A simple mechanism for stable oscillations in an intermediate complexity Earth System Model

The paper formulates the same delayed two-box salinity model, explicitly including the delayed feedback term σ(S1(t−τ)−S1(t))/V and m = k[β(S2−S1) − αT*], see Eqs. (1)–(2) in the PDF, and reports that without delay a subcritical Hopf occurs while with delay the Hopf can become supercritical, based on DDE-Biftool continuation and normal-form classification (thick/thin Hopf curves), matching Figures 3–4 and their discussion of criticality changes with (σ, τ) . The candidate solution independently derives the retarded characteristic operator Δ(λ) = λI − A0 − A1 e^{−λτ}, verifies Hopf conditions, and computes the first Lyapunov coefficient l1 for both σ=0 (subcritical) and some (σ, τ)>0 (supercritical) using standard ODE/RFDE normal-form formulas, then argues by smooth dependence that the supercritical set U is open. The paper’s methodology is numerical (with references to DDE-Biftool and normal-form computation), whereas the model’s solution is analytic/numerical with explicit l1 formulas, but their conclusions agree. The paper does not explicitly prove openness of the supercritical region or identify a Bautin curve l1=0, but its figures and discussion are consistent with this picture .