Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with Z4-symmetry

The paper proves, for the Z4–equivariant normal form (1.2) with b1<0 and Re(a0 b0)<0, the existence of a smooth heteroclinic stem γ=h(β,µ) (Theorem 1.2) and countably many open, disjoint parameter regions A−k (below the stem) with symmetric Lorenz attractors and A+k (above the stem) with Simó angels, all adjoining (h(0,µ),0) (Theorem 1.3). These claims are explicitly stated and proved via an analytic construction of the stem and a Shilnikov-like criterion developed in Section 5, including verification of the needed assumptions for system (1.2) (e.g., Proposition 5.4) . The candidate solution reaches the same conclusions by composing a local map near O with a global map and invoking the Afraimovich–Bykov–Shilnikov scheme and pseudohyperbolicity, matching the paper’s geometric framework (A1–A4) and definitions (Section 2) . Its mechanism (oscillatory quadratic forcing 2Re(b0 u^2), rotation θ̇≈β, exponentially small splitting after k half-turns) is a different but compatible proof narrative. Minor issues in the candidate solution include a small slip in identifying λ1+λ2 with tr/2 (it should equal tr) and sketch-level justification of pseudohyperbolicity. Overall, both the paper and model arrive at the same result; the paper’s proof is analytic and self-contained for the stated theorems, while the model provides a plausible geometric/normal-form derivation.