PERIOD TWO SOLUTION FOR A CLASS OF DISTRIBUTED DELAY DIFFERENTIAL EQUATIONS

The paper’s Theorem 5 states that, under (H): f odd and xf(x)>0 with f Lipschitz, a DDE SSPS (period 2, x(t−1) = −x(t)) corresponds to a period‑2 closed orbit of the Hamiltonian ODE x′ = −y, y′ = 2f(x), and conversely. The forward direction defines y(t)=∫_0^1 f(x(t−s))ds, computes y′(t)=f(x(t))−f(x(t−1)) and uses x(t−1)=−x(t) with f odd to get y′=2f(x), hence a period‑2 ODE orbit; this matches the model’s step (a) exactly (paper: Theorem 5 and its proof; see the definition of SSPS and (1.4)–(1.5) ). For the converse, the paper argues, via the Hamiltonian symmetry, that (x(t+1),y(t+1))=(−x(t),−y(t)) on a minimal period‑2 closed orbit, from which x(t−1)=−x(t) and y(t)=∫_{t−1}^t f(x(s))ds after fixing the integration constant by an odd‑about‑1/2 symmetry argument (Appendix A supports these symmetries) . The model’s step (b) is the same in substance: it uses the second‑order reduction x″=−2f(x), turning points with y=0 one unit apart, the uniqueness argument w(t)=−x(t+1)≡x(t), and then shows the integration constant vanishes by evaluating at a turning point, yielding the same identity y(t)=∫_0^1 f(x(t−s))ds. Both proofs rely on Lipschitz continuity for uniqueness and on oddness of f; the model also explicitly notes how xf(x)>0 yields turning points, consistent with the paper’s Hamiltonian picture. No substantive logical gaps were found; the paper’s symmetry step is concise but supported by Appendix A, while the model elaborates that step. Hence both are correct with substantially the same proof structure.