Uncertainty Quantification of Bifurcations in Random Ordinary Differential Equations

Kerstin Lux, Christian Kuehn

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:55 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

In Example 3.1 the paper reduces the system to the pitchfork normal form u̇ = −r1 r2 u^3 − a u − a^2 u + O(4) and states that the bifurcation is subcritical iff −r1 r2 > 0 . With r1 ∼ U(−1,3) and r2 ∼ Γ(3,1), the paper derives the product PDF/CDF via Mellin convolution and obtains P(−r1 r2 > 0) = Φr1r2(0) = 0.25 . The model’s solution observes that r2 > 0 a.s., hence the sign of −r1 r2 is the sign of −r1, giving P(r1 < 0) = 1/4. Both answers agree; the paper uses an analytic Mellin-based route, while the model uses a simpler sign-support argument.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper’s Mellin-transform-based derivation for the product r1 r2 in Example 3.1 is correct and culminates in P(−r1 r2 > 0) = 0.25, matching numerical checks. For pedagogy, the authors could note the immediate simplification that r2 > 0 a.s. implies P(−r1 r2 > 0) = P(r1 < 0), reinforcing the result without detracting from the general method’s value. Overall, the work is sound, clear, and useful for uncertainty quantification of bifurcation types.