“LARGE” STRANGE ATTRACTORS IN THE UNFOLDING OF A HETEROCLINIC ATTRACTOR

The paper constructs a first-return map F_λ with a logarithmic singularity and strong contraction in the stable direction, explicitly deriving F_λ(x,y) = (x + ξ + λΦ_1(x,y) − K_ω ln(y + λΦ_2(x,y)), (y + λΦ_2(x,y))^δ) + h.o.t., with δ>1, by composing local and global maps near a Bykov network (Theorem A), and reduces it to a one-dimensional singular limit ha(x) = x + ξ + a − K_ω ln Φ_2(x,0) after a scaling y→y/λ (Lemma 8.1) . It then verifies the Wang–Young rank‑one hypotheses (H1–H7) and applies the rank‑one theory to obtain a positive-Lebesgue-measure set of parameters with irreducible strange attractors supporting unique SRB measures, and positive Lyapunov exponents for Lebesgue-a.e. initial condition on the section (Theorem B) . Finally, it proves the existence of a sequence λ_n→0 with superstable 2-periodic orbits of the flow, each persisting on intervals (Theorem C) . The candidate solution mirrors these steps: it identifies the same rank‑one structure, uses the Morse property of ln Φ_2(·,0) for nonflat criticalities, checks parameter transversality via the −K_ω ln λ term, and invokes rank‑one results to obtain strange attractors and SRB measures. For sinks, the candidate offers two routes: a direct phase-matching using F_λ^2 leveraging the −2K_ω ln λ dependence, and an abstract parameter‑selection route in the 1D limit. The paper’s sink construction instead selects parameters via k(λ) = −K_ω ln λ modulo 2π, yielding λ_n→0 and superstable 2-cycles. These are substantively consistent but procedurally different. Minor differences include the candidate’s (nonessential) use of a stable foliation via HPS to justify the 1D quotient, whereas the paper stays entirely within the rank‑one framework and a two-parameter embedding reduction. Overall, both arguments are correct; the paper provides a complete, internally consistent proof, and the model solution gives a compatible but stylistically different proof sketch.