A MATHEMATICAL MODEL FOR WHORL FINGERPRINT

The paper studies the same system ẋ = y, ẏ = −x(x^2 − 1)^2 + θ y (x^2 − 1)^2 (its eq. (6)) and proves: E0 = (0,0) is stable if θ < 0, unstable if θ > 0, and a center if θ = 0 (Theorem 2.1), and that E1 = (−1,0), E2 = (1,0) are cusps (Theorem 2.2). These claims align with the model’s derivations, which compute the Jacobian at E0 and explicitly expand near ±1 to verify the cusp normal form. The paper’s cusp proof lightly overstates by setting b_n = 0, whereas a direct expansion shows the x^n y term is present with n = 2 for θ ≠ 0; however, the cited theorem already allows a cusp when n ≥ m with k = 2m, so the classification remains correct. Net: both reach the same conclusions; the model adds a clear expansion and notes the Hamiltonian structure at θ = 0. See the paper’s statements and normal-form criterion summarized in its Sections 1–2 and Theorems 2.1–2.2 .