Geometry of Chaos
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Effective S-Adic Symbolic Dynamical Systems
Effective S-adic symbolic dynamical systems Val´erieBerth´e,Thomas Fernique, and Mathieu Sablik? 1 IRIF, CNRS UMR 8243, Univ. Paris Diderot, France [email protected] 2 LIPN, CNRS UMR 7030, Univ. Paris 13, France [email protected] 3 I2M UMR 7373, Aix Marseille Univ., France [email protected] Abstract. We focus in this survey on effectiveness issues for S-adic sub- shifts and tilings. An S-adic subshift or tiling space is a dynamical system obtained by iterating an infinite composition of substitutions, where a substitution is a rule that replaces a letter by a word (that might be multi-dimensional), or a tile by a finite union of tiles. Several notions of effectiveness exist concerning S-adic subshifts and tiling spaces, such as the computability of the sequence of iterated substitutions, or the effec- tiveness of the language. We compare these notions and discuss effective- ness issues concerning classical properties of the associated subshifts and tiling spaces, such as the computability of shift-invariant measures and the existence of local rules (soficity). We also focus on planar tilings. Keywords: Symbolic dynamics; adic map; substitution; S-adic system; planar tiling; local rules; sofic subshift; subshift of finite type; computable invariant measure; effective language. 1 Introduction Decidability in symbolic dynamics and ergodic theory has already a long history. Let us quote as an illustration the undecidability of the emptiness problem (the domino problem) for multi-dimensional subshifts of finite type (SFT) [8, 40], or else the connections between effective ergodic theory, computable analysis and effective randomness (see for instance [14, 33, 44]). -
Strictly Ergodic Symbolic Dynamical Systems
STRICTLY ERGODIC SYMBOLIC DYNAMICAL SYSTEMS SHIZUO KAKUTANI YALE UNIVERSITY 1. Introduction We continue the study of strictly ergodic symbolic dynamical systems which was started in our earlier report [6]. The main tools used in this investigation are "homomorphisms" and "substitutions". Among other things, we construct two strictly ergodic symbolic dynamical systems which are weakly mixing but not strongly mixing. 2. Strictly ergodic symbolic dynamical systems Let A be a finite set consisting of more than one element. Let (2.1) X = AZ = H A, An = A forallne Z, neZ be the set of all two sided infinite sequences (2.2) x = {a"In Z}, an = A for all n E Z, where (2.3) Z = {nln = O, + 1, + 2,} is the set of all integers. For each n E Z, an is called the nth coordinate of x, and the mapping (2.4) 7r,: x -+ a, = 7En(x) is called the nth projection of the power space X = AZ onto the base space An = A. The space X is a totally disconnected, compact, metrizable space with respect to the usual direct product topology. Let q be a one to one mapping of X = AZ onto itself defined by (2.5) 7En(q(X)) = ir.+1(X) for all n E Z. The mapping p is a homeomorphism ofX onto itself and is called the shift trans- formation. The dynamical system (X, (p) thus obtained is called the shift dynamical system. This research was supported in part by NSF Grant GP16392. 319 320 SIXTH BERKELEY SYMPOSIUM: KAKUTANI Let X0 be a nonempty closed subset of X which is invariant under (p. -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Turbulence, Entropy and Dynamics
TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions -
Decomposing the Dynamics of the Lorenz 1963 Model Using Unstable
