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In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$ ( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

The Lotka-Volterra model plays an important role in the research area of population biology. This work presents the analysis of dynamical behaviours of a kind of three species GLV systemfrom the viewpoint of fractals. The definition of Julia set which describes the initial distribution of the three species’ densities is introduced. Then a gradient control method which contains both giant parameter and state feedback is applied to realize the control of Julia set. Coupling terms are designed to realize the synchronization of two Julia sets. Numerical examples are included to verify the conclusions

SIRS model is one of the most basic models in the dynamic warehouse model of infectious diseases, which describes the temporary immunity after cure. The discrete SIRS models with the Caputo deltas sense and the theories of fractional calculus and fractal theory provide a reasonable and sensible perspective of studying infectious disease phenomenon. After discussing the fixed point of the fractional order system, controllers of Julia sets are designed by utilizing fixed point, which are introduced as a whole and a part in the models. Then, two totally different coupled controllers are introduced to achieve the synchronization of Julia sets of the discrete fractional order systems with different parameters but with the same structure. And new proofs about the synchronization of Julia sets are given. The complexity and irregularity of Julia sets can be seen from the figures, and the correctness of the theoretical analysis is exhibited by the simulation results.

In this article, the fractal behavior in generalized alternated system is discussed. Take the classical fractal set as an example, the anti-synchronization of the Julia set between two different generalized alternated systems is discussed using the adaptive control method in this article. The anti-synchronization of the Julia set between different generalized alternated systems is also accomplished by synchronizing their iterative tracks. The simulations illustrate the effectiveness and correctness of this control method.

The purpose of this paper is to introduce the Mandelbrot and Julia sets by using Picard-Mann iteration procedure. Escape criteria is established which plays an important role to generate Mandelbrot and Julia sets. Also, numerous graphical pictures of these sets have been visualized and certain examples have been recognized. Presented results shows that fractal images generated by Picard-Mann iteration procedure are entirely different from those generated in Mann orbit.

The aim of this paper is to introduce a topological model of fractals. Self similar fractals will be approached as inverse limit of finite one dimensional topological spaces with alpha continuous bonding functions. The second approach is to investigate topological graphs in terms nano topological spaces for Lellis Thivagar. From these approximations, the dynamics of Julia sets as a special type of self similar fractals will be studied and some physical properties of fractals through their nano topological graphs will be applied.

This research was conducted to gain some patterns of fractals Julia set and Seirpinski that applied on Batik then creating Batik that has many varied motives and multifaceted. There are three steps in formulating the patterns of fractals of Julia set and Seirpinski. First, build the fractals by analyzing the function of fractals Julia set and determine the plane’s coordinate which you want to use. In this case, we use square and rectangle which will be created by using fractals patterns Seirpinski. Second, create a batik motives from fractals pattern Julia set and Seirpinski by using geometry transformation. The geometry transformation which will be used are translation, dilatation, reflection, and rotation. The last, combine some batik motives which were created by using image processing. It was summation of two images processing. The result is batik motives that has many variated and multifaceted.

The article describes an algorithm for generating fractals using polar coordinates. The classic Julia and Mandelbrot polynomial iteration applied to a complex number is replaced by an iteration with separate functions for distance and angle. A polynomial function is used for an angle and a power function for a distance. Varying the functions parameters allows to create a wide range of attractive pictures. Distance values ​​are used for coloring fractal images.

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

The aim of this paper is to modify the Jungck-S iterative scheme by adding the idea of $s$ -convexity. We define and analyze the modified Jungck-S orbit (MJSO) with $s$ -convex combination and derive the escape criterion for MJSO. Moreover, we establish the algorithms to visualize some Julia sets, Mandelbrot sets, and biomorphs in this orbit. In the biomorph generation algorithm, we did not fix the threshold radius of proposed orbit (i.e., MJSO) as fixed in literature earlier. We also discuss the graphical behavior of some complex polynomials in the generation of Julia sets, Mandelbrot sets, and biomorphs in MJSO.

The visual beauty, self-similarity, and complexity of Mandelbrot sets and Julia sets have made an attractive field of research. One can find many generalizations of these sets in the literature. One such generalization is the use of results from fixed-point theory. The aim of this paper is to provide escape criterion and generate fractals (Julia sets and Mandelbrot sets) via CR iteration scheme with s-convexity. Many graphics of Mandelbrot sets and Julia sets of the proposed three-step iterative process with s-convexity are presented. We think that the results of this paper can inspire those who are interested in generating automatically aesthetic patterns.

This paper introduces an effective image encryption method based on fractals for colored images. Fractals are now widely used in image security due to its randomness property and infinite boundaries. Since images are traversed over the public channel in day-to-day life confidentiality, integrity and authenticity must be pledged for prosperous transmission. This method capsulizes a larger key space and complex behavior to accomplish a productive confusion and diffusion techniques in a fruitful way. Julia set and Mandelbrot set are constructed and active confusion is exerted till the image pixels are altogether disordered and ensuing diffusion with XOR operation. Experimental results show the ability of the proposed system with respect to various statistical measures.

