Non-invertible case (δ_L<0, δ_R>0): one-component chaotic attractor in R^{(4)}_0

Establish that for parameters ξ in R^{(4)}_0 = { ξ ∈ Φ^{(4)} | φ_min(ξ) > 0, φ_min(g(ξ)) ≤ 0, α(ξ) < 0 }, with Φ^{(4)} = { ξ ∈ Φ | δ_L < 0, δ_R > 0 }, whenever the border-collision normal form f_ξ admits an attractor it is chaotic and has exactly one connected component.

Background

In the non-invertible regime with δ_L < 0 and δ_R > 0, the attractor is often destroyed by heteroclinic boundary crises not given by simple algebraic conditions, making rigorous characterisation challenging. Despite these complexities, the authors propose a conjecture for the single-component nature of the attractor in R^{(4)}_0.

They define R^{(4)}_0 via φ_min and the renormalisation operator g, with an additional constraint α(ξ) < 0 to avoid regions with stable low-period cycles.