On Shilnikov’s scenario with a homoclinic orbit in 3D - revised version

The paper establishes existence of entire trajectories with prescribed bi-infinite symbol sequences by constructing two angle-defined sets M0 and M1 and using curve-expansion plus a diagonal subsequence argument (Proposition 7.4 and Theorem 8.2 ). It explicitly does not claim a Smale horseshoe, Markov partition, conjugacy, or uniqueness of itineraries, noting this in the introduction . By contrast, the model assumes a uniform hyperbolic product structure with cone invariance, domain invariance Rε(D) ⊂ D, diffeomorphic strip mappings, and uniqueness of coding via shrinking curvilinear rectangles—none of which is derived in the paper. The paper’s quantitative bounds control the range and provide a near-identity estimate for K^{-1}∘E, but do not furnish the derivative hyperbolicity (nor the Markov/diffeomorphism assertions) used by the model (cf. Corollary 6.3 and Proposition 6.4 ). The model also identifies the first coordinate of Rε with an angle difference and claims images of entire vertical strips are horizontal full crossings—claims not supported by the paper’s framework. Hence: paper correct within scope; model overclaims and is unsupported by the cited results.