Generalized Mandelbrot Set Research Articles

The generalized M sets from a class complex mapping system

The method constructing the generalized Mandelbrot sets (the generalized M sets) for positive integer index number generated from the composition of two simple complex mapping which was put forward by Shirriff was expanded. Based on a class simple complex mapping system expanded by the author, a series of the generalized M sets for real index number were constructed. Using the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, the structural characteristics and evolutions of the generalized M sets were studied. The results show that the generalized M sets for integer index number have symmetry and fractal feature; while the generalized M sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle. And the author sets forth the physical meaning for the generalized M sets.

ON A BICOMPLEX DISTANCE ESTIMATION FOR THE TETRABROT

In this article, we present some distance estimation formulas that can be used to ray traced slices of the bicomplex Mandelbrot set and the bicomplex filled-Julia sets in dimension three. We also present a simple method to explore and infinitely approach these 3D fractals. Because of its rich fractal structure and symmetry, we emphasize our work on the generalized Mandelbrot set for bicomplex numbers in dimension three: the Tetrabrot.

Switched processes generalized mandelbrot sets for complex index number

According to the switched complex mapping proposed by the author, the method constructing the switched processes generalized M (Mandelbrot) sets was elaborated, and a series of the switched processes generalized M sets for complex index number were constructed. The construction characteristics of the generalized M sets were expounded according to the analysis of the algorithm constructing the generalized M sets. On the basis of what has already been achieved, the trajectories of a starting point in the complex C-plane under the switched mapping were researched into. The results show that the switched processes generalized M sets have the fractal feature, the construction characteristics of the switched processes generalized M sets are dependent on the complex index number w and the switched variable r0, and the reason which results in the discontinuity of the switched processes generalized M sets is the discontinuity of choice of the principal range of the phase angle.

M and J sets from Newton's transformation of the transcendental mapping F( z)=e zw+ c with vcps

Newton's transformation f w ( z)= z−1/( wz w−1 ) containing only one complex parameter w ( w≠0 or 1) is constructed from the transcendental mapping F( z)=e z w + c . Although the number of critical points of f w ( z) is countably infinite, a method based on the Valid Critical Point Set, vcps ={ z k∈C −π< arg(z k)⩽π, f′ w(z k)=0, k∈Z} , is discussed for the generation of the generalized Mandelbrot set M of f w ( z). The petal fragments, the multi-period fragments, and the classical “Mandelbrot set” fragments are found in M. The dynamical characteristics of f w ( z) for different values of w are analyzed. The relationship between the parameter w in a classical “Mandelbrot set” fragment and the structure of the corresponding filled-in Julia set is investigated. A large number of novel images of filled-in Julia sets are generated based on the rate of convergence of orbits.

A GENERALIZED MANDELBROT SET FOR BICOMPLEX NUMBERS

We use a commutative generalization of complex numbers called bicomplex numbers to introduce bicomplex dynamics. In particular, we give a generalization of the Mandelbrot set and of the "filled-Julia" sets in dimensions three and four. Also, we establish that our version of the Mandelbrot set with quadratic polynomial in bicomplex numbers of the form w2 + c is identically the set of points where the associated generalized "filled-Julia" set is connected. Moreover, we prove that our generalized Mandelbrot set of dimension four is connected.

An Abstract Mandelbrot Set Algorithm for zn + c

We present an algorithm, an extension of Lavaurs' algorithm for z2+c, for determining the combinatorial arrangement of the hyperbolic components of the generalized Mandelbrot set for zn+c. Several external rays are plotted to support the 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 &gt; 0) or unstable (α &lt; 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.

A new scaling along the spike of the Mandelbrot set

Using the zeros of a family of real polynomials, we locate a sequence of midgets along the spike of the Mandelbrot set. By this method we can readily find midgets (tiny copies of the Mandelbrot set) of high cycle number; we include a picture of a 200-cycle midget magnified by a factor of 5.68 × 10 238. The ratios of distances between the cardioid centers for successive midgets in this sequence exhibit an asymptotic scaling, as does the ratio of head-to-center distances. We present evidence supporting a pattern of scalings for some “generalized Mandelbrot sets.”

A generalized mandelbrot set and the role of critical points

We examine the Julia sets for certain cubic and quartic polynomials and observe the structure of the Julia set as determined by the orbits of the critical points. In examples where the orbits of some critical points diverge and others converge, or where orbits converge to different cycles, we note the local structure of the Julia set is dominated by the behavior of the nearest critical point. This information is encoded in a generalized Mandelbrot set reflecting the behavior of all the critical points.

Non-Differentiable Structure in the Generalized Mandelbrot Set

Abstract The generalized Mandelbrot set, described previously, contains - like the original Mandelbrot set - non-differentiable self-similar fractal structures. An example looking like a vortex structure is presented along with a corresponding generalized Julia set.

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

Abstract 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.