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.

Generation of Julia and Mandelbrot fractals for a generalized rational type mapping via viscosity approximation type iterative method extended with $ s $-convexity

<abstract><p>A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form $ S_{c}(z) = a z^{n}+\frac{b}{z^{m}}+c $, where $ a, b, c \in \mathbb{C} $ with $ |a| > 1 $ and $ n, m \in \mathbb{N} $ with $ n > 1 $. By applying viscosity approximation-type iteration processes extended with $ s $-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters.</p></abstract>

Emergent scalar symmetry in discrete dynamical systems

Discrete quadratic dynamical systems are frequently used to define models of reality. To understand these recursive systems is to know the behavior of their complex orbits. The Mandelbrot Set plays an important role in this: the behavior of the orbits of its points can be extrapolated to all quadratic polynomials. In calculating the map of periods for the orbits of the points belonging to the Mandelbrot Set by means of discrete (and finite) encoding, some singular points called symmetrical points appear on the map. Such points configure, in a surprisingly harmonic and regular manner, the values of the rest of points in the map of periods. Thus, knowing these points and their properties is extremely helpful to better understand the behavior of quadratic and discrete dynamical systems. Important properties of such points are described in this article. We emphasize the emergent scalar symmetry property, which appears as a consequence of the finiteness of discrete values with which we are inevitably limited to represent the continuum. Thanks to this property, the images created around these points are not altered when removing or selecting one of their several pixel rows and columns. The image can be sampled in its vicinity, at any scale, without losing information regarding the vicinity. We propose a justification on why the map of periods is highly hypersensitive to small changes in the parameters defining the calculation mode of the map.

Synchronization and Clustering in Complex Quadratic Networks.

Synchronization and clustering are well studied in the context of networks of oscillators, such as neuronal networks. However, this relationship is notoriously difficult to approach mathematically in natural, complex networks. Here, we aim to understand it in a canonical framework, using complex quadratic node dynamics, coupled in networks that we call complex quadratic networks (CQNs). We review previously defined extensions of the Mandelbrot and Julia sets for networks, focusing on the behavior of the node-wise projections of these sets and on describing the phenomena of node clustering and synchronization. One aspect of our work consists of exploring ties between a network's connectivity and its ensemble dynamics by identifying mechanisms that lead to clusters of nodes exhibiting identical or different Mandelbrot sets. Based on our preliminary analytical results (obtained primarily in two-dimensional networks), we propose that clustering is strongly determined by the network connectivity patterns, with the geometry of these clusters further controlled by the connection weights. Here, we first explore this relationship further, using examples of synthetic networks, increasing in size (from 3, to 4, to 20 nodes). We then illustrate the potential practical implications of synchronization in an existing set of whole brain, tractography-based networks obtained from 197 human subjects using diffusion tensor imaging. Understanding the similarities to how these concepts apply to CQNs contributes to our understanding of universal principles in dynamic networks and may help extend theoretical results to natural, complex systems.

Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration

The majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine Tc(z)=asin(zr)+bz+c and complex exponential Tc(z)=aezr+bz+c functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters ‘a’ and ‘b’, and the parameters involved in the series expansion of the sine and exponential functions.

First- and Second-Order Framings of Mandelbrot Sets and Structure of Fixed Points of Quadratic Polynomials

First- and Second-Order Framings of Mandelbrot Sets and Structure of Fixed Points of Quadratic Polynomials

On the viscosity approximation type iterative method and its non-linear behaviour in the generation of Mandelbrot and Julia sets

In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T(z)=zn+mz+r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(z) = z^{n} + mz + r$$\\end{document}, where n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} and m,r∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m, r \\in \\mathbb {C}$$\\end{document}, by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear.

On the quaternion Julia sets via Picard–Mann iteration

In recent years, an extensive study on the use of various iteration schemes from fixed point theory for the generation of Mandelbrot and Julia sets in complex space has been carried out. In this work, inspired by these progresses, we study the use of the Picard–Mann iteration scheme for the Julia sets in the quaternion space. Specifically, in our study, we prove the escape criterion of the Picard–Mann orbit and examine the symmetry of the Julia set for the quadratic function. Moreover, we present and discuss some 2D and 3D graphical examples of the sets generated using the Picard–Mann iteration scheme. We further analyse the influence of a parameter of interest used in the Picard–Mann iteration scheme on the average number of iterations for 2D cross sections of quaternion Julia sets of different degrees.

Formation of a nanostructured film at the “hydrocarbon — graphene nanofluid” interface

The interaction mechanisms between graphene nanoparticles (GN) and oil molecules are crucial for successful oil recovery. More than a hundred studies appear in the press each year showing the effectiveness of using nanofluids based on graphene-like nanoparticles to enhance oil recovery in various reservoirs. Increased oil recovery with nanofluid injection is explained by changes in wettability, reduction of interfacial tension and changes in viscosity. Therefore, knowledge of the interaction mechanisms between graphene nanoparticles and hydrocarbons is an urgent task of modern science, both fundamental and applied. A comprehensive study of the interaction of graphene nanoparticles and hydrocarbons was carried out in order to understand the mechanisms that affect the formation of microheterophase state at the interface of hydrocarbons and graphene nanofluids (GNF). Using the methods of X-ray analysis it was found that the structure of the microheterophase state is a nanocrystalline film. The paper presents the results of the film formation at the “hydrocarbon — graphene nanofluid” interface. It was found that both slow and fast growth of nanostructured films could be observed under different modes of heat sinking from the interface. At fast heat sinking a slow growth of the film with the formation of fractal structures of Mandelbrot set type is observed. With slow heat dissipation, rapid film growth is observed with the formation of a continuous homogeneous structure which is not a fractal.

