Synchronous Glacial Cycles in a Nonsmooth Conceptual Climate Model with Asymmetric Hemispheres

The paper proves that, under parameter choices ensuring P_s^− ∈ W(P_s^+) and P_s^+ ∈ W(P_s^−) together with R_± ∈ W(P_s^+) ∩ W(P_s^−), the Filippov system admits a unique attracting nonsmooth limit cycle for sufficiently small ε>0 by constructing half-return maps r^-_ε: V_+→V_- and r^+_ε: V_-→V_+ across the switching manifold Σ, showing each is a contraction, and composing them to obtain a contractive Poincaré map with a unique fixed point (Theorem 5.3) . The key ingredients—definition of Σ via h(w,η_S,η_N,ξ_N)=0, the crossing regions Σ^± determined by w≶h_±(η_N), repelling sliding Σ_SL, lines of fast equilibria `± and their intersections R_±, and the small-ε passage time along the slow direction—are all explicitly developed in the paper . The candidate solution follows the same structure: it identifies `± at ε=0, proves crossing via the signs of the Lie derivatives (equivalent to w≶h_±), constructs r^-_ε and r^+_ε, and shows contraction using fast–slow separation (Fenichel/Tikhonov phrasing) before composing to get a unique attracting fixed point/limit cycle. This mirrors the paper’s return-map proof; the differences are mainly stylistic (the model cites standard GSPT results where the paper uses ad hoc but equivalent estimates) and do not alter the logic or conclusions .