Multi-shooting parameterization methods for invariant manifolds and heteroclinics of 2 DOF Hamiltonian Poincaré maps, with applications to celestial resonant dynamics

The paper defines Wp via first oriented crossings of Σ (its Eq. (38)) and asserts that, because W solves the fixed-time invariance Φ_{τ(k)}(W(k,s)) = W(k+1,λ s) on |s|<D, the induced section parameterization Wp satisfies the Poincaré invariance P(Wp(k,s)) = Wp(k+1,λ s) locally, and is then globalized via iterates (its Eqs. (39)–(40)) . The candidate solution provides the same construction and adds a clean zero-crossing/shift argument proving local invariance, plus the same globalization and the heteroclinic-as-intersection equivalence the paper relies on in Section 5 (cf. Eq. (41)) . Thus both are aligned; the model formalizes steps that the paper states succinctly.