Infinitely Renormalizable Quadratic Polynomials 1
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Fractal Growth on the Surface of a Planet and in Orbit Around It
Wilfrid Laurier University Scholars Commons @ Laurier Physics and Computer Science Faculty Publications Physics and Computer Science 10-2014 Fractal Growth on the Surface of a Planet and in Orbit around It Ioannis Haranas Wilfrid Laurier University, [email protected] Ioannis Gkigkitzis East Carolina University, [email protected] Athanasios Alexiou Ionian University Follow this and additional works at: https://scholars.wlu.ca/phys_faculty Part of the Mathematics Commons, and the Physics Commons Recommended Citation Haranas, I., Gkigkitzis, I., Alexiou, A. Fractal Growth on the Surface of a Planet and in Orbit around it. Microgravity Sci. Technol. (2014) 26:313–325. DOI: 10.1007/s12217-014-9397-6 This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. 1 Fractal Growth on the Surface of a Planet and in Orbit around it 1Ioannis Haranas, 2Ioannis Gkigkitzis, 3Athanasios Alexiou 1Dept. of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada 2Departments of Mathematics and Biomedical Physics, East Carolina University, 124 Austin Building, East Fifth Street, Greenville, NC 27858-4353, USA 3Department of Informatics, Ionian University, Plateia Tsirigoti 7, Corfu, 49100, Greece Abstract: Fractals are defined as geometric shapes that exhibit symmetry of scale. This simply implies that fractal is a shape that it would still look the same even if somebody could zoom in on one of its parts an infinite number of times. -
An Image Cryptography Using Henon Map and Arnold Cat Map
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 05 Issue: 04 | Apr-2018 www.irjet.net p-ISSN: 2395-0072 An Image Cryptography using Henon Map and Arnold Cat Map. Pranjali Sankhe1, Shruti Pimple2, Surabhi Singh3, Anita Lahane4 1,2,3 UG Student VIII SEM, B.E., Computer Engg., RGIT, Mumbai, India 4Assistant Professor, Department of Computer Engg., RGIT, Mumbai, India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - In this digital world i.e. the transmission of non- 2. METHODOLOGY physical data that has been encoded digitally for the purpose of storage Security is a continuous process via which data can 2.1 HENON MAP be secured from several active and passive attacks. Encryption technique protects the confidentiality of a message or 1. The Henon map is a discrete time dynamic system information which is in the form of multimedia (text, image, introduces by michel henon. and video).In this paper, a new symmetric image encryption 2. The map depends on two parameters, a and b, which algorithm is proposed based on Henon’s chaotic system with for the classical Henon map have values of a = 1.4 and byte sequences applied with a novel approach of pixel shuffling b = 0.3. For the classical values the Henon map is of an image which results in an effective and efficient chaotic. For other values of a and b the map may be encryption of images. The Arnold Cat Map is a discrete system chaotic, intermittent, or converge to a periodic orbit. that stretches and folds its trajectories in phase space. Cryptography is the process of encryption and decryption of 3. -
WHAT IS a CHAOTIC ATTRACTOR? 1. Introduction J. Yorke Coined the Word 'Chaos' As Applied to Deterministic Systems. R. Devane
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties that characterize chaos. We also discuss how this term can be combined with the definition of an attractor. 1. Introduction J. Yorke coined the word `chaos' as applied to deterministic systems. R. Devaney gave the first mathematical definition for a map to be chaotic on the whole space where a map is defined. Since that time, there have been several different definitions of chaos which emphasize different aspects of the map. Some of these are more computable and others are more mathematical. See [9] a comparison of many of these definitions. There is probably no one best or correct definition of chaos. In this paper, we discuss what we feel is one of better mathematical definition. (It may not be as computable as some of the other definitions, e.g., the one by Alligood, Sauer, and Yorke.) Our definition is very similar to the one given by Martelli in [8] and [9]. We also combine the concepts of chaos and attractors and discuss chaotic attractors. 2. Basic definitions We start by giving the basic definitions needed to define a chaotic attractor. We give the definitions for a diffeomorphism (or map), but those for a system of differential equations are similar. The orbit of a point x∗ by F is the set O(x∗; F) = f Fi(x∗) : i 2 Z g. An invariant set for a diffeomorphism F is an set A in the domain such that F(A) = A. -
