Exploration of dynamics of Mandelbrot and Julia sets through a new four-step iteration algorithm

Julia sets and Mandelbrot sets in Noor orbit

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.

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.

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.

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.

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.

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a new approach to visualize Mandelbrot and Julia sets for complex polynomials of the form W(z)=zn+mz+r; n≥2 where m,r∈ℂ, and biomorphs for any complex function through a viscosity approximation method which is among the most widely used iterative methods for finding fixed points of non-linear operators. We derive novel escape criterion for generating Julia and Mandelbrot sets via proposed viscosity approximation method. Moreover, we visualize the sets using the escape time algorithm and the proposed iteration. Then, we discuss the shape change of the obtained sets depending on the parameters of the iteration using graphical and numerical experiments. The presented examples reveal that this change can be very complex, and we are able to obtain a great variety of shapes.

The Mandelbrot Set and Julia Sets are fractals that are generated employing the cyclic equation zn+1 = zn2 + C an infinite number of times, where zn and C are complex numbers. For the Mandelbrot set, the value of zn is static and the equation projects points that do not diverge to infinity for all possible values of C derived from coordinates on the complex plane. For Julia Sets, every value of C remains static while every value for zn that does not diverge to infinity is projected. Because the standard, 2D Mandelbrot set assumes the value of zn = 0, there is only one primary depiction of the Mandelbrot Set. For Julia Sets, the value of C is not assumed, and therefore, there are an infinite number of Julia Sets. In this study, the effect of changing the coordinate from which the value of C is derived in the complex plane for the equation zn+1 = zn2 + C on the qualitative characteristics of the coordinate’s corresponding, standard 2D Julia Set in relation to the standard, 2D Mandelbrot Set is observed —­ ­specifically, Julia Sets derived from points outside the Mandelbrot Set, in the main cardioid of the Mandelbrot Set, in the primary bulbs of the Mandelbrot Set, along the edges of the Mandelbrot set, along the real and imaginary axes.

Chapter 39 - A generalized Mandelbrot set and the role of critical points

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

Methods of computing Mandelbrot and Julia sets for a variety of nonlinear mappings are described. The original Mandelbrot set is constructed using a quadratic mapping. In this paper this is used as the first step in a numerical investigation of the properties of the Mandelbrot sets and the corresponding Julia sets for higher order mappings. The numerical results suggest several interesting relationships between the order of the mapping chosen and the rotational symmetries of the associated Mandelbrot and Julia sets.

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.

We consider the quadratic family of complex maps given by , where c is the centre of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded super-attracting cycle by adding one pole to each point in the cycle. When c = − 1, the Julia set of q − 1 is the well-known basilica and the perturbed map is given by , where are integers, and λ is a complex parameter such that |λ| is very small. We focus on the topological characteristics of the Julia and Fatou sets of f λ that arise when the parameter λ becomes non-zero. We give sufficient conditions on the order of the poles so that for small λ, the Julia sets consist of the union of homeomorphic copies of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on them.

In this paper, general Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method were discussed. The bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of these rational functions were given. The relations between general Mandelbrot sets and common Mandelbrot sets of zn + c (n ∈ Z, n ≥ 2), along with the relations between general Mandelbrot sets and their corresponding Julia sets were investigated. Consequently, the results were found in the study: there are similarities between the Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method and the Mandelbrot and Julia sets of zn + c (n ∈ Z, n ≥ 2).

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.