Analysis of Julia sets using viscosity approximation-based iteration processes

In today&#x2019;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 <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-convex combination. The Jungck-CR iteration process with <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-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.

In this paper, we propose a method for generating Julia sets and Mandelbrot sets by a three-step iteration process in which we use s-convex combination and a complex function ζ c (z) = a z p + b z q + log c t . The Julia sets are obtained as a collection of points determined through the escape criterion of the sequence established herein. It is the central concept in the study undertaken in this work. Mandelbrot sets are correspondingly determined in the parameter spaces. Visual patterns are generated for both the Julia and Mandelbrot sets for certain choices of parameters involved in the iteration. The changes in these patterns with the variations of the iteration parameters in certain ranges are qualitatively studied. Numerical experiments are performed using the MATLAB software. The average number of iterations (ANI) and the execution time in the course of the numerical experiments for the above two generation processes are noted and analyzed. In both cases, these two measures of the generation processes are observed to be in qualitative agreement.

Fixed point iterative procedures are the backbones of fractal geometry. In existing literature Julia sets, Mandelbrot sets and their variants have been studied using one - step, two - step, three - step and four - step iterative process. Recently, M. Abbas and T. Nazir (12) introduced a new iterative process (a four-step iterative process) which is faster than all of Picard, Mann and Agarwal processes. In this paper, we obtain further generalizations of Julia and Mandelbrot sets using this faster iterative process for quadratic, cubic and higher degree polynomials. Further, we analyze that few Julia and Mandelbrot sets took the shape of Lord Ganesha (name of Hindu God), Dragon and Urn.

A new technique for generating approximations of Julia sets, called reverse iteration, is proposed. The concept of reverse iteration is simple and based on the idea of solving an iterative map of the general form Z k =G(Z k-1 ,p) for Z k-1 instead of Z k . It is shown that if the process of reverse iteration is initiated at any singular point, the collection of inverse images must be members of the Julia set since singular points are in the Julia set and the Julia set is closed. This sequence of reverse iterates, say {Z k-1 }, is necessarily distributed throughout the basin boundaries. It is also shown that reverse iteration can have multiple inverse images and a tree structure for the Julia set but that the associated potential combinatorial computational demand is easily resolved by exploiting the fractal nature of any Julia set. From this, practical ways generating initial values that converge to solutions to the given model equations are proposed. Several examples and geometric illustrations are used to elucidate key concepts.

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

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

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T ( x ) = x n + m x + r where m , r ∈ C and n ≥ 2 . Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals.

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)st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

Nowadays, many researchers are employing various iterative techniques to analyse the dynamics of fractal patterns. In this paper, we explore the formation of Mandelbrot and Julia sets using the Picard-Thakur iteration process, extended with s-convexity. To achieve this, we establish an escape criterion using a complex polynomial of the form [Formula: see text], where k≥1 and x,c∈ℂ. Based on our proposed algorithms, we provide graphical illustrations of the Mandelbrot and Julia sets. Additionally, we extend our research to examine the relationship between the sizes of Mandelbrot and Julia sets and the iteration parameters, utilising some well-known methods from the literature.

&lt;abstract&gt;&lt;p&gt;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| &amp;gt; 1 $ and $ n, m \in \mathbb{N} $ with $ n &amp;gt; 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.&lt;/p&gt;&lt;/abstract&gt;

In this article, we study and explore novel variants of Julia set patterns that are linked to the complex exponential function $W(z)=pe^{z^n}+qz+r$, and complex cosine function $T(z)=\cos({z^n})+dz+c$, where $n\geq 2$ and $c,d,p,q,r\in \mathbb{C}$ by employing a generalized viscosity approximation type iterative method introduced by Nandal et al. (Iteration process for fixed point problems and zero of maximal monotone operators, Symmetry, 2019) to visualize these sets. We utilize a generalized viscosity approximation type iterative method to derive an escape criterion for visualizing Julia sets. This is achieved by generalizing the existing algorithms, which led to visualization of beautiful fractals as Julia sets. Additionally, we present graphical illustrations of Julia sets to demonstrate their dependence on the iteration parameters. Our study concludes with an analysis of variations in the images and the influence of parameters on the color and appearance of the fractal patterns. Finally, we observe intriguing behaviors of Julia sets with fixed input parameters and varying values of $n$ via proposed algorithms.

Escape time algorithm is a classical algorithm to calculate the Julia sets, but it has the disadvantage of dull color and cannot record the iterative process of the points. In this paper, we present the equipotential point algorithm to calculate the Julia sets by recording the strike frequency of the points in the iterative process. We calculate and analyze the Julia sets in the complex plane by using this algorithm. Finally, we discuss the iteration trajectory of a single point.

The visual beauty, self-similarity, and complexity of Mandelbrot sets and Julia sets have made&#13;\nan attractive eld of research. One can nd many generalizations of these sets in the literature. One such&#13;\ngeneralization is the use of results from xed-point theory. The aim of this paper is to provide escape&#13;\ncriterion and generate fractals (Julia sets and Mandelbrot sets) via CR iteration scheme with s-convexity.&#13;\nMany graphics of Mandelbrot sets and Julia sets of the proposed three-step iterative process with s-convexity&#13;\nare presented. We think that the results of this paper can inspire those who are interested in generating&#13;\nautomatically aesthetic patterns.

In this article, we study the use of the generalized viscosity approximation-type iterative methods in the generation of fractals as Julia and Mandelbrot sets for generalized rational map of the form + + sin(ε♭), where p ≥ 2, p, q , ϑ, φ , ε \{0} and [1, ∞). Utilizing the proposed iterative methods, we establish a novel escape criterion and implement it within the escape time algorithms to generate and visualize Julia and Mandelbrot sets. This criterion is essential for terminating the iterative process and is the key to producing the resulting captivating fractal patterns. Through graphical and numerical experiments, we analyze how the iteration parameters influence the shape, size and color of the fractals. This work aims to inspire the application of fractal patterns in textile design and printing.

In recent years, researchers have studied the use of different iteration processes from fixed point theory for the generation of complex fractals. Examples are the Mann, the Ishikawa, the Noor, the Jungck-Mann and the Jungck-Ishikawa iterations. In this paper, we present a generalisation of complex fractals, namely Mandelbrot, Julia and multicorn sets, using the Jungck-CR implicit iteration scheme. This type of iteration does not reduce to any of the other iterations previously used in the study of complex fractals; thus, this generalisation gives rise to new fractal forms. We prove a new escape criterion for a polynomial of the following form zm − az + c, where a, c ∈ ℂ, and present some graphical examples of the obtained complex fractals.