Periodic Center Manifolds and Normal Forms for DDEs in the Light of Suns and Stars

The paper rigorously establishes a periodic, finite‑dimensional center manifold for translated time‑dependent DDEs via sun–star calculus, using (i) an exponential trichotomy (Hypothesis 1), (ii) a crucial embedding hypothesis (Hypothesis 2), (iii) a carefully constructed Green operator K^η_s, and (iv) a δ‑modification of the nonlinearity to obtain a uniform (small) global Lipschitz constant needed for a contraction on the two‑sided weighted space BC^η_s(R,X). This yields Theorem 14 in the abstract setting and Corollary 17 for classical DDEs, including periodicity and C^k/C^{k−1} smoothness claims . By contrast, the candidate solution omits the δ‑modification (Section 3.3) and directly sets f=j^{-1}R, which is not justified since R(t,u) need not lie in j(X); in the paper, j^{-1} is used only after projecting into subspaces guaranteed by Hypothesis 2 so that the integrals take values in j(X) . Without the δ‑modification, a uniform contraction on BC^η_s(R,X) is not secured because the weighted norm does not control |u(t)| pointwise for all t, so a merely local Lipschitz constant at 0 does not suffice for the two‑sided Lyapunov–Perron map (the paper’s Proposition 6 and Theorem 8 address exactly this gap) .