TRACKING NONAUTONOMOUS ATTRACTORS IN SINGULARLY PERTURBED SYSTEMS OF ODES WITH DEPENDENCE ON THE FAST TIME

The paper’s main theorem under Assumptions 3.1 and 5.2 proves two claims: (i) along ε→0 subsequences the slow components converge to a solution of the averaged differential inclusion (4.1), and (ii) on long fast-time windows the fast components track δ-inflations of pullback-attractor fibers along pieces of orbits in the omega-limit set; this is exactly Theorem 5.9 with the tracking estimate (5.3). The candidate solution reconstructs the same structure: Grönwall uniform bounds and existence, Arzelà–Ascoli compactness for x^ε, identification of the limit via invariant measures as in Artstein’s theorem, and a blockwise comparison/tracking argument using pullback attraction on the skew-product over O(g)×K0. The steps and hypotheses align with the paper’s setup (Assumptions 3.1 and 5.2, existence of global attractors for the layered semiflows, and the forward-attraction to inflated pullback attractors), and reproduce the conclusion (5.3). Minor technical gaps (e.g., an explicit diagonal choice to ensure ε_k T_ℓ→0) are standard and easily patched. Overall, both are correct and follow substantially the same proof strategy, with the paper citing Artstein for the DI limit and providing a streamlined proof for tracking via Theorem 5.8 and 5.9. Key support: statement of Theorem 5.9 including (5.3) and hypotheses (), the averaged DI (4.1) (), the fast-slow model and Assumption 3.1 (), Assumption 5.2 and the O(g) framework (), existence of global attractors for the layered semiflows (), and forward attraction to inflated pullback attractors (Theorem 5.8) ().