Remark on the paper “The hybrid matching of Hurwitz systems” by Luis Fernando Mello and Paulo Santana

This note shows that the linear branch-fixing construction used in the proof and subsequent application of Lemma 1 in the paper by Mello and Santana is invalid for a genuinely broken switching line, and that Theorem A (a) is false as stated. If 𝜌 > 0 , every linear map fixing pointwise both non-collinear branches of the switching set Σ 𝜌 is the identity, so the proposed piecewise linear reduction cannot transform arbitrary Hurwitz matrices to the asserted normal form while preserving the switching set and reset map. An explicit node–focus hybrid system satisfying all hypotheses of Theorem A is then constructed. Its return map is 𝑃 ⁡ ( 𝜉 ) = 𝐾 ⁢ 𝜉 with 𝐾 ≈ 4 0 . 5 2 > 1 , and hence the origin is not even Lyapunov stable. A valid monomial return-map formula is also recorded under explicit branch-to-branch transition assumptions. The straight-line case 𝜌 = 0 , including the limit-cycle example of the original paper, remains unaffected.