On the dynamics of non-autonomous systems in a neighborhood of a homoclinic trajectory

The paper rigorously proves Theorems 4.2 and 4.3—existence/regularity of forward/backward Poincaré maps and sharp power-law/log-time estimates—under F0–F2, K, G, by constructing barrier curves, using exponential dichotomy, and a sequence of fixed-point and comparison arguments; see the theorem statements and definitions of the exponents σ and Σ (Theorem 4.2, Theorem 4.3, and (2.9)) . The Cr nature of the event times and maps follows from the flow smoothness and an implicit-function argument around the transversal crossings (Lemma 3.6 and Remark 3.7) . The proofs in §5 rely on robust exponential dichotomy and carefully controlled estimates across the four subpaths of the loop, culminating in the stated bounds . The candidate model gives a different but standard approach: nonautonomous Hadamard–Perron, implicit-function theorem for event times, Dulac-type estimates on each side, and composition; it recovers exactly the same exponents (including Σfwd = Σfwd+ + σfwd+/λ−u) and the trajectory–trajectory bounds via a terminal-value variation-of-constants estimate. Minor omissions in the model (notably the barrier-set machinery and parameter-order bookkeeping ϖ,β,δ,µ used in the paper) do not affect correctness. Hence both are correct, with substantially different proof strategies.