SLOW-FAST NORMAL FORMS ARISING FROM PIECEWISE SMOOTH VECTOR FIELDS

Otavio Henrique Perez, Gabriel Rondón, Paulo Ricardo da Silva

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorems A–C establish: (a) loss of normal hyperbolicity at ϕ′=0, (b) overshoot |ϕ|>1 enlarges the regularized sliding region and projects C0 into sewing, (c) the reduced slow flow projects to Filippov’s ZΣ wherever f1≠g1, and (d) GSPT cannot define sliding on Σ\D(ZΣ) unless C0 contains horizontal lines; and, for linear regularizations, only normally hyperbolic, SF-fold and SF-transcritical singularities are obtainable—SF-pitchfork is impossible—while nonlinear regularizations can realize pitchforks. The model’s solution derives the same conclusions and adds an alternative obstruction for pitchfork in the linear case (Fxy∝ϕ′), whereas the paper uses a different contradiction based on ϕ‴(0). Aside from a minor notational slip (ϕ(0) vs ϕ(x) in Fy), the model aligns with the paper’s results.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper cleanly systematizes which slow–fast singularities can arise from linear versus nonlinear regularizations of planar piecewise-smooth systems, proving clear theorems (A–C) and illustrating them with explicit transition functions and examples. The results agree with and extend established links between regularization and GSPT, and the impossibility of linear pitchfork versus the realizability under nonlinear regularization is convincingly demonstrated. Minor clarifications would further improve readability (e.g., consistently highlighting the jet conditions and the role of D(ZΣ)).