Planar chemical reaction systems with algebraic and non-algebraic limit cycles

The paper’s Theorem 3 constructs the same polynomial system, proves its degree n=2 n_h+1, shows it is mass-action realizable, and establishes N algebraic limit cycles along the ovals of h=0 with cofactor s(x,y) = -xy ||∇h||^2. The candidate independently derives all these facts and further uses the Darboux cofactor identity dh/dt = s h to exclude any other periodic orbits in the positive quadrant and compute the Poincaré multiplier. The paper proves existence and stability via a transversality/annulus argument (Theorem 2), while the model proves stability and uniqueness via the cofactor integral—two correct but different proof routes. The model’s mass-action realization matches the constructive approach (one reaction per monomial) presented earlier in the paper.