CHAOTIC MOTION AND SINGULARITY STRUCTURES OF FRONT SOLUTIONS IN MULTI-COMPONENT FITZHUGH-NAGUMO-TYPE SYSTEMS

Martina Chirilus-Bruckner, Peter van Heijster, Jens D.M. Rademacher

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper rigorously derives a reduced nilpotent-chain speed ODE on a center manifold and proves (i) universality of scalar singularities for N=1 with the sharp nondegeneracy condition τ1/(2√d1) ≠ √2/3, and (ii) Shil’nikov-type chaos for N=3 under a quadratic nondegeneracy on F3. The candidate solution reaches the same conclusions via a more direct center-manifold/normal-form route using explicit solvability projections and Allen–Cahn identities. Apart from a mild overstatement that one always has N additional small eigenvalues (the paper carefully qualifies that N′ ≤ N depends on parameter choices), the model’s arguments align with the paper’s results.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This work offers a clean, rigorous route from a singularly perturbed PDE with one fast Allen–Cahn field and N slow diffusive fields to a nilpotent-chain speed ODE capturing dynamic front motion. The strategy combines Evans/NLEP/SLEP analysis, Weierstrass preparation, and center-manifold reduction to control both linear and nonlinear terms of the reduced ODE. The highlights—universal scalar singularities for N=1 and provable Shil’nikov chaos for N=3—are well motivated, precise, and of notable interest. Minor clarifications on parameter tuning and coefficient identification would strengthen accessibility.