On the Numerical Construction Of
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On the Mandelbrot Set for I2 = ±1 and Imaginary Higgs Fields
Journal of Advances in Applied Mathematics, Vol. 6, No. 2, April 2021 https://dx.doi.org/10.22606/jaam.2021.62001 27 On the Mandelbrot Set for i2 = ±1 and Imaginary Higgs Fields Jonathan Blackledge Stokes Professor, Science Foundation Ireland. Distinguished Professor, Centre for Advanced Studies, Warsaw University of Technology, Poland. Visiting Professor, Faculty of Arts, Science and Technology, Wrexham Glyndwr University of Wales, UK. Professor Extraordinaire, Faculty of Natural Sciences, University of Western Cape, South Africa. Honorary Professor, School of Electrical and Electronic Engineering, Technological University Dublin, Ireland. Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa. Email: [email protected] Abstract We consider the consequence√ of breaking with a fundamental result in complex analysis by letting i2 = ±1 where i = −1 is the basic unit of all imaginary numbers. An analysis of the Mandelbrot set for this case shows that a demarcation between a Fractal and a Euclidean object is possible based on i2 = −1 and i2 = +1, respectively. Further, we consider the transient behaviour associated with the two cases to produce a range of non-standard sets in which a Fractal geometric structure is transformed into a Euclidean object. In the case of the Mandelbrot set, the Euclidean object is a square whose properties are investigate. Coupled with the associated Julia sets and other complex plane mappings, this approach provides the potential to generate a wide range of new semi-fractal structures which are visually interesting and may be of artistic merit. In this context, we present a mathematical paradox which explores the idea that i2 = ±1. -
Fast Visualisation and Interactive Design of Deterministic Fractals
Computational Aesthetics in Graphics, Visualization, and Imaging (2008) P. Brown, D. W. Cunningham, V. Interrante, and J. McCormack (Editors) Fast Visualisation and Interactive Design of Deterministic Fractals Sven Banisch1 & Mateu Sbert2 1Faculty of Media, Bauhaus–University Weimar, D-99421 Weimar (GERMANY) 2Department of Informàtica i Matemàtica Aplicada, University of Girona, 17071 Girona (SPAIN) Abstract This paper describes an interactive software tool for the visualisation and the design of artistic fractal images. The software (called AttractOrAnalyst) implements a fast algorithm for the visualisation of basins of attraction of iterated function systems, many of which show fractal properties. It also presents an intuitive technique for fractal shape exploration. Interactive visualisation of fractals allows that parameter changes can be applied at run time. This enables real-time fractal animation. Moreover, an extended analysis of the discrete dynamical systems used to generate the fractal is possible. For a fast exploration of different fractal shapes, a procedure for the automatic generation of bifurcation sets, the generalizations of the Mandelbrot set, is implemented. This technique helps greatly in the design of fractal images. A number of application examples proves the usefulness of the approach, and the paper shows that, put into an interactive context, new applications of these fascinating objects become possible. The images presented show that the developed tool can be very useful for artistic work. 1. Introduction an interactive application was not really feasible till the ex- ploitation of the capabilities of new, fast graphics hardware. The developments in the research of dynamical systems and its attractors, chaos and fractals has already led some peo- The development of an interactive fractal visualisation ple to declare that god was a mathematician, because mathe- method, is the main objective of this paper. -
