Doubling bifurcations of invariant closed curves in 3D maps

The paper’s main claims are: (i) under assumptions A1–A4, if a 2D attracting manifold M containing the resonant ICC exists and is tangent along the ICC to the eigenvectors associated with the eigenvalue approaching −1, then the topology of M (cylinder vs Möbius) predicts loop- vs length-doubling; and (ii) an even-period resonant ICC can only length-double. Both are stated and argued (primarily by a geometric/topological sketch and computational evidence) in the paper’s Main result and Section 3.3, together with a practical eigenvector-based proxy for detecting the topology of the tangent space . The candidate solution reaches the same conclusions via a more formal line-bundle argument (Stiefel–Whitney w1 classification) and a short cocycle sign-proof for the even-period obstruction. Aside from a minor notational slip (treating α(x) as constant rather than constant sign), the model’s proof is rigorous and complements the paper’s sketch. The paper’s existence claim for M is plausible under A1–A4 but not fully proved; still, its conclusions match the model’s, and its computational protocol aligns with the model’s proxy. Hence, both are correct, with different proof styles.