The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps.

Both the paper and the model use the same renormalisation scheme based on the substitution (L,R) → (RR, LR), yielding an affine conjugacy between the two-step return f^2 and the original BCNF with the renormalised parameter map g(ξ) = (τ_R^2−2δ_R, δ_R^2, τ_Lτ_R−δ_L−δ_R, δ_Lδ_R) (equation (4.1), with the conjugacy h_ξ on Π_ξ given in (4.5)–(4.6)) . In the orientation-preserving region Φ(1) (δ_L>0, δ_R>0) , the base case R(1)_0 is controlled by ϕ_+(ξ) (equation (5.3)), giving a connected chaotic Milnor attractor (Theorem 5.1) and the regions R(1)_n are defined by ϕ_+(g^n(ξ))>0, ϕ_+(g^{n+1}(ξ))≤0 (equation (5.5)) . Then, via Proposition 5.2 and the component-doubling pullback (Proposition 4.4), the paper concludes Theorem 5.3: exactly 2^n connected components for ξ∈R(1)_n . The model’s three-step outline tracks this structure closely. Minor issues: the model asserts the Milnor property without noting the (technical) δ_R<1 condition invoked for (iii) in Theorem 5.1 (the paper comments it is likely unnecessary) , and it informally interchanges ALAR with ARAL in one place; since only tr(·) and det(·) are used, the renormalised parameters are unaffected. Overall, both arguments agree in substance and conclusion.