Arxiv:1808.10408V1 [Math.DS] 30 Aug 2018 Set Md ⊂ C Is Defined As the Set of C Such That the Julia Set J(Fc) Is Con- Nected
ON THE INHOMOGENEITY OF THE MANDELBROT SET YUSHENG LUO Abstract. We will show the Mandelbrot set M is locally conformally inhomogeneous: the only conformal map f defined in an open set U intersecting @M and satisfying f(U \ @M) ⊂ @M is the identity map. The proof uses the study of local conformal symmetries of the Julia sets of polynomials: we will show in many cases, the dynamics can be recovered from the local conformal structure of the Julia sets. 1. Introduction Given a monic polynomial f(z), the filled Julia set is n K = K(f) = fz 2 C :(f (z))n2N is boundedg 2 and the Julia set is J = @K. For quadratic family fc(z) = z + c, the Mandelbrot set M can be defined as the subset in the parameter plane such that the Julia set Jc is connected, M = fc 2 C : Jc is connectedg Let A be a compact subset of C. We call an orientation preserving home- omorphism H : U −! V a local conformal symmetry of A if H is conformal, U is connected and H sends U \ A onto V \ A. We say a local conformal symmetry is trivial if it is the identity map or U \ A = ;. In this paper, we will show Theorem 1.1. The boundary of the Mandelbrot set @M admits no non- trivial local conformal symmetries. d Remark 1.2. For a fixed d ≥ 2, and let fc(z) = z + c. The Multibrot arXiv:1808.10408v1 [math.DS] 30 Aug 2018 set Md ⊂ C is defined as the set of c such that the Julia set J(fc) is con- nected. -
Quaternion Julia Set Shape Optimization
Eurographics Symposium on Geometry Processing 2015 Volume 34 (2015), Number 5 Mirela Ben-Chen and Ligang Liu (Guest Editors) Quaternion Julia Set Shape Optimization Theodore Kim Media Arts and Technology Program University of California, Santa Barbara (a) (b) (c) Figure 1: (a) The original Bunny. (b) Julia set of the 331-root rational map found by our optimization. (c) Highly intricate surface obtained by translating the roots by 1.41 in the z direction. Image is high-resolution; please zoom in to see details. Abstract We present the first 3D algorithm capable of answering the question: what would a Mandelbrot-like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi-resolution strategies. The resulting shapes are not in- variant under common operations such as translation, and instead undergo intricate, non-linear transformations. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations 1. Introduction variants [LLC∗10], and similar methods such as hypertextur- ing [EMP∗02]. These methods are widely employed to add The problem of robustly generating smooth geometry with details once the overall geometry has been finalized, e.g. -
Laplacians on Julia Sets of Rational Maps
LAPLACIANS ON JULIA SETS OF RATIONAL MAPS MALTE S. HASSLER, HUA QIU, AND ROBERT S. STRICHARTZ Abstract. The study of Julia sets gives a new and natural way to look at fractals. When mathematicians investigated the special class of Misiurewicz's rational maps, they found out that there is a Julia set which is homeomorphic to a well known fractal, the Sierpinski gasket. In this paper, we apply the method of Kigami to give rise to a new construction of Laplacians on the Sierpinski gasket like Julia sets with a dynamically invariant property. 1. Introduction Recall that the familiar Sierpinski gasket (SG) is generated in the following way. Starting with a triangle one divides it into four copies, removes the central one, and repeats the iterated process. One aspect of the study of this fractal stems from the analytical construction of a Laplacian developed by Jun Kigami in 1989 [10]. Since then, the analysis on SG has been extensively investigated from various viewpoints. The theory has been extended to some other fractals [11], too. And for some of them one could say they are more invented like SG than discovered. A standard reference to this topic is the book of Robert S. Strichartz [15]. When mathematicians studied the behaviour of polynomial maps under itera- tion, at first, they did not think about fractals. Given a starting point z0 2 C and a polynomial P (z) they wanted to know, whether the sequence of iterations P (z0);P (P (z0))::: converges. They named the set of points that show this behaviour the filled Julia set and its boundary the Julia set after the French mathematician Gaston Julia. -
