Mandelbrot Set Research Articles

Application of CR Iteration Scheme in the Generation of Mandelbrot Sets of zp+logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + \\log c^t$$\\end{document} Function

Mandelbrot set is one of the most fascinating objects in mathematics, so in the literature, one can find many studies related to this set. Moreover, there are many different extensions and generalizations of the classical Mandelbrot set. The two most popular extensions are the use of various kinds of functions and iterations. In this paper, firstly, we replace the constant c in the classical zp+c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + c$$\\end{document} function used in the definition of Mandelbrot set with logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log c^t$$\\end{document}, where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} and t≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\ge 1$$\\end{document}. Secondly, we replace the Picard iteration with the CR iteration scheme. For this combination of function and iteration, we prove the escape criterion for the escape-time algorithm used to generate some images of the proposed sets. Moreover, we study the dependency between the iteration’s parameters and two numerical measures (average number of iterations and generation time). We show that this dependency is complex and non-linear.

Exploring the Complex Dynamics of Julia and Mandelbrot Sets for the Complex-valued Mapping sin(z<sup>k</sup>) + az + c Using Four-Step Iterative Scheme with s-Convexity

In this paper, we investigate unique variations of the Julia and Mandelbrot sets, establishing criteria for the escape of a function Tcz = sin(zk) + az + c for all z ∈ C, where k ∈ N \ {1} and a, c ∈ C with a̸ = 0 by utilizing a four-step iterative scheme extended with s-convexity, we analyze the dynamics of these fractals by formulating and implementing the algorithms. Our exploration focuses on understanding how distinct parameters impact these fractals’ color, dynamics, and overall visual characteristics. Our findings reveal that specific fractal patterns resemble exquisite natural objects like flowers and spiders. These fractal designs are also employed as visually striking artistic motifs on garments and textile materials. The captivating and complex properties of fractal patterns within dynamic systems make them an enticing and visually appealing area of research.

Determining optimal parameters of algebraic fractals in zero-knowledge authentication protocols

Most information systems, especially on the Internet, have a distributed architecture with remote access and an insecure communication channel. In such systems, the tasks of permanent authorization mean implementing time intervals of user work without re-authentication are especially actual. The problem is that repeatedly sending a password increases the likelihood of it corruption. One solution is to use zero-knowledge protocols. In these protocols, passwords are not transmitted over the channel but are included in the algorithms as parameters. However, the computational complexity, as well as the finite number of passwords, limit their use, ensuring the relevance of further research. Focusing on object of the exchange protocols security, the use of algebraic fractal sets has been proposed as a potentially infinite source of data for passwords. In this work, algorithms were developed, implemented, and tested, which proved the higher reliability of fractal protocols in comparison with the reference generator of random bits (with an error probability of 0.5). It was also noted that the calculation operations have an insignificant influence on the overall time complexity of the exchange protocol as a whole. Practical recommendations for the use of fractals with a Hausdorff dimensionally of about 1.6 on the boundary of the Mandelbrot set are given. The paper also highlights the advantages of including color information in fractal sets, which gives about 3 times improving of confidential security indicators of communication protocol. The proposed algorithms do not require specialized software and can be implemented in the majority of network information systems as an additional module.

The Natural Pattern-based Fuzzy Investment System

Research background: The underlying research problem concerns simplifying nature’s infinite sustainability pattern into a usable system to prove self-care in the financial field. Many disciplines have discovered self-sustaining systems, implying one should also exist in the portfolio- or asset management field. Purpose of the article: This paper models and applies a self-sustaining investment system based on nature’s sustainability pattern to present a usable and easily adaptable financial self-care model that can contribute to societal and individual well-being. Methods: This interdisciplinary research incorporates fractal patterns from nature and integrates these into an innovative investment decision system via a working and utilizable model. The article approaches the problem in three ways. First, it applies an unprecedented, proprietary fuzzy heuristic approach to parse the Mandelbrot set hiding nature’s growth code. Second, it subjects the resulting model system to static, dynamic, and iterative methods. Finally, it tests the above in practice in a focus group research project based on individual decisions in a portfolio collision. Findings & value added: The paper brings manageable order to investment decision processes using a specific econophysics approach to provide a complex whole of frames, alternatives, and dynamic wealth management functions. The paper attempts to demonstrate the self-sustaining power of the natural order in investment portfolio returns, behavioral finance, and wealth management decisions. The theory of an efficient, well-functioning, self-sustaining investment decision system based on heuristic criteria is developable and has been proven. The study made the organizational dominance measurable and – through fuzzy logic – simplified the complexity of the Mandelbrot set. The research also shows the scalability of the fuzzy symmetry framework, implying that it is transferable to other disciplines.

