^3 + x = 0, both the paper and the model apply first-order Krylov–Bogoliubov averaging to obtain the identical amplitude equation ṙ = −(ε/16) r^3 (r^2 − 6) and φ̇ = 0, concluding a unique, asymptotically stable limit cycle with amplitude √6 (to leading order). The paper further argues—by example—that the LLS condition C3 (F(0,0) < 0) is not necessary, while the model makes the same point and also proposes a more precise relaxed condition. The paper’s broader “C3 ≤ 0” proposal is suggestive rather than a proved general theorem, but the central example and its analysis are correct and match the model’s derivation.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} note/short/other
\textbf{Justification:}
The paper convincingly demonstrates—via a transparent example analyzed with standard KB averaging—that the LLS condition C3 is not necessary, thereby clarifying a frequent misconception. The computations are correct for small ε, and the illustrative numerics support the claims. However, the broader suggestion to relax C3 to F(0,0) ≤ 0 is not proved as a general theorem; the paper would benefit from clarifying sufficiency vs. necessity and citing a persistence/averaging theorem to make the existence claim from the averaged system mathematically precise.