In today's world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of s -convex combination. The Jungck-CR iteration process with s-convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria.

This chapter describes how to use intersection and closest-hit shaders to implement real-time visualizations of complex fractals using distance functions. The Mandelbulb and Julia sets are used as examples.

The fractional Potts model on diamond-like hierarchical lattices is introduced in this manuscript, which is a fractional rational system in the complex plane. Then, the fractal dynamics of this model is discussed from the fractal viewpoint. Julia set of the fractional Potts model is given, and control items of this fractional model are designed to control the Julia set. To associate two different Julia sets of the fractional model with different parameters and fractional orders, nonlinear coupling items are taken to make one Julia set change to another. The simulations are provided to illustrate the efficacy of these methods.

The work belongs to the direction of experimental mathematics, which investigates the properties of mathematical objects by the computing facilities of a computer. The base is an exponential map, its topological properties (Cantor's bouquets) differ from properties of polynomial and rational complex-valued functions. The subject of the study are the character and features of the Fatou and Julia sets, as well as the equilibrium points and orbits of the zero of three iterated complex-valued mappings: $f:z \to (1+ \mu) \exp (iz)$, $g : z \to \big(1+ \mu |z - z^*|\big) \exp (iz)$, $h : z \to \big(1+ \mu (z - z^* )\big) \exp (iz)$, with $z,\mu \in \mathbb{C}$, $z^* : \exp (iz^*) = z^*$. For a quasilinear map g having no analyticity characteristic, two bifurcation transitions were discovered: the creation of a new equilibrium point (for which the critical value of the linear parameter was found and the bifurcation consists of "fork" type and "saddle"-node transition) and the transition to the radical transformation of the Fatou set. A nontrivial character of convergence to a fixed point is revealed, which is associated with the appearance of "valleys" on the graph of convergence rates. For two other maps, the monoperiodicity of regimes is significant, the phenomenon of "period doubling" is noted (in one case along the path $39\to 3$, in the other along the path $17\to 2$), and the coincidence of the period multiplicity and the number of sleeves of the Julia spiral in a neighborhood of a fixed point is found. A rich illustrative material, numerical results of experiments and summary tables reflecting the parametric dependence of maps are given. Some questions are formulated in the paper for further research using traditional mathematics methods.

In memory of Sibe Mardesic, our friend. Sibe Mardesic has enriched algebraic topology developing shape and strong shape theories with important constructions and theorems. This paper relates computational topology to shape theory. We have developed some algorithms and implementations that under some conditions give a shape resolution of some Julia sets. When a semi-flow is induced by a rational map g of degree d defined on the Riemann sphere, one has the associated Julia set J ( g ) . The main objective of this paper is to give a computational procedure to study the shape of the compact metric space J ( g ) . Our main contribution is to provide an inverse system of cubic complexes approaching J ( g ) by using implemented algorithms based in the notion of spherical multiplier. This inverse system of cubical complexes is used to: (i) obtain nice global visualizations of the fractal structure of the Julia set J ( g ) ; (ii) determine the shape of the compact metric space J ( g ) . These techniques also give the possibility of applying overlay theory (introduced by R. Fox and developed among others by S. Mardesic) to study the symmetry properties of the fractal geometry of the Julia set J ( g ) .

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen’s formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite contracting iterated function systems. Similarly, we show that Bowen’s formula holds for the limit set of a contracting conformal iterated function system without the cone condition.

Mandelbrot and Julia sets are examples of fractal patterns generated in the complex plane. In the literature we can find many generalizations of those sets. One of such generalizations is the use of switching process. In this paper we introduce some switching processes to another type of complex fractals, namely polynomiographs. Polynomiograph is an image presenting the visualization of the complex polynomial's root finding process. The proposed switching processes will be divided into four groups, i.e., switching of: the root finding methods, the iterations, the polynomials and the convergence tests. All the proposed switching processes change the dynamics of the root finding process and allowed us to obtain new and diverse fractal patterns.

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

We present eigenvalue data and pictures of eigenfunctions of the classic and quadratic snowflake fractal and of quadratic filled Julia sets. Furthermore, we approximate the area and box-counting dimension of selected Julia sets to compare the eigenvalue counting function with the Weyl term., 29 pages, 143 figures

It is of crucial significance to study infectious disease phenomenon by using the discrete SIRS model with the Caputo deltas sense and fractal viewpoint. In this paper, Julia set of the discrete fractional SIRS model is established to analyze the fractal dynamics of this model. Then three different controllers, which are, respectively, added to different parts of the model as a whole, a part, and a product factor, are designed to change the Julia set, and the graphs illustrate the complexity of the model. Simulation results show the efficacy of these methods.

A new method of fractals construction based on Fatou-Julia iteration is proposed. We develop a non-ordinary way to map fractal into a new fractal, where the mapping function is involved in the dynamics of the generated formula such that the modification does not violate the Julia and Fatou recipe for creating fractals. The method is applied for both Mandelbrot and Julia sets, and we follow the same technique of determining the fractal set using in the original iteration. The results of this paper are expected to have beneficial effects on many fields such as computer graphics and 3D printing technologies. Ministry of Education, Libya

Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the M&ouml, bius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented.