The Boundedness Locus and Baby Mandelbrot Sets for Some Generalized McMullen Maps

In this paper, we study rational functions of the form [Formula: see text] with [Formula: see text] fixed and at least [Formula: see text], and hold either [Formula: see text] or [Formula: see text] fixed while the other varies. We locate some homeomorphic copies of the Mandelbrot set in the [Formula: see text]-parameter plane for certain ranges of [Formula: see text], as well as in the [Formula: see text]-plane for some [Formula: see text]-ranges. We use techniques first introduced by Douady and Hubbard [1985] that were applied for the subfamily [Formula: see text] by Devaney [2006]. These techniques involve polynomial-like maps of degree two.

Some Properties of the Mandelbrot Sets M(Q_α)

The aim of this work is to study the concept of the parameter plane, so we are interested in introducing the Mandebrot set,as well as the Mandelbrot set is a fractals ,whereas, if we zoom in on any small piece in the border regions of the shape, we notice that it is similar to the original shape, that is the Mandelbrot set, with special topological specifications, and we introduce of some properties of the Mandelbrot sets for the map in the form , where is a complex constant, and we prove that the conjecture is the Duadi-Hubbard theorem . By studying of the parameter plane that the Mandelbrot set is a component form in one of its details in the dynamical plane it is the Julia set , in other words every small detail of the Mandelbrot set represents the Julia set, so there is more than one characteristic of the Julia set, we will limit ourselves only to presenting the connection of the Julia set for .

Local Connectivity of the Mandelbrot Set at Some Satellite Parameters of Bounded Type

Local Connectivity of the Mandelbrot Set at Some Satellite Parameters of Bounded Type

Stability, Data Dependence, and Convergence Results with Computational Engendering of Fractals via Jungck–DK Iterative Scheme

We have developed a Jungck version of the DK iterative scheme called the Jungck–DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck–DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck–DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration.

Modeling GPU Dynamic Parallelism for self similar density workloads

Dynamic Parallelism (DP) is a GPU programming abstraction that can make parallel computation more efficient for problems that exhibit heterogeneous workloads. With DP, GPU threads can launch kernels with more threads, recursively, producing a subdivision effect where resources are focused on the regions that exhibit more parallel work. Doing an optimal subdivision process is not trivial, as the combination of different parameters play a relevant role in the final performance of DP. Also, the current programming abstraction of DP relies on kernel recursion, which has performance overhead. This work presents a new subdivision cost model for problems that exhibit self similar density (SSD) workloads, useful for finding efficient subdivision schemes. Also, a new subdivision implementation free of recursion overhead is presented, named Adaptive Serial Kernels (ASK). Using the Mandelbrot set as a case study, the cost model shows that optimal performance is achieved when using {g∼32,r∼2,B∼32} for the initial subdivision, recurrent subdivision and stopping size, respectively. Experimental results agree with the theoretical parameters, confirming the usability of the cost model. In terms of performance, the ASK approach runs up to ∼60% faster than DP in the Mandelbrot set, and up to 12× faster than a basic exhaustive implementation, whereas DP is up to 7.5× faster. In terms of energy efficiency, ASK is up to ∼2× and ∼20× more energy efficient than DP and the exhaustive approach, respectively. These results put the subdivision cost model and the ASK approach as useful tools for analyzing the potential improvement of subdivision based approaches and for developing more efficient GPU-based libraries or fine-tune specific codes in research teams.

Fractals: The Secret Code of Creation

Fractals: The Secret Code of Creation

Mandelbrot set and Julia sets of fractional order

In this paper, the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputo-like discrete fractional differences, for $$q\in (0,1)$$ , are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for $$q\uparrow 1$$ , contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for $$q\downarrow 0$$ . Also, we conjecture that for $$q\downarrow 0$$ , the fractional-order Mandelbrot set is similar to the integer-order Mandelbrot set, while for $$q\downarrow 0$$ and $$c=0$$ , one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.

On the Mandelbrot set of [formula omitted] via the Mann and Picard–Mann iterations

Since its introduction, the Mandelbrot set has been studied and generalized in various directions. Some authors generalized it by using iterations from fixed point theory, whereas others characterized it by using different complex functions or polynomials. In this paper, we replace the constant c in the classical zp+c function with logct, where t∈R and t≥1. Moreover, we prove escape criteria for the Mann and Picard–Mann iterations in which we use the modified function. Then, we present graphical and numerical examples showing the behaviour of the generated sets depending on the parameters of the iterations and the parameter t. Using the proposed approach, we can generate a great variety of fascinating fractal patterns, and when t∈N the sets form rosette patterns.

A Properties of Mandelbrot Set of Quartic Rational Maps

A Properties of Mandelbrot Set of Quartic Rational Maps

Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations

In this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine the alteration in generated graphical images and study the consequences of underlying parameters on the variation of dynamics, colour, time of generation, and shape of generated fractals. The black points in the obtained fractals are the “non-chaotic” points and the dynamical behaviour in the black area is easily predictable. The coloured points are the points that “escape”, that is, they tend to infinity under one of iterative methods based on a four-step fixed-point iteration scheme extended with s-convexity. The different colours tell us how quickly a point escapes. The order of escaping of coloured points is red, orange, yellow, green, blue, and violet, that is, the red point is the fastest to escape while the violet point is the slowest to escape. Mostly, these generated fractals have symmetry. The Julia set, where we find all of the chaotic behaviour for the dynamical system, marks the boundary between these two categories of behaviour points. The Mandelbrot set, which was originally observed in 1980 by Benoit Mandelbrot and is particularly important in dynamics, is the collection of all feasible Julia sets. It perfectly sums up the Julia sets.

On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.