Limit Cycles in Piecewise Quadratic Kolmogorov Systems

The paper proves that some Kolmogorov quadratic piecewise systems K2 admit at least six small crossing limit cycles by explicitly constructing a sixth‑order weak focus on the switching line and unfolding it versally within K2 (Theorem 1.1; see the definition of K2 in (1) and the return‑map/difference‑map setup) . It also shows weak foci of order eight exist, though within K2 they need not unfold to eight cycles . The candidate solution outlines the same high‑level strategy: normalize near a monodromic pseudo‑equilibrium on a straight switching line, use analyticity of the (half‑)return map and Lyapunov quantities, tune parameters so the first five coefficients vanish and the sixth does not, then apply higher‑order (pseudo‑Hopf) unfolding to obtain six crossing cycles. This aligns with the paper’s framework for the difference map and Lyapunov quantities Wk (Sections 2.1–2.4) and its realization of a sixth‑order weak focus with versal unfolding in K2 (Section 4) . Minor issues in the candidate’s write‑up include (i) saying the translated system “remains tangent to the axes” (the Kolmogorov multiplicative structure is not preserved by translation), (ii) using a displacement map Π(r)−r vs. the paper’s difference map, and (iii) asserting a general any‑order versal unfolding “inside the same class” (the paper cautions that order‑8 weak foci in K2 need not versally unfold to eight cycles). None of these affect the core existence claim, which matches the paper.