Generalized Mandelbrot Set Research Articles

FRACTAL PROPERTIES OF THE GENERALIZED MANDELBROT SET WITH COMPLEX EXPONENT

Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized [Formula: see text] set with different kinds of exponent [Formula: see text] ([Formula: see text] set), properties are hard to understand when [Formula: see text] is a complex number. In this paper, fractal property of generalized [Formula: see text] set with complex exponent [Formula: see text] is studied. First, a relation is constructed between generalized [Formula: see text] set with complex and real exponent. Then, distribution of [Formula: see text] set on complex plane is researched. Meanwhile, symmetry of generalized [Formula: see text] set is proved. Finally, graphics, generated by escape time algorithm, are the validated results of this paper.

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

An effective algorithm to compute Mandelbrot sets in parameter planes

In McMullen (2000) it was proven that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.

The dynamical André–Oort conjecture: Unicritical polynomials

We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set M2 (or generalized Mandelbrot set Md for degree d>2), we classify all curves C⊂A2 defined over C with Zariski-dense subsets of points (a,b)∈C, such that both zd+a and zd+b are simultaneously PCF for a fixed degree d≥2. Our result is analogous to the famous result of André regarding plane curves which contain infinitely many points with both coordinates being complex multiplication parameters in the moduli space of elliptic curves and is the first complete case of the dynamical André–Oort phenomenon studied by Baker and DeMarco.

Colored HOMFLY and generalized Mandelbrot set

Mandelbrot set is a closure of the set of zeroes of $resultant_x(F_n,F_m)$ for iterated maps $F_n(x)=f^{\circ n}(x)-x$ in the moduli space of maps $f(x)$. The wonderful fact is that for a given $n$ all zeroes are not chaotically scattered around the moduli space, but lie on smooth curves, with just a few cusps, located at zeroes of $discriminant_x(F_n)$. We call this phenomenon the Mandelbrot property. If approached by the cabling method, symmetrically-colored HOMFLY polynomials $H^{\cal K}_n(A|q)$ can be considered as linear forms on the $n$-th "power" of the knot ${\cal K}$, and one can wonder if zeroes of $resultant_{q^2}(H_n,H_m)$ can also possess the Mandelbrot property. We present and discuss such resultant-zeroes patterns in the complex-$A$ plane. Though $A$ is hardly an adequate parameter to describe the moduli space of knots, the Mandelbrot-like structure is clearly seen -- in full accord with the vision of arXiv:hep-th/0501235, that concrete slicing of the Universal Mandelbrot set is not essential for revealing its structure.

On a visualization of the convergence of the boundary of generalized Mandelbrot set to (n-1)-sphere

In this article we analyze the generalized Mandelbrot set in higher-order hyper- complex number spaces following both the Cayley-Dickson construction algebraic spaces and the spaces defined by Clifford algebras. The particular case of the generalized 3D quasi-Mandelbrot set was also considered. In particular, we investigated the increase of power of the iterated variable and proved that when this power tends to infinity, the Man- delbrot set is convergent to the unit circle. The same is true for the generalized Mandelbrot sets in higher-dimensional hypercomplex number spaces, i.e. when the power of iterated variable tends to infinity, the generalized Mandelbrot set is convergent to the unit ( n-1)- sphere. The results of our investigation were visualized for the generalized Mandelbrot set in a complex number space and the generalized quasi-Mandelbrot set in a 3D Euclidean space.

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.

On numerical approximations of the area of the generalized Mandelbrot sets

In the present work, the area of the generalized Mandelbrot sets is defined as the double limit of the areas of the plotted generalized Mandelbrot sets in a given square lattice, using the finite escape algorithm, while the lattice resolution and the number of iteration counts, used to plot them, tends to infinity. The asymptotic behavior of the areas of the generalized Mandelbrot sets in terms of their degree growth is investigated. Finally, numerical approximations of the area of the Mandelbrot set are proposed by using tools from regression analysis.

