Saddle Invariant Objects and their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

The paper computes and summarizes the manifold organization near homoclinic flip bifurcations of case B in Sandstede’s model, including the case-B eigenvalue conditions, the codimension-one loci Ho, Ht, 2Ho, SNP, PD, the fold F of q→0 heteroclinics with quadratic tangency, and the loci CC± where Floquet multipliers coalesce; it also documents the distinct S* intersection pictures at BI versus Bo and shows topological equivalence of IF and OF away from the codimension-two points (e.g., conditions and case-B unfolding in 2.1; BI/Bo pictures and “Results” summary; fold F and CC±; infinite heteroclinic orbits for case B) . The candidate solution reconstructs these findings via a theoretical (Lin-map/return-map) proof sketch, aligning on all major claims (BI quadratic self-tangency, Bo closure at one Ŵss(0) point with accumulation on Ŵs(q), the same codimension-one curves and region structure, and regions with infinitely many heteroclinic orbits tied to the non-orientable Möbius geometry). Minor nuance: in region 1, the paper sometimes describes the basin boundary on S* as including both Ŵs(0) and the two points Ŵs(q), whereas the model summary states ∂B(Γa)=Ws(0); this is a boundary-description nuance on S*, not a substantive conflict in the phase-space organization (compare the region-1 descriptions) .