A proof of the uniqueness of the limit cycle of a quasi-homogeneous system

The paper proves there is a unique parameter α* at which the two separatrices at infinity coincide (a heteroclinic on the Poincaré equator) and, consequently, exactly one limit cycle exists for α in (α*,0) and none otherwise; see the Poincaré-chart formulation dv/dτ = v fα(v) + z^2, dz/dτ = z fα(v), fα(v)=v^3−αv^2−1, and Theorem 3.1 summarizing the existence/uniqueness range (α*,0) with no cycles for α≤α* or α≥0 . The candidate solution establishes the same result by: (i) showing the family is a positively rotated vector field via ∂θ/∂α = (M ∂αN − N ∂αM)/(M^2+N^2) with M=y, ∂αN=x^2y, giving x^2y^2≥0 (the same rotated-vector-field property highlighted in the paper) ; (ii) giving a direct Bendixson–Dulac argument for nonexistence when α≥0; and (iii) using the compactified dynamics in the (v,z) chart to define α* and produce a trapping annulus for α>α*. The approaches differ in technique (the paper uses a careful comparison framework for the separatrices to show a unique α* and a contradiction-via-ᾱ argument for existence, while the model uses a trapping-annulus Poincaré–Bendixson argument), but they reach the same conclusions and are mutually consistent with the paper’s chart, equilibria at infinity, and uniqueness argument for α* .