MATH34042: Discrete Time Dynamical Systems David Broomhead (Based on Original Notes by James Montaldi) School of Mathematics, University of Manchester 2008/9
MATH34042: Discrete Time Dynamical Systems David Broomhead (based on original notes by James Montaldi) School of Mathematics, University of Manchester 2008/9 Webpage: http://www.maths.manchester.ac.uk/∼dsb/Teaching/DynSys email: [email protected] Overview (2 lectures) Dynamical systems continuous/discrete; autonomous/non-autonomous. Iteration, orbit. Applica- tions (population dynamics, Newton’s method). A dynamical system is a map f : S S, and the set S is called the state space. Given an initial point x0 S, the orbit of x0 is the sequence x0,x1,x2,x3,... obtained by repeatedly applying f: ∈ → n x1 = f(x0), x2 = f(x1), x3 = f(x2),...,xn = f (x0), ... Basic question of dynamical systems: given x0, what is behaviour of orbit? Or, what is behaviour of most orbits? + + Example Let f : R R be given by f(x)= √x. Let x0 = 2. Then x1 = f(2)= √2 1.4142. And ≈ x = √2 1.1892 and x = √1.1892 1.0905. 2 p → 3 The orbit of≈2 therefore begins {2.0, 1.4142,≈ 1.1892, 1.0905, ...}. The orbit of 3 begins {3.0, 1.7321, 1.3161, 1.1472, ...}. The orbit of 0.5 begins {0.5, 0.7071, 0.8409, .9170, .9576, ...}. All are getting closer to 1. Regular dynamics fixed points, periodic points and orbits. Example f(x)= cos(x). Globally attracting fixed point at x = 0.739085 . ··· Chaotic dynamics Basic idea is unpredictability. There is no “typical’ behaviour. Example f(x)= 4x(1 − x), x [0,1]: Split the interval into two halves: L = [0, 1 ] and R = [ 1 ,1]. -
MA427 Ergodic Theory
MA427 Ergodic Theory Course Notes (2012-13) 1 Introduction 1.1 Orbits Let X be a mathematical space. For example, X could be the unit interval [0; 1], a circle, a torus, or something far more complicated like a Cantor set. Let T : X ! X be a function that maps X into itself. Let x 2 X be a point. We can repeatedly apply the map T to the point x to obtain the sequence: fx; T (x);T (T (x));T (T (T (x))); : : : ; :::g: We will often write T n(x) = T (··· (T (T (x)))) (n times). The sequence of points x; T (x);T 2(x);::: is called the orbit of x. We think of applying the map T as the passage of time. Thus we think of T (x) as where the point x has moved to after time 1, T 2(x) is where the point x has moved to after time 2, etc. Some points x 2 X return to where they started. That is, T n(x) = x for some n > 1. We say that such a point x is periodic with period n. By way of contrast, points may move move densely around the space X. (A sequence is said to be dense if (loosely speaking) it comes arbitrarily close to every point of X.) If we take two points x; y of X that start very close then their orbits will initially be close. However, it often happens that in the long term their orbits move apart and indeed become dramatically different. -
Attractors and Orbit-Flip Homoclinic Orbits for Star Flows
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 8, August 2013, Pages 2783–2791 S 0002-9939(2013)11535-2 Article electronically published on April 12, 2013 ATTRACTORS AND ORBIT-FLIP HOMOCLINIC ORBITS FOR STAR FLOWS C. A. MORALES (Communicated by Bryna Kra) Abstract. We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be C1 approximated by vector fields with orbit-flip homoclinic orbits. 1. Introduction The notion of attractor deserves a fundamental place in the modern theory of dynamical systems. This assertion, supported by the nowadays classical theory of turbulence [27], is enlightened by the recent Palis conjecture [24] about the abun- dance of dynamical systems with finitely many attractors absorbing most positive trajectories. If confirmed, such a conjecture means the understanding of a great part of dynamical systems in the sense of their long-term behaviour. Here we attack a problem which goes straight to the Palis conjecture: The fini- tude of the number of attractors for a given dynamical system. Such a problem has been solved positively under certain circunstances. For instance, we have the work by Lopes [16], who, based upon early works by Ma˜n´e [18] and extending pre- vious ones by Liao [15] and Pliss [25], studied the structure of the C1 structural stable diffeomorphisms and proved the finitude of attractors for such diffeomor- phisms. His work was largely extended by Ma˜n´e himself in the celebrated solution of the C1 stability conjecture [17]. On the other hand, the Japanesse researchers S. -
