BIFURCATIONS IN ASYMPTOTICALLY AUTONOMOUS HAMILTONIAN SYSTEMS UNDER MULTIPLICATIVE NOISE

Oskar A. Sultanov

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorems 2–3 (case (13)) and their proofs are correct and complete via carefully constructed stochastic Lyapunov functions after the normal-form reduction (10) (see the statement and proofs around Definition 1, Theorems 2–3, and formulas (33)–(37)) . The model’s outline reproduces the correct thresholds and weights but its stability proof (Step 2A) contains a substantive gap: it asserts a Doob/BDG control of exp(sup M) to bound sup_t |z(t)|γ(t) without establishing the requisite exponential-martingale or localization estimates over an infinite time horizon. This invalidates the claimed probability bound for the weighted stability, whereas the paper’s argument avoids this pitfall by using a nonnegative supermartingale Lyapunov function directly .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript gives a clear, rigorous account of stochastic stability/instability bifurcations for asymptotically autonomous planar Hamiltonian systems with multiplicative noise that decays in time. The combination of averaging, a normal-form reduction, and bespoke Lyapunov functions is executed carefully and yields sharp thresholds, including the role of the Itô correction at the critical decay rate σ = n/q. The results are well illustrated by examples. Some small clarifications (e.g., mapping assumption (16) through the change of variables) would further help readers.