THE STRONG UNSTABLE MANIFOLD AND PERIODIC SOLUTIONS IN DIFFERENTIAL DELAY EQUATIONS WITH CYCLIC MONOTONE NEGATIVE FEEDBACK

The paper’s Theorem 4.2 (items a–d) is established via Lemma 3.2 (differences of points in W lie in Σ, hence pr is injective on W and W\{0} ⊂ Σ), Corollary 3.3 (global graph structure and openness of pr(W)), Lemma 3.5 (ω-limit sets on W\{0} are periodic and lie in Σ ∩ (W\W)), and a clean topological argument that pr(W) is exactly the Jordan interior of Γ = pr(O) (hence ∂W = O) . The statement of Theorem 4.2 itself matches the solver’s target precisely . The candidate (model) solution proves the same four items using the Mallet–Paret–Sell zero-count Lyapunov functional, the local strong unstable manifold (Theorem 3.1), and the Poincaré–Bendixson theory. Its structure is sound and aligns with the paper, but a few justifications are lighter: (i) it assumes pr is injective on W rather than deriving it from Lemma 3.2 + Σ ∩ (X3 ⊕ Q) = ∅ (Lemma 2.10(b)) ; (ii) the claimed “strong order-preserving” property on W with respect to the rank-2 cone K is not explicitly justified (the paper instead uses open basins and connectedness to obtain uniqueness) ; (iii) openness of pr(W) is better handled as in Corollary 3.3 using the auxiliary map h and an open-mapping argument . Despite these presentation gaps, the model’s route reaches the same conclusions and can be tightened by citing the paper’s lemmas where needed. The spectral decomposition and V-level facts used by the model also match the paper’s setup (X = Xuu ⊕ X3 ⊕ Q, V = 1 on Xuu\{0}, V = 3 on X3\{0}) .