A General View on Double Limits in Differential Equations

The paper defines the two-parameter fast–slow normal form (Xtc): x' = (x−y)(x+y) + ε^2/δ, y' = ε, sets sections Σ− = {x = −2, y ∈ [1,3]} and Σ+ = {x = 2, y ∈ [−1,1]}, and states the three-case classification: (I) δ(ε) = ε(1 + O(|ε|^p)) ⇒ γ intersects Σ−, (II) δ(ε) = ε(1 − O(|ε|^p)) ⇒ γ intersects Σ+, (III) δ(ε) = ε(1 ± O(e^{−K/ε})) ⇒ γ never intersects Σ± (canard) . The candidate solution proves this classification via a detailed GSPT-based proof sketch: outer Fenichel manifolds, inner scaling x = √ε X, y = √ε Y with κ = ε/δ, a Riccati variation for Z = X−Y, monotone dependence on κ near κ = 1 (δ = ε), and exponential amplification e^{(y^2−y_0^2)/ε} along the repelling slow manifold. This aligns with the paper’s claims and known results on transcritical canards. The paper provides a high-level statement with references, while the model provides a compatible proof sketch with standard tools; hence both are correct, but the approaches differ.