2012.05005
Multi-Delay Differential Equations: A Taylor Expansion Approach
Philip Doldo, Jamol Pender
correcthigh confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem 3.1 derives the neutral-approximation critical delay by setting r = iω in r = α0 + A0 e^{-rΔ*} + A1 r e^{-rΔ*}, separating real/imaginary parts, solving for cos(ωΔ*), and eliminating θ to obtain ω^2 = (A0^2 − α0^2)/(1 − A1^2), which yields Δ_cr^approx = (1/ω) arccos((−α0 A0 + A1 ω^2)/(A0^2 + A1^2 ω^2))—exactly the model’s steps and final expressions (see equations (3.28)–(3.31) and (3.25) in the paper) .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The neutral-approximation calculation of the critical delay matches standard practice and is executed correctly. The result is useful for approximating stability transitions in multi-delay systems via a tractable single-delay surrogate, and the numerics support the approach. Minor clarifications on parameter conditions (ensuring real ω), branch choices for arccos, and a brief note on Hopf transversality would improve rigor and readability.