Feedback control and synchronization of Mandelbrot sets

The movement of a particle could be depicted by the Mandelbrot set from the fractal viewpoint. According to the requirement, the movement of the particle needs to show different behaviors. In this paper, the feedback control method is taken on the classical Mandelbrot set. By amending the feedback item in the controller, the control method is applied to the generalized Mandelbrot set and by taking the reference item to be the trajectory of another system, the synchronization of Mandelbrot sets is achieved.

ON A GENERALIZED FATOU–JULIA THEOREM IN MULTICOMPLEX SPACES

In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ2n of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tricomplex numbers.

A boundary equation of the principal period-2 component in the degree-n bifurcation set

The degree-n bifurcation set is a generalized Mandelbrot set for the complex polynomial P c (z)=z n +c. The boundary of the principal period-2 component in the degree-n bifurcation set is first defined and then formulated by a parametrization of its image, which is the unit circle under the multiplier map. We investigate the boundary equation using the geometric symmetry of the degree-n bifurcation set with respect to rays of symmetry in the complex plane. In addition, an algorithm drawing the boundary curve with Mathematica codes is proposed.

Groebner basis, resultants and the generalized Mandelbrot set

This paper demonstrates how the Groebner basis algorithm can be used for finding the bifurcation points in the generalized Mandelbrot set. It also shows how resultants can be used to find components of the generalized Mandelbrot set.

The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative

Adopting the experimental mathematics method combining complex variable function theory with computer aided drawing, this paper researches on the structural characteristic and the fission–evolution law of the generalized Mandelbrot set (generalized M set in short) perturbed by composing noise of additive and multiplicative, analyzes the effect of random perturbation to the generalized M set.

RENDERING OF THE INSIDE STRUCTURE OF THE GENERALIZED M SET PERIOD BULBS BASED ON THE PRE-PERIOD

This paper modifies the escape-time method, analyzes the process that the points inside a period bulb of the generalized Mandelbrot set (the generalized M set) after iterating a certain pre-period process finally enter into the periodic orbit. All the points with same pre-period form a circularity structure which shapes similar as the period bulb; as pre-period grows the circularity structure diffuses towards the period bulb boundary until crams the whole period bulb. Besides, we present a particular experimental observation of the rule of the pre-period bulb graphic of the generalized M set and its growing process and offer the theory proof.

Noise perturbed generalized Mandelbrot sets

Adopting the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, in this paper on the structure characteristics and the discontinuity evolution law of the additive noise perturbed generalized Mandelbrot sets (M-sets) was studied. On the influence of stochastic perturbed parameters of the structure of generalized M-sets was analyzed. The physical meaning of the additive noise perturbed generalized M-sets was expounded.

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

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.

Visual presentation of dynamic systems with hyperbolic planar symmetry

Hyperbolic symmetric mappings defined on hyperbolic tilings are investigated. Ljapunov exponents of the dynamic systems are computed with the Euclidean distance. The parameter combinations with great impact on the characteristics of the dynamic systems were chosen as the window coordinates for construction of generalized Mandelbrot sets. The accelerated direct search algorithm is used to search for the set of the critical points in the fundamental region. The parameter space is separated into chaotic, periodic and mixed regions by the Ljapunov exponents of the critical points. The generalized Mandelbrot sets (M-set), which are the cross-sections of the parameter space, were constructed. Three different types of hyperbolic symmetry patterns, which are chaotic attractors, filled-in Julia sets and mixed images composed of an attractor and a filled-in Julia set from the same set of parameters, were created by using parameters from this kind of M-sets.

Critical points and dynamic systems with planar hexagonal symmetry

Abstract In this investigation, we detect and utilize critical points of functions with hexagonal symmetry in order to study their dynamics. The asymmetric unit in a parallelogram lattice is chosen as the initial searching region for a critical point set in a dynamic plane. The accelerated direct search algorithm is used within the parallelogram lattice to search for the critical points. Parameter space is separated into regions (chaotic, periodic or mixed) by the Ljapunov exponents of the critical points. Then the generalized Mandelbrot set (M-set), which is a cross-section of the parameter space, is constructed. Many chaotic attractors and filled-in Julia sets can be generated by using parameters from this kind of M-sets.