Self-Similarity of the Mandelbrot Set
Self-similarity of the Mandelbrot set Wolf Jung, Gesamtschule Aachen-Brand, Aachen, Germany. 1. Introduction 2. Asymptotic similarity 3. Homeomorphisms 4. Quasi-conformal surgery The images are made with Mandel, a program available from www.mndynamics.com. Talk given at Perspectives of modern complex analysis, Bedlewo, July 2014. 1a Dynamics and Julia sets 2 n Iteration of fc(z) = z + c . Filled Julia set Kc = fz 2 C j fc (z) 6! 1g . Kc is invariant under fc(z) . n fc (z) is asymptotically linear at repelling n-periodic points. 1b The Mandelbrot set Parameter plane of quadratic polynomials 2 fc(z) = z + c . Mandelbrot set M = fc 2 C j c 2 Kcg . Two kinds of self-similarity: • Convergence of rescaled subsets: geometry is asymptotically linear. • Homeomorphisms between subsets: non-linear, maybe quasiconformal. 2. Asymptotic similarity 134 Subsets M ⊃ Mn ! fag ⊂ @M with 2a Misiurewicz points asymptotic model: 'n(Mn) ! Y in 2b Multiple scales Hausdorff-Chabauty distance. 2c The Fibonacci parameter 2d Elephant and dragon Or convergence of subsets of different 2e Siegel parameters Julia sets: c 2 M , K ⊂ K with n n n cn 2f Feigenbaum doubling asymptotics n(Kn) ! Z. 2g Local similarity 2h Embedded Julia sets Maybe Z = λY . 2a Misiurewicz points n Misiurewicz point a with multiplier %a. Blowing up both planes with %a gives λ(M − a) ≈ (Ka − a) asymptotically. Classical result by Tan Lei. Error bound by Rivera-Letelier. Generalization to non-hyperbolic semi-hyperbolic a. Proof by Kawahira with Zalcman Lemma. - 2b Multiple scales (1) −n Centers cn ! a with cn ∼ a + K%a and small Mandelbrot sets of diameter −2n j%aj . -
Fractal Geometry: the Mandelbrot and Julia Sets
Fractal Geometry: The Mandelbrot and Julia Sets Stephanie Avalos-Bock July, 2009 1 Introduction The Mandelbrot set is a set of values c ∈ C with certain important proper- ties. We will examine the formal definition of the set as well as many of its interesting, strange, and beautiful properties. The Mandelbrot set is most well known outside of mathematics as a set of beautiful images of fractals; this is partially thanks to the work of Heinz-Otto Peitgen and Peter Richter. The Mandelbrot set is relevant to the fields of complex dynamics and chaos theory, as well as the study of fractals. Here we will delve into the mathematics behind the phenomenon of the Mandelbrot set. 2 Formal Definitions First we will give some definitions of the Mandelbrot set as a whole, and then of some related concepts and characteristics. Definition 2.1. First we must define a set of complex quadratic polynomials 2 Pc : z → z + c. The Mandelbrot set M is the set of all c-values such that the sequence 0,Pc(0),Pc(Pc(0)),... does not escape to infinity, z, c ∈ C. 2.1 Alternate definitions of the Mandelbrot set n Definition 2.2. The Mandelbrot set is the set M = {c ∈ C|∃s ∈ R, ∀n ∈ N, |Pc (0)| ≤ s} n where Pc (z) is the nth iterate of Pc(z). (Note that the same result is achieved by replacing s with 2 because the Mandelbrot set falls entirely within a circle of radius 2 and centered at the origin on the complex plane.) Definition 2.3. -
An Introduction to Fatou and Julia Sets
