Perturbations of the quadratic family of order two

Define the quadratic family of order two as F-mu (x, y) = (y, -x(2) + mux), where mu is a real parameter. The boundary of the basin of attraction of t he fixed point at infinity is an invariant curve for mu < 4, and is a Canto r set for mu > 4. Perturbations of F-mu with mu not equal 4 were studied in Romero et al (2001 Discrete Continuous Dynam. Syst. 7 35) (also in higher dimension), where it was proved that these situations persist. Now we study perturbations of the bifurcation point mu = 4, where the explosion of the basin, B-infinity, occurs. We prove that either there exists a connected in variant curve J contained in the boundary of the basin, or the set of criti cal points is a subset of B-infinity and the boundary has uncountably many components accumulated by the pre-images of the analytic continuation of th e fixed point at the origin. The curve J undergoes a fractalization process until it ceases to exist.