Orientation-reversing case: uniqueness and form of the chaotic attractor in R^{(2)}_0

Establish that for the two-dimensional border-collision normal form f_ξ(x,y) = (τ_L x + y + 1, −δ_L x) for x ≤ 0 and (τ_R x + y + 1, −δ_R x) for x ≥ 0, with parameters ξ = (τ_L, δ_L, τ_R, δ_R) in the orientation-reversing region R^{(2)}_0 = { ξ ∈ Φ^{(2)} | φ^−(ξ) > 0, φ^+(g(ξ)) ≤ 0, α(ξ) < 0 }, the map has a unique chaotic attractor with one connected component equal to the closure of the unstable manifold W^u(X) of the right-half-plane saddle fixed point X = (−1/(τ_R − δ_R − 1), δ_R/(τ_R − δ_R − 1)). Here Φ^{(2)} = { ξ ∈ Φ | δ_L < 0, δ_R < 0 }, Φ = { ξ | τ_L > |δ_L + 1|, τ_R < −|δ_R + 1| }, g(ξ) = (τ_R^2 − 2δ_R, δ_R^2, τ_L τ_R − δ_L − δ_R, δ_L δ_R), α(ξ) = τ_L τ_R + (δ_L − 1)(δ_R − 1), φ^+(ξ) = δ_R − (τ_R + δ_L + δ_R − (1 + τ_R)λ_L^u)λ_L^u, and φ^−(ξ) = δ_R − (δ_R + τ_R − (1 + λ_R^u)λ_L^u)λ_L^u, with λ_L^u and λ_R^u the unstable eigenvalues of A_L and A_R respectively.

Background

The paper studies the four-parameter two-dimensional border-collision normal form (BCNF) and its chaotic attractors across orientation-preserving, orientation-reversing, and non-invertible regimes. In the orientation-reversing case (δ_L < 0, δ_R < 0), numerical evidence indicates that the chaotic attractor coincides with the closure of the unstable manifold of the right-half-plane saddle fixed point X and is destroyed at a heteroclinic bifurcation where T = C.

The authors define regions in parameter space via a renormalisation operator g and bifurcation functions φ+ and φ−. For R{(2)}_0, they conjecture existence and uniqueness of a one-component chaotic attractor equal to cl(Wu(X)), extending partial rigorous results previously established on a subset of this region.

References

We conjecture that throughout R{(2)}_0 the map has a unique chaotic attractor with one connected component equal to the closure of the unstable manifold of X, as in Fig.~\ref{fig:typ_pp_2}-a.

— The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps  (2402.05393 - Ghosh et al., 2024) in Section 7 (The orientation-reversing case)

Still, the internal geometry of \ell remains unresolved. As noted above, the equality \ell = {P,P'} holds on a subset of the parameter region \mathfrak{R}, making it plausible that this equality might hold throughout \mathfrak{R}. However, in a private communication, Y. Ishii and D. Sands brought to the author's attention examples of parameters suggesting that, in general, \ell could have a more complex structure.

— Basins of attraction and escape in the Lozi map  (2609.11540 - Kvaternik, 10 Sep 2026) in Section 1, Introduction; Section 6, Concluding remarks

Thus, although the present work, together with the previous one in , establishes that \ell is an attractor of the Lozi map and, at the same time, contains all of its entropy-inducing dynamics, the structure of this set remains an open question. A deeper understanding of its geometry and dynamics requires further research and may ultimately lead to the identification of a new class of attractors for Lozi maps, distinct from the examples known to date.

— Basins of attraction and escape in the Lozi map  (2609.11540 - Kvaternik, 10 Sep 2026) in Section 6, Concluding remarks