The Construction for Generalized Mandelbrot Sets of the Frieze Group

By an analysis of symmetric features of equivalent mappings of the frieze group, a definition of their generalized Mandelbrot sets is given and a novel method for constructing generalized Mandelbrot sets of equivalent mappings of frieze group is presented via utilizing the Ljapunov exponent as the judgment standard. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

Two Partitioning Algorithms for Generating of M Sets of the Frieze Group

Symmetric features of the frieze group equivalent mappings were analysed, and two partitioning algorithms are given for constructing generalized Mandelbrot sets of frieze group equivalent mappings in order to study the characteristics of generalized Msets. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

RESEARCH FRACTAL STRUCTURES OF GENERALIZED M-J SETS USING THREE ALGORITHMS

Extreme modulus escaping time algorithm, decomposition algorithm and fisheye algorithm are analyzed in this thesis, and we construct a series of generalized Mandelbrot-Julia (M-J) sets using these three algorithms. By studying the structural character of generalized M-J sets, we find: (1) extreme modulus escaping time algorithm and decomposition algorithm are simple modifications of classic escaping time algorithm, they can both construct the structure of non-boundary areas of generalized M-J sets; (2) non-boundary areas of generalized M-J sets have fractal characters; (3) generalized M-J sets have symmetry, and the process of evolvement depends on range of phase angle; (4) we can observe not only the whole structure of generalized Mandelbrot sets but also the details of some parts by using fisheye algorithm.

FRACTAL IMAGES OF GENERALIZED MANDELBROT SETS

The transformation function z ← zα+βi+c, with both α and β being either positive or negative integers or real numbers, is used to generate families of mostly new fractal images in the complex plane [Formula: see text]. The calculations are restricted to the principal value of zα+βi and the obtained fractal images are called the generalized Mandelbrot sets, ℳ (α, β). Three general classes of ℳ (α, β) are considered: (1) α ≠ 0 and β = 0; (2) α ≠ 0 and β ≠ 0; and (3) α = 0 and β ≠ 0. Our results demonstrate that the shapes of fractal images representing ℳ (α, 0) are usually significantly deformed when β ≠ 0, and that the size of either stable (α > 0) or unstable (α < 0) regions in the complex plane may increase as a result of non-zero β. It is also shown that fractal images of the generalized Mandelbrot sets ℳ (0, β) are significantly different than those obtained with a non-zero α.

Smooth Decomposition of Generalized Fatou Set Explains Smooth Structure in Generalized Mandelbrot Set

Generalized Mandelbrot sets arise in perturbed (non-analytic) versions of the complex logistic map. Numerically, it contains smooth portions as shown previously. To exclude that this result is specific to particular initial conditions only, the structure of the analogue to the Fatou set is looked at in the region in question. The set of non-divergent points is being "eaten up" by a smooth invading boundary. Therefore, the same type of decomposition applies independent of position in parameter space, in the region in question.