Generation of Mandelbrot and Julia sets by using M-iteration process

In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form xk+1+c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard–Mann iterations. Moreover, we study the dependency between the iterations’ parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations.

From ambiguous sets to single-valued ambiguous complex numbers: Applications in Mandelbrot set generation and vector directions

From ambiguous sets to single-valued ambiguous complex numbers: Applications in Mandelbrot set generation and vector directions

Fractals as Julia and Mandelbrot Sets via S-iteration

To understand the phenomena of expanding symmetries Fractals patterns are an important tool which exhibit similar patterns for different scales. In the present paper, establishing an escape criteria by using S-iteration process to visualize fractals namely Julia and Mandelbrot sets for the function F(w)=aewp+c where c,a∈ C and p≥2. The result obtain is a generalization of the existing algorithm and technique providing fractals for different parameter values. Also, the time taken to obtain fractals for different parameters by using computer software MATLAB is computed in seconds.

Performing a multi-stage mathematical information task "Framing the Mandelbrot set of families of polynomials of the third degree and remarkable curves"

In this article, within the framework of a multi-stage mathematical information task, the methodology for students to study Mandelbrot sets and frames of Mandelbrot sets of a family of polynomials of the third degree is indicated. Algorithms for constructing these sets in various environments are described. The connections of the frames of the Mandelbrot set with remarkable curves are studied: lemniscate, epicycloid and others. This work is aimed at developing students' creativity and research competencies. When performing a multi-stage mathematical and informational task, the student acts as a mathematician, programmer and computer artist, which is aimed at developing his creativity and increasing motivation for mathematics and programming.

From hyperbolic to parabolic parameters along internal rays

For the quadratic family f c ( z ) = z 2 + c f_{c}(z) = z^2+c with c c in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter c c converges to a parabolic parameter c ^ {\hat {c}} radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on c ^ {\hat {c}} . We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at c ^ {\hat {c}} with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].

Mandelbrot set for fractal n-gons and zeros of power series

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset G⊂C. We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set G of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set Mn for fractal n-gons. We prove that Mn is connected and locally connected for any n.

Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form QC(p)=apn+mp+c, where n≥2. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.

The Julia and Mandelbrot sets for the function [formula omitted] exhibit Mann and Picard–Mann orbits along with [formula omitted]-convexity

This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form F(z)=zp−qz2+rz+sincw for all z∈ℂ, where p∈N∖{1}, q∈ℂ, r,c∈ℂ∖{0} and w∈[1,∞). The Mann and Picard–Mann iteration schemes with s-convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal’s shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations.

Archetypal Resonances Between Realms: The Fractal Interplay of Chaos and Order

The intricate balance between chaos and order has fascinated humans for centuries, serving as a recurring theme in culture, psychology, and science. Carl Jung’s insights into archetypes reveal cultural patterns that emerge nearly universally across the globe, independent of history. This paper connects archetypal psychology with contemporary mathematical and quantum concepts. We focus on profound interconnectedness at the interface between chaos and order, by demonstrating how the Mandelbrot set from contemporary fractal geometry circles round to embody the Ouroboros, ancient symbol of recursive dynamics. Our research bridges micro and macro realms, symbolic meanings, and empirical data, as well as psychological and physical dimensions. By merging age-old symbols with cutting-edge mathematical and quantum ideas, we shed light on the intricate dance of chaos and to underscore the unified nature of existence.

Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration

Fractals are a common characteristic of many artificial and natural networks having topological patterns of a self-similar nature. For example, the Mandelbrot set has been investigated and extended in several ways since it was first introduced, whereas some authors characterized it using various complex functions or polynomials, others generalized it using iterations from fixed-point theory. In this paper, we generate Mandelbrot sets using the hybrid Picard S-iterations. Therefore, an escape criterion involving complex functions is proved and used to provide numerical and graphical examples. We produce a wide range of intriguing fractal patterns with the suggested method, and we compare our findings with the classical S-iteration. It became evident that the newly proposed iteration method produces novel images that are more spontaneous and fascinating than those produced by the S-iteration. Therefore, the generated sets behave differently based on the parameters involved in different iteration schemes.

A Method and Algorithm for Implementing the Mandelbrot Set for Processing Complex Structured Images

The purpose of research. In our life, fractals are ubiquitous, for example, in nature, there are a huge number of figures similar to themselves and built according to certain laws. Fractals are used in computer graphics, economics, radio engineering, physics and many other fields. The use of fractals in various fields of science can open up many new possibilities. Therefore, the development of a software product designed to visualize the Mandelbrot set is an urgent task. The purpose of the research is to develop a software product designed for processing complex structured images based on fractal methods and algorithms for visualizing the Mandelbrot set.Methods. The Java programming language was used to implement the software product in order to visualize the Mandelbrot set. The SQLite database was used as a local data store. Libraries were also used: JavaFX ‒ to create a user interface; SQLite JDBC 3.21.0 ‒ to work with the SQLite database; imgscalr ‒ to scale images; Apache Commons IO 2.6 to work with the file system; ControlsFX v8.40.14 ‒ to create a user interface.Results. The developed software product based on fractal methods makes it possible to significantly highlight the textural features of images, on the basis of which the clustering operation of selected objects of interest in images is performed in order to export them for further classification.Conclusion. During the implementation of the project, a software product for visualization of the Mandelbrot set was implemented, which can be used to model objects having a fractal shape, in relation to the processing of complexly structured images containing invisible hidden features of objects in the images.

Exploring the Julia and Mandelbrot sets of [formula omitted] using a four-step iteration scheme extended with [formula omitted]-convexity

Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form Q(z)=zp+logct for all z∈ℂ, where p∈N∖{1},t∈[1,∞),c∈ℂ∖{0}, using a four-step iteration scheme extended with s-convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments.

Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order

This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined.

Fractional Mandelbrot sets with impulse

This paper proposes fractional Mandelbrot sets with impulses based on a fractional difference map, and preliminarily investigates their fractal dynamic characteristics by numerical methods and graphical explorations. The influence of impulse and parameters on the novel fractal sets is performed by dimension analysis. It confirms that the symmetry of the Mandelbrot set can be preserved under piecewise linear random impulse. Several extensive numerical experiments are done to support the main conclusions.

Generation of Mandelbrot and Julia sets for generalized rational maps using SP-iteration process equipped with [formula omitted]-convexity

In this paper, we introduce a generalized rational map to develop a theory of escape criterion via the SP-iteration process equipped with s-convexity. Furthermore, we develop algorithms for the exploration of unique kinds of Mandelbrot as well as Julia sets. We demonstrate graphically the change in colour, size, and shape of images with the change in values of the considered iteration’s parameters. The new fractals thus obtained are visually very pleasing and attractive. Most of these newly generated fractals resemble natural objects around us. Moreover, we numerically study the dependence between the iteration’s parameters and the set size. The experiments show that this dependency is non-linear. We believe that the obtained conclusions will motivate researchers who are interested in fractal geometry.

Archetypal Resonances Between Realms: The Fractal Interplay of Chaos and Order

The intricate balance between chaos and order has fascinated humans for centuries, serving as a recurring theme in culture, psychology, and science. Carl Jung’s insights into archetypes reveal cultural patterns that emerge nearly universally across the globe, independent of history. This paper connects archetypal psychology with contemporary mathematical and quantum concepts. We focus on the profound interconnectedness at the interface between chaos and order, by demonstrating how the Mandelbrot set from contemporary fractal geometry circles round to embody the Oroboros, an ancient symbol of recursive dynamics. Our research bridges micro and macro realms, symbolic meanings, and empirical data, as well as psychological and physical dimensions. By merging age-old symbols with cutting-edge mathematical and quantum ideas, we shed light on the intricate dance of chaos and to underscore the unified nature of existence.