The dud canard: Existence of strong canard cycles in R3

The paper proves Theorem 1.1 in the analytic setting via a two-chart blow-up analysis that constructs (i) small Hopf-born cycles in the ε̄=1 chart using an analytic slow manifold and a Melnikov-type return map, and (ii) intermediate cycles in the x̄=−1 chart via transition maps and then glues these families to obtain the full branch Γ_{h,ε} with µ(h,ε) and Hausdorff convergence to singular canard cycles; see the setting (3)–(5) and the statement of Theorem 1.1 (µ=µ_H(√ε), µ_H(0)=0) and its proof architecture in Sections 2–4 . By contrast, the model’s solution asserts a global Poincaré map that is a small C^1 perturbation of the identity and even posits that the singular return map is the identity, from which it applies an IFT in µ. These steps sidestep the zero-Hopf nonuniformity and the need for analyticity stressed in the paper (for extending a normally elliptic slow manifold and setting up a uniform return map near the Hopf) and they omit the chart-wise construction and gluing that the paper makes essential . The model also does not justify the claimed transversality of the return map’s µ-dependence at ε=0 nor the O(√ε) near-identity estimate for the global map. Hence, while the model states the correct high-level result, its proof outline relies on unjustified or false claims (e.g., singular map ≡ identity) and misses key hypotheses (analyticity), making it incorrect as a proof.