Limit cycles in mass-conserving deficiency-one mass-action systems

The candidate’s classification of rank‑two bimolecular mass‑action systems that can exhibit periodic solutions matches Theorem 6 exactly, including the two normal forms (Lotka–Volterra/Kolmogorov in 2D with b′ ≤ b and the antisymmetric cyclic 3D system on x+y+z=const), the shared sign conditions, the uniqueness of the positive equilibrium in each class, the fact that all other trajectories in the class are periodic (centers), and the nonexistence of limit cycles. The paper proves this via a multidimensional Bendixson–Dulac argument using the Dulac factor 1/∏xi and a case analysis that forces a_k≡0 and c_k≡0, leading to ẋ_k = x_k(r_k + Σ_{i≠k} b_{k i} x_i) and then to the same two normal forms; see Theorem 6 and its proof. The candidate employs a different route (Carathéodory reduction of reaction directions plus Bendixson–Dulac and explicit first integrals), but arrives at the same conclusions. Minor gaps in the candidate’s derivations (e.g., the asserted sign constancy of div(μF) outside the two degeneracies and the handling of n≥4) do not affect the final classification, which coincides with the paper’s result and implications (no limit cycle) as stated in Theorem 6 and Corollary 7 of the paper .