Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

The paper states Proposition 4.2 (A→B→C implies infinitely many structurally stable A→C orbits; and, when W^s(A) cuts the small sphere S* transversely, infinitely many curves in Ŵ^s(C) accumulate backward onto Ŵ^s(A) with Ŵ^s(B) as a geometric accumulation curve) and gives only a brief justification: “The proof follows from the λ-lemma and is a variation of the proof given in [13].” This is exactly the configuration they already discuss in region 3, where Γo→Γt→0 yields infinitely many Γo→0 connections and infinitely many curves in Ŵ^s(0) accumulating onto Ŵ^s(Γt) and then onto Ŵ^s(Γo) as seen on S* . The candidate solution provides the standard λ-lemma construction with explicit 2D sections and hitting maps to S*, which is the natural detailed version of the paper’s sketch. Both arguments hinge on the same mechanism (λ-lemma plus transversality in a section), so they are substantially the same proof; the model adds the missing details the paper omits.