LIMIT CYCLES OF PIECEWISE SMOOTH DIFFERENTIAL SYSTEMS WITH NILPOTENT CENTER AND LINEAR SADDLE

The paper’s Theorem 2.1 proves exactly the same classification as the candidate solution for the two-zone system (1.14): period annulus iff µ=0 and k≥0; for µ≠0 and δ≠0 with k<0, the number of crossing limit cycles is 0, 1, or 2 according as k>−δ^2/µ^2, k=−δ^2/µ^2, or k<−δ^2/µ^2. The paper’s proof uses the equalities F(0,y1)=F(0,y2) and H(0,y1)=H(0,y2), factorization, and elimination (equation (2.3)), which is algebraically equivalent to the model’s “line-and-circle intersection” argument on the section x=0. Thus, the proofs are the same in substance, though the model adds a geometric Poincaré-map interpretation. One minor flaw in the model is an incorrect exclusion of the y2=−y1 branch for k<0 on orientation grounds; this branch is instead excluded by H(0,y1)=H(0,y2) forcing µ=0, so it does not affect the final classification for µ≠0. Overall, both are correct and aligned with Theorem 2.1 and its proof. See Theorem 2.1 and its proof steps and examples in the PDF , and the system/first integrals definitions in (1.10)–(1.13) .