On the Combinatorics of External Rays in the Dynamics of the Complex Henon
ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Ricardo Antonio Oliva May 1998 c Ricardo Antonio Oliva 1998 ALL RIGHTS RESERVED Addendum. This is a slightly revised version of my doctoral thesis: some typing and spelling mistakes have been corrected and a few sentences have been re-worded for better legibility (particularly in section 4.3). Also, to create a nicer pdf document with hyperref, the title of section 3.3.2 has been made shorter. The original title was A model for a map with an attracting fixed point as well as a period-3 sink: the (3-1)-graph. ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. Ricardo Antonio Oliva , Ph.D. Cornell University 1998 We present combinatorial models that describe quotients of the solenoid arising from the dynamics of the complex H´enon map 2 2 2 fa,c : C → C , (x, y) → (x + c − ay, x). These models encode identifications of external rays for specific mappings in the H´enon family. We investigate the structure of a region of parameter space in R2 empirically, using computational tools we developed for this study. We give a combi- natorial description of bifurcations arising from changes in the set of identifications of external rays. Our techniques enable us to detect, predict, and locate bifurca- tion curves in parameter space. We describe a specific family of bifurcations in a region of real parameter space for which the mappings were expected to have sim- ple dynamics. -
ORBIT EQUIVALENCE of ERGODIC GROUP ACTIONS These Are Partial Lecture Notes from a Topics Graduate Class Taught at UCSD in Fall 2
1 ORBIT EQUIVALENCE OF ERGODIC GROUP ACTIONS ADRIAN IOANA These are partial lecture notes from a topics graduate class taught at UCSD in Fall 2019. 1. Group actions, equivalence relations and orbit equivalence 1.1. Standard spaces. A Borel space (X; BX ) is a pair consisting of a topological space X and its σ-algebra of Borel sets BX . If the topology on X can be chosen to the Polish (i.e., separable, metrizable and complete), then (X; BX ) is called a standard Borel space. For simplicity, we will omit the σ-algebra and write X instead of (X; BX ). If (X; BX ) and (Y; BY ) are Borel spaces, then a map θ : X ! Y is called a Borel isomorphism if it is a bijection and θ and θ−1 are Borel measurable. Note that a Borel bijection θ is automatically a Borel isomorphism (see [Ke95, Corollary 15.2]). Definition 1.1. A standard probability space is a measure space (X; BX ; µ), where (X; BX ) is a standard Borel space and µ : BX ! [0; 1] is a σ-additive measure with µ(X) = 1. For simplicity, we will write (X; µ) instead of (X; BX ; µ). If (X; µ) and (Y; ν) are standard probability spaces, a map θ : X ! Y is called a measure preserving (m.p.) isomorphism if it is a Borel isomorphism and −1 measure preserving: θ∗µ = ν, where θ∗µ(A) = µ(θ (A)), for any A 2 BY . Example 1.2. The following are standard probability spaces: (1) ([0; 1]; λ), where [0; 1] is endowed with the Euclidean topology and its Lebegue measure λ. -
Efficient Method for Detection of Periodic Orbits in Chaotic Maps And
Efficient Method for Detection of Periodic Orbits in Chaotic Maps and Flows A Thesis by Jonathan Crofts in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Department of Mathematics University of Leicester March 2007. c Jonathan J Crofts, 2007. arXiv:0706.1940v2 [nlin.CD] 13 Jun 2007 Acknowledgements I would like to thank Ruslan Davidchack, my supervisor, for his many suggestions and constant support and understanding during this research. I am also thankful to Michael Tretyakov for his support and advice. Further, I would like to acknowledge my gratitude to the people from the Department of Mathematics at the University of Leicester for their help and support. Finally, I would like to thank my family and friends for their patience and sup- port throughout the past few years. In particular, I thank my wife Lisa and my daughter Ellen, without whom I would have completed this research far quicker, but somehow, it just would not have been the same. At this point I would also like to reassure Lisa that I will get a real job soon. Leicester, Leicestershire, UK Jonathan J. Crofts 31 March 2007 i Contents Abstract iv List of figures v List of tables vi 1 Introduction 1 1.1 History,theoryandapplications . 