The spin-spin model and the capture into the double synchronous resonance

The paper’s Theorem 3 establishes analytic continuation of the conservative odd 2π–periodic orbit Θ*(t) to a dissipative 2π–periodic solution Ψ*(t,δ) for small |δj|, and proves asymptotic stability when the spin–spin couplings |Λ_{m1m2}| are also sufficiently small. This is done by (i) showing strong linear stability of the conservative variational equation under conditions (38)–(40) (Theorem 2), which implies 1 is not a Floquet multiplier, and then (ii) applying a standard continuation theorem for periodic solutions (Proposition 2); asymptotic stability is obtained by reducing first to the uncoupled dissipative spin–orbit case (43) and perturbing by small coupling . The candidate solution proves the same claims by a different, functional-analytic route: it sets up a Banach-space map H(Φ,δ)=0, proves the conservative linearization L0 is invertible on the odd, 2π–periodic space X using a Poincaré–Wirtinger argument plus the smallness bound (38), invokes the analytic IFT to obtain Ψ*(·,δ), and then uses Floquet/Liouville arguments for the uncoupled damped case and continuity of multipliers under small couplings. Minor slips (an unnecessarily strong Poincaré constant) do not affect correctness. Hence both are correct but use different proofs.