Entrée-sortie dans le halo d’une courbe lente semi-stable

The paper defines Sm, the constrained C-trajectory, and the exit map Sρ via the integral thresholds {+2ρ,0,−2ρ}, and proves the approximation theorem (Theorem 1.5) using nonstandard “halo” arguments and the exponential magnifying change of variables z = [y]ε (the “loupe exponentielle”) to analyze the entry–exit through y = 0. See the system and hypotheses in (4) and §1.2, the C-trajectory and Sρ in Def. 1.1–1.3 and (6), and Theorem 1.5 itself . The paper’s passage near y = 0 derives the thresholds by tracking z and shows that exits occur when ∫ f crosses −2ρ, 0, or +2ρ (e.g., equations (12), (17), (18) in §2.3 and the formalization referred to in App. B.4) . It also proves rapid approach to the slow curve away from zeros of f (barrier-domain argument in §2.1 and App. B.2) and develops the z-dynamics in §2.2 . The candidate model solution reaches the same conclusions but via a different route: exact r± identities leading to U' = (2/ε)(f−y), classical contraction-to-graph estimates away from zeros (invoking Fenichel, though a direct variation-of-constants argument suffices here), and a standard entry–exit balance yielding the same ∫ f thresholds. Minor issues in the model include dropping the integration constant in y(x) = m sinh((1/ε)∫(f−y)), which is only correct when the entry point has y=0; and an unnecessary invocation of Fenichel in a nonautonomous setting (dx/dt = 1), where standard linear estimates already yield O(ε)-tracking. Nevertheless, the model’s reasoning is substantively correct and matches the paper’s choices of segments V, C, H and exit locations Sρ.