Geometric decomposition of planar vector fields with a limit cycle

The paper derives the Hamilton–Jacobi-type PDE (∇H)^T ẋ − p(H)|∇H|^2 = 0 directly from the κ∇H decomposition (Eq. 6) by left-multiplying with (∇H)^T and using the antisymmetry of R⊥ (their Eqs. 6, 28/38, 29/39) . It also observes p = 0 on the limit cycle and that κ is skew there, so the flow is tangent to the level set of H (discussion around Eqs. 6, 9, 13 and the paragraph on p(HLC) = 0) . The candidate solution replicates these steps and further proves the converse: if the PDE holds with ∇H ≠ 0, then ẋ − p(H)∇H is orthogonal to ∇H and equals wR⊥∇H, giving the same κ = p(H)I + wR⊥. The paper also proposes tracing level sets via ∇Ĥ⊥ = R⊥ κ^{-1} ẋ, exactly as used by the candidate (their Eqs. 30–31) . On the radial example ẋ = (x(β − r^2) − ωy, y(β − r^2) + ωx), both choose H = r^2/2, p(H) = β − 2H, w ≡ ω, yielding HLC = β/2 and r = √β (their Eq. 9 and accompanying discussion) . Net: same framework and conclusions; the model adds a clean 2D “⇐” argument not written explicitly in the paper.