Non-invertible case (δ_L>0, δ_R<0): existence of a one-component chaotic attractor in R^{(3)}_0

Establish that for the border-collision normal form f_ξ with parameters ξ in the non-invertible region R^{(3)}_0 = { ξ ∈ Φ^{(3)} | φ_min(ξ) > 0, φ_min(g(ξ)) ≤ 0, α(ξ) < 0 }, where Φ^{(3)} = { ξ ∈ Φ | δ_L > 0, δ_R < 0 } and φ_min(ξ) = min[φ^+(ξ), φ^−(ξ)], the map has a unique chaotic attractor with one connected component.

Background

In the non-invertible regime with δ_L > 0 and δ_R < 0, the attractor can be destroyed either by the homoclinic boundary φ^+(ξ) = 0 or the heteroclinic boundary φ^−(ξ) = 0. The region R^{(3)}_0 is defined using φ_min and the renormalisation operator g to capture where the attractor persists.

The authors’ numerical component-counting results suggest a single-component chaotic attractor throughout R^{(3)}_0, leading them to pose this conjecture for rigorous verification.