AN INTRODUCTION TO JULIA AND FATOU SETS SCOTT SUTHERLAND ABSTRACT. We give an elementary introduction to the holomorphic dynamics of mappings on the Riemann sphere, with a focus on Julia and Fatou sets. Our main emphasis is on the dynamics of polynomials, especially quadratic polynomials. 1. INTRODUCTION In this note, we briefly introduce some of the main elementary ideas in holomorphic dynamics. This is by no means a comprehensive coverage. We do not discuss the Mandelbrot set, despite its inherent relevance to the subject; the Mandelbrot set is covered in companion articles by Devaney [Dev13, Dev14]. For a more detailed and and comprehensive introduction to this subject, the reader is refered to the book [Mil06] by John Milnor, the survey article [Bla84] by Blanchard, the text by Devaney [Dev89], or any of several other introductory texts ([Bea91], [CG93], etc.) Holomorphic dynamics is the study of the iterates of a holomorphic map f on a complex man- ifold. Classically, this manifold is one of the complex plane C, the punctured plane C∗, or the Riemann sphere Cb = C [ f1g. In these notes, we shall primarily restrict our attention to the last case, where f is a rational map (in fact, most examples will be drawn from quadratic polynomials f(z) = z2 + c, with z and c 2 C). Based on the behavior of the point z under iteration of f, the Riemann sphere Cb is partitioned into two sets • The Fatou set Ff (or merely F), on which the dynamics are tame, and • The Julia set Jf (or J ), where there is sensitive dependence on initial conditions and the dynamics are chaotic. -
Contents 5 Introduction to Complex Dynamics
Dynamics, Chaos, and Fractals (part 5): Introduction to Complex Dynamics (by Evan Dummit, 2015, v. 1.02) Contents 5 Introduction to Complex Dynamics 1 5.1 Dynamical Properties of Complex-Valued Functions . 1 5.1.1 Complex Arithmetic . 2 5.1.2 The Complex Derivative and Holomorphic Functions . 3 5.1.3 Attracting, Repelling, and Neutral Fixed Points and Cycles . 6 5.1.4 Orbits of Linear Maps on C ..................................... 9 5.1.5 Some Properties of Orbits of Quadratic Maps on C ........................ 10 5.2 Julia Sets For Polynomials . 11 5.2.1 Denition of the Julia Set and Basic Examples . 12 5.2.2 Computing the Filled Julia Set: Escape-Time Plotting . 13 5.2.3 Computing the Julia Set: Backwards Iteration . 16 5.3 Additional Properties of Julia Sets, The Mandelbrot Set . 18 5.3.1 General Julia Sets . 18 5.3.2 Julia Sets and Cantor Sets . 19 5.3.3 Critical Orbits and the Fundamental Dichotomy . 21 5.3.4 The Mandelbrot Set . 22 5.3.5 Attracting Orbits and Bulbs of the Mandelbrot Set . 23 5 Introduction to Complex Dynamics In this chapter, our goal is to discuss complex dynamical systems in the complex plane. As we will discuss, much of the general theory of real-valued dynamical systems will carry over to the complex setting, but there are also a number of new behaviors that will occur. We begin with a brief review of the complex numbers and dene the complex derivative d of a complex-valued C dz function, and then turn our attention to the basic study of complex dynamical systems: xed points and periodic points, attracting and repelling xed points and cycles, behaviors of neutral xed points, attracting basins, and orbit analysis. -
The Sets of Julia and Mandelbrot for Multi- Dimensional Case of Logistic Mapping
Central Asian Problems of Modern Science and Education Volume 2020 Issue 3 Central Asian Problems of Modern Article 4 Science and Education 2020-3 11-19-2020 THE SETS OF JULIA AND MANDELBROT FOR MULTI- DIMENSIONAL CASE OF LOGISTIC MAPPING R.N. Ganikhodzhaev National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan, [email protected] Sh.J Seytov National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan, [email protected] I.N Obidjonov Urgench State University, [email protected] L.O Sadullayev Urgench State University, [email protected] Follow this and additional works at: https://uzjournals.edu.uz/capmse Part of the Physical Sciences