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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

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincare plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

It is known that Julia sets, as a rule, have a fractal structure. In this paper, we give examples of smooth Julia sets, among them: a circle, a segment, an infinite interval, a straight line, and the complex plane. It is shown that the functions studied in the paper are chaotic on their Julia sets. The results obtained by analytical research are visualized using computer programs. The algorithms for constructing the Julia sets being considered are indicated.

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.

This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system’s Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

Julia set is one of the most important branches in fractal theory, and has achieved much progress on both theoretical analysis and application research. Nevertheless, most of the existing investigations relied on the systems with known parameters. Few works have been done on the inverse problem about how to determine the system parameter based on a given Julia set’s shape. This work aims to address this issue by starting with the most classical polynomial map. First, by constructing a proper fitness function measuring the Julia sets’ area error, the Julia sets’ parameter estimation is formulated as a kind of optimization problem. According to the known conditions of Julia set to be estimated, the optimization problem is classified into two cases. For one case, when both the shape and size of Julia set are given, we only need to estimate the system’s own parameter [Formula: see text]. For the other case, when the Julia set has only a shape, we redesign the problem into three dimensions by adding a new scaling factor parameter [Formula: see text] and applying the partial coverage principle. Second, a particle swarm optimization (PSO) approach including dynamically adjustment strategy is adopted to solve the proposed optimization problem. At last, we present seven groups of simulations in which both randomly generated Julia sets and those in existing literature (or on the internet) are included. The simulation results verify the effectiveness of the proposed parameter estimation scheme.

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.

This paper reports the investigation of a fractional Lotka–Volterra model from the fractal viewpoint. The fractal characteristic of the Julia set is discussed after introducing the discrete version of this model. Then, control of the model’s Julia set is studied via two different control strategies. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

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 paper reports the investigation of a fractional Lotka–Volterra model from the fractal viewpoint. A Julia set of a discrete version of this model is introduced and its state feedback control is realized. Coupled terms are designed to realize the synchronization of two Julia sets with different parameters. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

In the article, the formation of the creative qualities of students is analyzed using the example of the Mandelbrot set and the Julia sets. The connection between the points of the Mandelbrot set and the Julia sets for the degree of complex argument 2 is considered. A program that demonstrates this link is developed. An algorithm for constructing the Mandelbrot set of an arbitrary natural degree is given.

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.

Special cases of otherwise intuitive mathematical theories often seem to exhibit a “pathological behavior” that requires a technically difficult special treatment. The argument and the illustrations in the present paper relate “Siegel discs” to Peano curves, hence ought to show that the Seigel disc behavior is in fact eminently reasonable. The key to the argument resides in the concept of Peano “plane-filling curve”, which the fractal geometry of nature retrieved from among the monster sets and showed to have great concrete importance. The point of the present paper is that the same “Peano” behavior is characteristic of certain sequences of Julia sets.

This study updates a novel technique for evolving parameters that specify fractal images. Example parameter sets are provided as an information resource to evolution, following an earlier study. Instead of choosing parameters with high average compatibility with all other parameters, this study clusters the parameters using a graph clustering algorithm within a network where the adjacency relation of the network is derived from co-fertility, i.e. genetic compatibility values. The result of using the new types of sets of parameters as information resources is studied and compared to evolution that uses the previous type of information resource. The new technique of selecting information resources presented here yields higher fitness values. The new results are on the high end of the fitness distribution, and so the new information resources tested give similar improvements in fitness. However, their variability vary substantially and the resulting fractals have different appearances.

In recent times, charismatic nature of Fractals and the study of their aspects are fascinating and thoughtprovoking concepts of research in the field of mathematics and without the notions of Mandelbrot and Julia sets we can’t image the fractal theory. This paper examines the various newborn M-sets (Mandelbrot sets) for the complex general polynomial $z^{n}+C$ for $n\geq 2$ in GK-orbit. The main focus of this paper is to generate new Mandelbrot sets for above said polynomial and Figure out their mathematical tendencies.

Recent days, Fractals, superfractals and examination of their dynamic nature are interesting and important notions of research in the field of Mathematics. In Fractal theory, Julia sets and Mandelbrot sets are two remarkable concepts for the researchers with many practical applications. This paper explores the various newborn J-sets (Julia sets) for the complex polynomial, $z^{n}+c, n\geq 2$ in GK-orbit. The main centre of point of this research paper is to originate new J-sets (Julia sets) with the understanding of their mathematical characteristics.

The fractal concept is the generation of images graphically with self-similarity properties produced by recursive or iterative algorithms to produce a new image form. Fractal structure is a common tool for describing the visual effects of one or more objects. In this research, we will design a system that can design batik fractal motifs using the Julia set. Therefore, a programmable tool to generate computerized batik motifs is needed. In this research, Julia set was used as the basis for generating batik motifs. In this system, the process of generating fractal batik images consists of 3 (three) steps: (1) determining the shape of the Julia set function for some batik motifs; (2) visualizing the results of the first step using a Pythonbased program; and (3) designing the fractal batik motifs using Julia set obtained in the second step. From experiments conducted in generating batik fractal, some conclusions were obtained as follows. First, for the greater number of iterations, the resulting image will be more detailed and the number of colors will increase. Second, the result image is influenced by the value of the complex c and the number of colors or gray-levels indicated by the number of iterations, i.e., the greater the number of iterations, the more colors are generated. Third, several variations of input parameters produce images that approach traditional batik motifs. In this research, traditional motifs that can be approached by Julia-set functions are: Parangkusumo, Nitik, Batik Liong, Mega Mendung, and Ceplok motifs. Fourth, changes in the number of iterations and the value of c applied to Julia set are better able to generate the variations of motifs. Thus, the Julia set function can help the batik designer in making fractal batik motifs.

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 spatial Julia set, which is the natural extension of planar Julia set, has also attracted significant interest in recent years.

Recently, the properties and applications of Julia sets from more general complex maps were investigated. Saitoh et al. [210] discussed a complex system extended from Logistic map, and got the condition under which the Julia set is Cantor set. Fornaess [211] studied the complex Henon map and pointed out that the complex Henon map’s maximum entropy set equals to its Julia set. By extending the complex map from integral expression into rational expression, Blanchard et al. [212, 213] investigated the fractal behavior of the following system