On the Limit Cycles of a Quartic Model for Evolutionary Stable Strategies

The paper establishes, for X in X2: (a) at most two centers in Δ and existence of examples with one or two centers, via a no-contact segment argument and orientation constraints; (b) existence of at least five nested limit cycles from a high-order weak focus (Proposition 9); and (c) existence of four limit cycles as two disjoint nests via a reversible subfamily (Proposition 10). These claims and proofs appear correct and self-contained in the uploaded PDF. In particular, Theorem 1(a–c) is stated explicitly, and the proofs of (b) and (c) are traced to Propositions 9 and 10, respectively . The model’s solution correctly paraphrases (b) and (c), including the small-amplitude argument ensuring cycles lie inside Δ. However, its treatment of (a) contains a substantive error: it asserts that one can take a quadratic Hamiltonian core Y and then obtain X by multiplying components by the factors (4x^2−1) and (4y^2−1), claiming centers of Y remain centers for X. The paper explicitly warns that transforming Y=(f,g) to X=(uf,vg) with u≠v can change the type of singularity (centers may become nodes or foci, and vice versa). Indeed, det DX(p)=u(p)v(p) det DY(p) while the trace—and hence the focus/center character—need not be preserved unless u=v locally; inside Δ we have u(p),v(p)≠0 but generally u(p)≠v(p), so a center of Y need not remain a center of X . The model’s “explicit examples” for exactly one or two centers rely on this incorrect invariance and are not justified. By contrast, the paper’s proof of (a) uses a contact-point/Bezout-based no-contact segment argument culminating in the opposite orientations conclusion and the two-center bound, with a careful line-contact analysis that the model omits .