and Mathematics Commons Recommended Citation Ganikhodzhaev, R.N.; Seytov, Sh.J; Obidjonov, I.N; and Sadullayev, L.O (2020) "THE SETS OF JULIA AND MANDELBROT FOR MULTI-DIMENSIONAL CASE OF LOGISTIC MAPPING," Central Asian Problems of Modern Science and Education: Vol. 2020 : Iss. 3 , Article 4. Available at: https://uzjournals.edu.uz/capmse/vol2020/iss3/4 This Article is brought to you for free and open access by 2030 Uzbekistan Research Online. It has been accepted for inclusion in Central Asian Problems of Modern Science and Education by an authorized editor of 2030 Uzbekistan Research Online. For more information, please contact [email protected]. UDK 51 (075) The sets of Julia and Mandelbrot for multi-dimensional case of logistic mapping Ganikhodzhaev R.N. National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan e-mail: [email protected] Seytov Sh.J. National University of Uzbekistan named after Mirzo Ulugbek Tashkent, Uzbekistan e-mail: [email protected] Obidjonov I.N Urgench State University, Urgench, Uzbekistan e-mail: [email protected] Sadullayev L.O. -
Generation of Julia and Mandelbrot Sets Via Fixed Points
S S symmetry Article Generation of Julia and Mandelbrot Sets via Fixed Points Mujahid Abbas 1,2, Hira Iqbal 3 and Manuel De la Sen 4,* 1 Department of Mathematics, Government College University, Lahore 54000, Pakistan; [email protected] 2 Department of Medical Research, China Medical University No. 91, Hsueh-Shih Road, Taichung 400, Taiwan 3 Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Lahore 54000, Pakistan; [email protected] 4 Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain * Correspondence: [email protected] Received: 27 November 2019; Accepted: 26 December 2019; Published: 2 January 2020 Abstract: 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) = xn + mx + r where m, r 2 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. Keywords: iteration; fixed points; fractals MSC: Primary: 47H10; Secondary:47J25 1. Introduction Fixed point theory provides a suitable framework to investigate various nonlinear phenomena arising in the applied sciences including complex graphics, geometry, biology and physics [1–4]. -
Self-Similarity of the Mandelbrot Set
Self-similarity of the Mandelbrot set Wolf Jung, Gesamtschule Aachen-Brand, Aachen, Germany. 1. Introduction 2. Asymptotic similarity 3. Homeomorphisms 4. Quasi-conformal surgery The images are made with Mandel, a program available from www.mndynamics.com. Talk given at Perspectives of modern complex analysis, Bedlewo, July 2014. 1a Dynamics and Julia sets 2 n Iteration of fc(z) = z + c . Filled Julia set Kc = fz 2 C j fc (z) 6! 1g . Kc is invariant under fc(z) . n fc (z) is asymptotically linear at repelling n-periodic points. 1b The Mandelbrot set Parameter plane of quadratic polynomials 2 fc(z) = z + c . Mandelbrot set M = fc 2 C j c 2 Kcg . Two kinds of self-similarity: • Convergence of rescaled subsets: geometry is asymptotically linear. • Homeomorphisms between subsets: non-linear, maybe quasiconformal. 2. Asymptotic similarity 134 Subsets M ⊃ Mn ! fag ⊂ @M with 2a Misiurewicz points asymptotic model: 'n(Mn) ! Y in 2b Multiple scales Hausdorff-Chabauty distance. 2c The Fibonacci parameter 2d Elephant and dragon Or convergence of subsets of different 2e Siegel parameters Julia sets: c 2 M , K ⊂ K with n n n cn 2f Feigenbaum doubling asymptotics n(Kn) ! Z. 2g Local similarity 2h Embedded Julia sets Maybe Z = λY . 2a Misiurewicz points n Misiurewicz point a with multiplier %a. Blowing up both planes with %a gives λ(M − a) ≈ (Ka − a) asymptotically. Classical result by Tan Lei. Error bound by Rivera-Letelier. Generalization to non-hyperbolic semi-hyperbolic a. Proof by Kawahira with Zalcman Lemma. - 2b Multiple scales (1) −n Centers cn ! a with cn ∼ a + K%a and small Mandelbrot sets of diameter −2n j%aj .