We introduce shrubs in this paper in order to present a first approach to studying the structure of the Mandelbrot set. Primary, secondary ... and N-ary shrubs are analyzed. We have experimentally obtained formulae to calculate the periods of the hyperbolic components representatives of the structural branches, and the preperiods and periods of both the… (More)
In this paper we present the alternated Julia sets, obtained by alternated iteration of two maps of the quadratic family 2 1 , 1,2 n n i z z c i and prove analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou-Julia theorem in the case of polynomials of degree greater than… (More)
Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated… (More)
This article describes a new family of cryptographically secure pseudorandom number generators, based on coupled chaotic maps, that will serve as keystream in a stream cipher. The maps are a modification of a piecewise linear map, by dynamic changing of the coefficient values and perturbing its lesser significant bits.
— Both the computer drawing of the complement of the Mandelbrot-like set of a one-parameter dependent complex exponential family of maps and the computer drawing of the Julia sets of the maps of this family, grow with the maximal number of iterations we choose. Some graphic examples of this growth, which evoke the image of a garden, are shown here.
Hyperbolic components and Misiurewicz points of the chaotic region of the Mandelbrot set are located in what we call shrubs. Each shrub has two well-differentiated parts. The first one, shrub 0 ; corresponds to the main branch of the shrub. The second one, shrub r ; corresponds to all the other branches of the shrub. In this paper, we have focused on the… (More)