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′ = z2 - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-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 $\overline{T}: x^{\prime }=x^{2}+y-c$; y′ = γ y + 4x2y, γ ≥ 0. For $\gamma = 0, \overline{T}$ 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. [ABSTRACT FROM AUTHOR]