Grazing bifurcations and transitions between periodic states of the PP04 model for the glacial cycle.

The paper proves that the grazing set G on a fixed Poincaré section is a smooth 2D manifold except at isolated degeneracies, and that each leaf is locally (nearly) planar with nearly constant grazing time tg, deriving the leading-order condition −λ1(c^T f(tg)+d)=c^T ḟ(tg) and its trigonometric specialization. The candidate solution reproduces the same structure: it sets up G via the constraints F(tg)=Ḟ(tg)=0, uses an Implicit Function Theorem argument for local manifold structure, employs a spectral-gap approximation e^{LΔ}≈e^{−λ1Δ}M to show leaves are planar with normal n=c^TM, derives the same scalar condition for tg, shows tg is nearly constant along each leaf, and obtains the same trigonometric identity. These steps align with Lemma 5.2, Lemma 5.3, Theorem 5.1, and Corollary 5.5 of the paper, including the multi-leaf structure via push-backs/push-forwards. No material contradictions were found; differences are stylistic and in rigor level, not substance.