Decomposing the Dynamics of the Lorenz 1963 model using Unstable Periodic Orbits: Averages, Transitions, and Quasi-Invariant Sets Chiara Cecilia Maiocchi∗ and Valerio Lucarini† Centre for the Mathematics of Planet Earth, University of Reading and Department of Mathematics and Statistics, University of Reading Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems. They allow one to extract information from a system and to distill its dynamical structure. We consider here the Lorenz 1963 model with the classic parameters’ value and decompose its dynamics in terms of UPOs. We investigate how a chaotic orbit can be approximated in terms of UPOs. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that, somewhat unexpectedly, longer period UPOs overwhelmingly provide the best local approximation to the trajectory, even if our UPO-detecting algorithm severely undersamples them. Second, we construct a finite-state Markov chain by studying the scattering of the forward trajectory between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. We then study the transitions between the different states. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets. I. INTRODUCTION would suffice in obtaining an accurate approximation of ergodic averages [70–72]. These results are proven to be Unstable periodic orbits (UPOs) play an important valid for dynamical systems exhibiting strong chaoticity role in the analysis of dynamical systems that exhibit [73, 74], such as hyperbolic and Axiom A systems chaotic behaviour and in some cases they provide a [75, 76]. -
Bolles E.B. Einstein Defiant.. Genius Versus Genius in the Quantum
Selected other titles by Edmund Blair Bolles The Ice Finders: How a Poet, a Professor, and a Politician Discovered the Ice Age A Second Way of Knowing: The Riddle of Human Perception Remembering and Forgetting: Inquiries into the Nature of Memory So Much to Say: How to Help Your Child Learn Galileo’s Commandment: An Anthology of Great Science Writing (editor) Edmund Blair Bolles Joseph Henry Press Washington, DC Joseph Henry Press • 500 Fifth Street, NW • Washington, DC 20001 The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public. Joseph Henry was one of the founders of the National Academy of Sciences and a leader in early American science. Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Bolles, Edmund Blair, 1942- Einstein defiant : genius versus genius in the quantum revolution / by Edmund Blair Bolles. p. cm. Includes bibliographical references. ISBN 0-309-08998-0 (hbk.) 1. Quantum theory—History—20th century. 2. Physics—Europe—History— 20th century. 3. Einstein, Albert, 1879-1955. 4. Bohr, Niels Henrik David, 1885-1962. I. Title. QC173.98.B65 2004 530.12′09—dc22 2003023735 Copyright 2004 by Edmund Blair Bolles. All rights reserved. Printed in the United States of America. To Kelso Walker and the rest of the crew, volunteers all. -
Chao-Dyn/9402001 7 Feb 94
chao-dyn/9402001 7 Feb 94 DESY ISSN Quantum Chaos January Einsteins Problem of The study of quantum chaos in complex systems constitutes a very fascinating and active branch of presentday physics chemistry and mathematics It is not wellknown however that this eld of research was initiated by a question rst p osed by Einstein during a talk delivered in Berlin on May concerning Quantum Chaos the relation b etween classical and quantum mechanics of strongly chaotic systems This seems historically almost imp ossible since quantum mechanics was not yet invented and the phenomenon of chaos was hardly acknowledged by physicists in While we are celebrating the seventyfth anniversary of our alma mater the Frank Steiner Hamburgische Universitat which was inaugurated on May it is interesting to have a lo ok up on the situation in physics in those days Most I I Institut f urTheoretische Physik UniversitatHamburg physicists will probably characterize that time as the age of the old quantum Lurup er Chaussee D Hamburg Germany theory which started with Planck in and was dominated then by Bohrs ingenious but paradoxical mo del of the atom and the BohrSommerfeld quanti zation rules for simple quantum systems Some will asso ciate those years with Einsteins greatest contribution the creation of general relativity culminating in the generally covariant form of the eld equations of gravitation which were found by Einstein in the year and indep endently by the mathematician Hilb ert at the same time In his talk in May Einstein studied the -
The Mathematical Heritage of Henri Poincaré
