From the box-within-a-box bifurcations organization to the Julia set. Part II: Bifurcation routes to different Julia sets from an indirect embedding of a quadratic complex map

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ: z = z2 − c, c being a real parameter, −1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called “boxwithin- a-box ”), generated by the map x = x2 − c with x a real variable. Here, the second part deals with an embedding of TZ into the two-dimensional noninvertible map T : x = x2 + y − c; y = γy + 4x2y, γ ≥ 0. For γ = 0, T is semiconjugate to TZ in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0.