Dynamics near a class of nonhyperbolic fixed points

The paper proves that, under positive definiteness of the symmetric coefficient tensors for ∂P/∂x, ∂Q/∂y, and the mixed forms ∂P/∂x±∂P/∂y and ∂Q/∂y±∂Q/∂x, the local stable and unstable sets are Lipschitz graphs, using cone invariance and a vertical-arc argument (Lemma 3.1, Proposition 3.1, Theorem 1.1) . The candidate solution proves the same conclusion via a quantitative Hadamard–Perron graph-transform approach with explicit constants. The assumptions match (positivity of A,B,C,D,E,H; higher-order X,Y ensuring smallness near the origin), and the conclusions (existence, uniqueness on vertical/horizontal fibers, and Lipschitz bounds) agree with the paper’s theorem. Thus both are correct, with different methods; the model’s proof is more quantitative, while the paper’s is geometric/topological. Key inequalities used in the paper, derived from positive definiteness and higher-order smallness (2.1)-(2.2), align with the model’s scale-invariant bounds used to obtain contraction and vertical expansion/uniqueness .