From the Box-within-a-Box bifurcation Structure 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′ = z − 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′ = x − c with x a real variable. Here, the second part deals with an embedding of TZ into the two-dimensional noninvertible map T : x′ = x + y− c; y′ = γy + 4xy, 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.