http://dx.doi.org/10.1090/pspum/039.1 THE MATHEMATICAL HERITAGE of HENRI POINCARE PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 39, Part 1 THE MATHEMATICAL HERITAGE Of HENRI POINCARE AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 39 PROCEEDINGS OF THE SYMPOSIUM ON THE MATHEMATICAL HERITAGE OF HENRI POINCARfe HELD AT INDIANA UNIVERSITY BLOOMINGTON, INDIANA APRIL 7-10, 1980 EDITED BY FELIX E. BROWDER Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 79-22916 1980 Mathematics Subject Classification. Primary 01-XX, 14-XX, 22-XX, 30-XX, 32-XX, 34-XX, 35-XX, 47-XX, 53-XX, 55-XX, 57-XX, 58-XX, 70-XX, 76-XX, 83-XX. Library of Congress Cataloging in Publication Data Main entry under title: The Mathematical Heritage of Henri Poincare\ (Proceedings of symposia in pure mathematics; v. 39, pt. 1— ) Bibliography: p. 1. Mathematics—Congresses. 2. Poincare', Henri, 1854—1912— Congresses. I. Browder, Felix E. II. Series: Proceedings of symposia in pure mathematics; v. 39, pt. 1, etc. QA1.M4266 1983 510 83-2774 ISBN 0-8218-1442-7 (set) ISBN 0-8218-1449-4 (part 2) ISBN 0-8218-1448-6 (part 1) ISSN 0082-0717 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit librar• ies acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in re• views provided the customary acknowledgement of the source is given. -
Τα Cellular Automata Στο Σχεδιασμό
τα cellular automata στο σχεδιασμό μια προσέγγιση στις αναδρομικές σχεδιαστικές διαδικασίες Ηρώ Δημητρίου επιβλέπων Σωκράτης Γιαννούδης 2 Πολυτεχνείο Κρήτης Τμήμα Αρχιτεκτόνων Μηχανικών ερευνητική εργασία Ηρώ Δημητρίου επιβλέπων καθηγητής Σωκράτης Γιαννούδης Τα Cellular Automata στο σχεδιασμό μια προσέγγιση στις αναδρομικές σχεδιαστικές διαδικασίες Χανιά, Μάιος 2013 Chaos and Order - Carlo Allarde περιεχόμενα 0001. εισαγωγή 7 0010. χάος και πολυπλοκότητα 13 a. μια ιστορική αναδρομή στο χάος: Henri Poincare - Edward Lorenz 17 b. το χάος 22 c. η πολυπλοκότητα 23 d. αυτοοργάνωση και emergence 29 0011. cellular automata 31 0100. τα cellular automata στο σχεδιασμό 39 a. τα CA στην στην αρχιτεκτονική: Paul Coates 45 b. η φιλοσοφική προσέγγιση της διεπιστημονικότητας του σχεδιασμού 57 c. προσομοίωση της αστικής ανάπτυξης μέσω CA 61 d. η περίπτωση της Changsha 63 0101. συμπεράσματα 71 βιβλιογραφία 77 1. Metamorphosis II - M.C.Escher 6 0001. εισαγωγή Η επιστήμη εξακολουθεί να είναι η εξ αποκαλύψεως προφητική περιγραφή του κόσμου, όπως αυτός φαίνεται από ένα θεϊκό ή δαιμονικό σημείο αναφοράς. Ilya Prigogine 7 0001. 8 0001. Στοιχεία της τρέχουσας αρχιτεκτονικής θεωρίας και μεθοδολογίας προτείνουν μια εναλλακτική λύση στις πάγιες αρχιτεκτονικές μεθοδολογίες και σε ορισμένες περιπτώσεις υιοθετούν πτυχές του νέου τρόπου της κατανόησής μας για την επιστήμη. Αυτά τα στοιχεία εμπίπτουν σε τρεις κατηγορίες. Πρώτον, μεθοδολογίες που προτείνουν μια εναλλακτική λύση για τη γραμμικότητα και την αιτιοκρατία της παραδοσιακής αρχιτεκτονικής σχεδιαστικής διαδικασίας και θίγουν τον κεντρικό έλεγχο του αρχιτέκτονα, δεύτερον, η πρόταση μιας μεθοδολογίας με βάση την προσομοίωση της αυτο-οργάνωσης στην ανάπτυξη και εξέλιξη των φυσικών συστημάτων και τρίτον, σε ορισμένες περιπτώσεις, συναρτήσει των δύο προηγούμενων, είναι μεθοδολογίες οι οποίες πειραματίζονται με την αναδυόμενη μορφή σε εικονικά περιβάλλοντα. -
Symbolic Dynamics and Markov Partitions
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 1, January 1998, Pages 1–56 S 0273-0979(98)00737-X SYMBOLIC DYNAMICS AND MARKOV PARTITIONS ROY L. ADLER Abstract. The decimal expansion of real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by sym- bolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by sym- bolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely, hyperbolic automorphisms of the two dimensional torus. While there are some results in higher dimensions, this area remains a fertile one for research. 1. Introduction We address the question: how and to what extent can a dynamical system be represented by a symbolic one? The first use of infinite sequences of symbols to describe orbits is attributed to a nineteenth century work of Hadamard [H]. How- ever the present work is rooted in something very much older and familiar to us all: namely, the representation of real numbers by infinite binary expansions. As the example of Section 3.2 shows, such a representation is related to the behavior of a special partition under the action of a map. Partitions of this sort have been linked to the name Markov because of their connection to discrete time Markov processes. -
