Detecting normally hyperbolic invariant tori in periodic non-autonomous differential equations

The paper proves Theorem A: for ẋ = Σ_{i=1}^N ε^i F_i(t,x) + ε^{N+1} F̃(t,x,ε) with f_0 = ⋯ = f_{ℓ−1} = 0 and a guiding system ż = (1/T) f_ℓ(z) possessing an attracting hyperbolic limit cycle γ, there exists (for ε small) a T‑periodic solution γ_int and a normally hyperbolic attracting invariant torus that converges to γ × S^1; this is stated and proved via averaging, a moving frame, and a method-of-continuation (Theorem A and Theorem B) and a Brouwer fixed-point argument for γ_int . The candidate solution correctly applies higher-order averaging and a stroboscopic map expansion and uses normal hyperbolicity to persist an invariant closed curve for the map, and a degree argument to obtain a T‑periodic orbit. However, it then incorrectly asserts that M_ε := Γ_ε × S^1 is invariant for the continuous-time flow merely because it is invariant for the time‑T map. Invariance under the stroboscopic map does not by itself imply invariance under the full flow; the paper instead constructs the torus as the graph of a phase-dependent function h(σ,τ) (not a product set), ensuring flow invariance in the extended space . Hence the paper is correct, while the model’s proof contains a critical flaw in its final step.