1 1.2 Periodicorbits............................... 3 1.2.1 Periodicorbittheory . 4 1.2.2 Efficient detection of UPOs . 6 1.3 Extendedsystems............................. 9 1.4 Anoteonnumerics............................ 10 1.4.1 Intervalarithmetic . 12 1.5 Overview.................................. 14 1.6 Thesisresults ............................... 16 2 Conventional techniques for detecting periodic orbits 18 2.1 Specialcases................................ 18 2.1.1 One-dimensionalmaps . -
Geography of the Cubic Connectedness Locus : Intertwining Surgery
ANNALES SCIENTIFIQUES DE L’É.N.S. ADAM EPSTEIN MICHAEL YAMPOLSKY Geography of the cubic connectedness locus : intertwining surgery Annales scientifiques de l’É.N.S. 4e série, tome 32, no 2 (1999), p. 151-185 <http://www.numdam.org/item?id=ASENS_1999_4_32_2_151_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1999, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. sclent. EC. Norm. Sup., 4e serie, t 32, 1999, p. 151 a 185. GEOGRAPHY OF THE CUBIC CONNECTEDNESS LOCUS: INTERTWINING SURGERY BY ADAM EPSTEIN AND MICHAEL YAMPOLSKY ABSTRACT. - We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction intertwining two quadratics is realized by means of quasiconformal surgery. The associated asymptotic geography of the cubic connectedness locus is discussed in the Appendix. © Elsevier, Paris RESUME. - Nous trouvons des produits de Fensemble de Mandelbrot dans 1'espace a deux variables complexes des polynomes cubiques. La renormalisation d'un polyn6me cubique appartenant a un tel produit donne deux polyn6mes quadratiques. Le precede inverse qui entrelace deux polynomes quadratiques est obtenu par chirurgie quasiconforme. -
1D Dynamics: Symbolic Coding
1D Dynamics: Symbolic Coding 1. Full shift of two symbols – Let X = {s =(s0s1s2 ··· si ··· ); si =0 or 1} be the collection of all sequences of 0 and 1. – For s =(s0s1 ··· si ··· ), t =(t0t1 ··· ti ··· ), let ∞ |s − t | d(s, t) = i i X i i=0 2 Claim: (i) (X,d) is a metric space. s t 1 (ii) d( , ) ≤ 2n iff si = ti for all i < n. Proof: (i) d(s, t) ≥ 0 and d(s, t) = d(t, s) are obvious. For the triangle inequality, observe ∞ ∞ ∞ |si − ti| |si − vi| |vi − ti| d(s, t) = ≤ + X 2i X 2i X 2i i=0 i=0 i=0 = d(s, v)+ d(v, t). (ii) is straight forward from definition. – Let σ : X → X be define by σ(s0s1 ··· si ··· )=(s1s2 ··· si+1 ··· ). Claim: σ : X → X is continuous. Proof: 1 For ε > 0 given let n be such that 2n < ε. Let 1 s t s t δ = 2n+1 . For d( , ) <δ, d(σ( ), σ( )) < ε. – (X,σ) as a dynamical system: Claim (a) σ : X → X have exactly 2n periodic points of period n. (b) Union of all periodic orbits are dense in X. (c) There exists a transitive orbit. Proof: (a) The number of periodic point in X is the same as the number of strings of 0, 1 of length n, which is 2n. (b) ∀s =(s0s1 · · · sn · · · ), let sn =(s0s1 · · · sn; s0s1 · · · sn; · · · ). We have 1 d(sn, s) < . 2n−1 So sn → s as n →∞. (c) List all possible strings of length 1, then of length 2, then of length 3, and so on. -
Multiple Periodic Solutions and Fractal Attractors of Differential Equations with N-Valued Impulses
mathematics Article Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses Jan Andres Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic; [email protected] Received: 10 September 2020; Accepted: 30 September 2020; Published: 3 October 2020 Abstract: Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5). Keywords: impulsive differential equations; n-valued maps; Hutchinson-Barnsley operators; multiple periodic solutions; topological fractals; Devaney’s chaos on attractors; Poincaré operators; Nielsen number MSC: 28A20; 34B37; 34C28; Secondary 34C25; 37C25; 58C06 1. Introduction The theory of impulsive differential equations and inclusions has been systematically developed (see e.g., the monographs [1–4], and the references therein), among other things, especially because of many practical applications (see e.g., References [1,4–11]).