Strange Attractors in a Multisector Business Cycle Model
Journal of Economic Behavior and Organization 8 (1987) 397411. North-Holland STRANGE ATTRACTORS IN A MULTISECTOR BUSINESS CYCLE MODEL Hans-Walter LORENZ* Uniuersity of Giittingen, 34 Giittingen, FRG Received October 1985. final version received December 1986 Three coupled oscillating sectors in a multisector Kaldor-type business cycle model can give rise to the occurrence of chaotic motion. If the sectors are linked by investment demand interdependencies, this coupling can be interpreted as a perturbation of a motion on a three- dimensional torus. A theorem by Newhouse, Ruelle and Takens implies that such a perturbation may possess a strange attractor with the consequence that the flow of the perturbed system may become irregular or chaotic. A numerical investigation reveals that parameter values can be found which indeed lead to chaotic trajectories in this cycle model. 1. Introduction Recent work on chaos and strange attractors in non-linear dynamical systems has raised the question whether a ‘route to turbulence’ can be traced in a continuous-time dynamical system by means of a steady increase of a control parameter and successive bifurcations with a change of the topo- logical nature of the trajectories. A famous example for such a route to turbulence was provided by Ruelle and Takens (1971)‘: Starting with an asymptotically stable fixed point for low values of the parameter, the system undergoes a Hopf bifurcation if the control parameter is sufficiently in- creased. While a second Hopf bifurcation implies a bifurcation of the generated limit cycle to a two-dimensional torus, a third Hopf bifurcation can lead to the occurrence of a strange attractor and hence of chaos. -
Arxiv:Math/0611322V1 [Math.DS] 10 Nov 2006 Sisfcsalte Rmhdudt Os,Witnbtent Between Int Written Subject the Morse, Turn to to Intention Hedlund His Explicit
ON THE GENESIS OF SYMBOLIC DYNAMICS AS WE KNOW IT ETHAN M. COVEN AND ZBIGNIEW H. NITECKI Abstract. We trace the beginning of symbolic dynamics—the study of the shift dynamical system—as it arose from the use of coding to study recurrence and transitivity of geodesics. It is our assertion that neither Hadamard’s 1898 paper, nor the Morse-Hedlund papers of 1938 and 1940, which are normally cited as the first instances of symbolic dynamics, truly present the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the beginning of symbolic dynamics in a paper published by Hedlund in 1944. Symbolic dynamics, in the modern view [LM95, Kit98], is the dynamical study of the shift automorphism on the space of bi-infinite sequences of symbols, or its restriction to closed invariant subsets. In this note, we attempt to trace the begin- nings of this viewpoint. While various schemes for symbolic coding of geometric and dynamic phenomena have been around at least since Hadamard (or Gauss: see [KU05]), and the two papers by Morse and Hedlund entitled “Symbolic dynamics” [MH38, MH40] are often cited as the beginnings of the subject, it is our view that the specific, abstract version of symbolic dynamics familiar to us today really began with a paper,“Sturmian minimal sets” [Hed44], published by Hedlund a few years later. The outlines of the story are familiar, and involve the study of geodesic flows on surfaces, specifically their recurrence and transitivity properties; this note takes as its focus a letter from Hedlund to Morse, written between their joint papers and Hedlund’s, in which his intention to turn the subject into a part of topology is explicit.1 This letter is reproduced on page 6.