In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping $z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. We demonstrate with several examples that this technique successfully reveals the rich and intricate dynamical features of Julia sets and their fractal structure., Comment: 12 pages, 7 figures. Submitted

This research aims to: (1) study and explain the definition of the Julia set, (2) study and explain the characteristics of the Julia set, (3) construct and visualize the Julia set in computer. This research is a qualitative descriptive research in the form of literature study. The method used in this research is examine various scientific literatures such as books and scientific journals about the definition, characteristics, and the method of constructing and visualizing Julia's set. The main reference of this research is the book Fractal Geometry Mathematical Foundations and Applications edition by Kenneth Falconer (2003). The results of this research are as follows: (1) the Julia set is the boundary of filled- in Julia sets, which is the set of points where in every neighbourhood of there are points and with and ; (2) there are two characteristics of Julia set based on parameters , namely Julia set in a totally disconnected curve and Julia set in a simple closed curve ; (3) the steps in constructing the Julia set are to take the function of the Julia set, choose the value, determine the number of iterations to be performed, do the calculations according to the iteration that has been determined, then visualize. In visualizing the Julia set, it is needed computer application assistance, one of which is Matlab, so that the results for the Julia set can be modeled in an interesting form. Keywords: Complex Quadratic Functions; Julia Set; Iteration

Since the introduction of complex fractals by Mandelbrot they gained much attention by the researchers. One of the most studied complex fractals are Mandelbrot and Julia sets. In the literature one can find many generalizations of those sets. One of such generalizations is the use of the results from fixed point theory. In this paper we introduce in the generation process of Mandelbrot and Julia sets a combination of the S -iteration, known from the fixed point theory, and the s -convex combination. We derive the escape criteria needed in the generation process of those fractals and present some graphical examples.

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness) which characterizes the standard Julia sets. Via computer graphics, we unveil these connectivity domains which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected., Comment: 7 figures

In this paper, we propose two new methods to realize drive-response system synchronization control and parameter identification for two kinds of sine-function Julia sets. By means of these two methods, the zero asymptotic sliding variables and the stability theory in difference equations are applied to control the fractal identification. Furthermore, the problem of synchronization control is solved in the case of a drive system with unknown parameters, where the unknown parameters of the drive system can be identified in the asymptotic synchronization process. The results of simulation examples demonstrate the effectiveness of the new methods.

We studied the creation of fractal images with polygonal rotation symmetry. As in Loocke's method[13] we start with IFS of affine functions that create polygonal fractals and extends the IFS by adding functions that create Julia sets instead of adding square root functions. The resulting images are rotationally symmetric and Julia set shaped. Also we can improve fractal images by modifying probabilistic IFS algorithm, and we suggest a method of deforming Julia set by changing exponent value.Keywords : Fractal, Rotational symmetry, Iterated function systems, Julia set Received: Nov. 24, 2014 Accepted: Dec. 10, 2014 Corresponding Author: Yeong-Deok Han (Woosuk University)E-mail: ydhan@woosuk.ac.krISSN: 1598-4540 / eISSN: 2287-8211Ⓒ The Korea Game Society. All rights reserved. This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.otg/licenses/by-nc/3.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The intent of this paper is to study the fractal patterns of one dimensional complex logistic map by finding the optimum values of the control parameter using Ishikawa iterative scheme. The logistic map is shown to have bounded and stable behaviour for larger values of the control parameter. This is well depicted via time series analysis and interesting fractal patterns as well are presented

The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting field of research nowadays. We introduce in this paper the dynamics of polynomials z n z + c = 0 for n 2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. In order to solve this function by Jungck –type iterative schemes, we write it in the form of Sz = Tz, where the function T, S are defined as Tz = z n + c and Sz = z. Only mathematical explanations are derived by applying Jungck Ishikawa Iteration for polynomials in the literature but in this paper we have generated Relative Mandelbrot sets and Relative Julia sets.

Complex graphics of dynamical system have been a subject of intense research nowadays. The fractal geometry is the base of these beautiful graphical images. Many researchers and authors have worked to study the complex nature of the two most popular sets in fractal geometry, the Julia set and the Mandelbrot set, and proposed their work in various forms using existing tools and techniques. The generation of fractals and study of the dynamics of transcendental function is one of the emerging and interesting fields of research nowadays. Recently, Ashish Negi, Rajeshri Rana and Yashwant S. Chauhan are among those researchers who have contributed a lot in the area of Fractal Geometry applications. In this paper we review the recently done work on sine and inverse tangent functions.

Today Fractal geometry is completely new area of research in the field of computer science and engineering. It has wide range of applications. Fractals in nature are so complicated and irregular that it is hopeless to model them by simply using classical geometry objects. Benoit Mandelbrot, the father of fractal geometry, from his book The Fractal Geometry of Nature, 1982. This paper present construction of famous fractal images- Mandelbrot set and Julia set using 3D iterated function system which gives real look and feel of complex natural fractal images.