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the user’s approach to topological methods in 3d dynamical systems TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk the user’s approach to topological methods in 3d dynamical systems mario a natiello lund university, sweden hernán g solari universidad de buenos aires, argentina World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE USER’S APPROACH TO TOPOLOGICAL METHODS IN 3-D DYNAMICAL SYSTEMS Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-270-380-4 ISBN-10 981-270-380-2 Printed in Singapore. EH - The User's Approach.pmd 1 7/12/2007, 11:42 AM April10,2007 14:40 WSPC/BookTrimSizefor9inx6in wsbook Hernán dedicates this effort to his extended family: • my daughter Flor, who is the essence of life • Princesa and Leboni (dogs), that taught me that being useful to the pack implies leadership, and that logic ap- plies to dogs, claims of leadership do not make us useful. • Brillito and Luna (cats), that remind me that freedom cannot be negotiated, and love is something we do not exchange, we just give it away. • Shoot (horse), who taught me that horse and rider are one and at the same time they are the mirror reflexion of the other. • and to Ba´rbara, who made them all exist. I dedicate this effort to mia amata moglie Patrizia and min älskade dotter Saffo que iluminan cada d´ıa de mi vida haciendo que dé gusto vivirlo, and to the forests and lakes of Patagonia and Scandinavia for letting me be part of them every now and then. Mario v April10,2007 14:40 WSPC/BookTrimSizefor9inx6in wsbook TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk April10,2007 14:40 WSPC/BookTrimSizefor9inx6in wsbook Preface During most of the Twentieth Century, physicists have been mainly con- cerned with linear dynamics. Despite the works of Poincaré, Birkhoff and von Neumann, the paradigm in physics was linear dynamics. Courses in ClassicalMechanicssystematicallyignoredintrinsicallynon-linearphenom- enaandchaos,restrictingMechanicstoIntegrableSystems,i.e., dynamical systems with an underlying Lie group structure, having dynamics that are exponentials of linear algebras. During the second half of the 70’s the interest in nonlinear dynamics gradually emerged in physics fueled by the possibility of enriching our intuition using increasingly powerful (as well as popular and affordable) computers. The chaos paradigm took form, with new problems and new ways to analyze nature. An intense development followed the introduction of graphic workstations in the 1980s. Questions such as: How to charac- terize systems presenting chaotic dynamics? How to compare models with experiments? were then included within the valid questions of the chaos paradigm. Bythattimeitbecameclearthatalthoughthereexistonlyafewdiffer- ent ways of displaying linear behaviour (always present in widely different classesof problems), nonlinearproblems presenteda largevariety ofdiffer- ent patterns, as well as other specific features such as sensitivity to initial conditions. The urge to generate some comprehensive understanding of chaos (are there different classes of chaotic behaviour?) became evident. Duringthe’80s,therewereseveralattemptstosolvetheclassificationprob- lem. Earlier attempts focused in the routes to chaos, the sequence of bi- furcations as a function of a single control parameter, that lead to chaos in a particular system. By the middle ’80s this attempt had proven to be of limited use: there were infinitely many routes to chaos in simple two- vii April10,2007 14:40 WSPC/BookTrimSizefor9inx6in wsbook viii The User’sApproach toTopological Methods in 3-D Dynamical Systems parameter systems. The chaos community then turned its hopes towards fractal dimensions, i.e., a measure of the geometrical imprint (in phase- space) of a chaotic attractor. By the end of the ’80s this path had also proven to be almost useless for the characterization/classificationproblem (althoughsomeinterestingfeaturessuchasBarnsley’sfractalpicturesspun off this effort). The two main directions taken by the chaos community that we just described were not the only explored directions. Around 1987, a third programme aiming to classify low dimensional (3-D) systems using topological orbit organization began. This project in Physics was preceded by at least two important developments in Mathematics: (a) results from Birman-Williams-Holmes (1983–) developed to extract the knot content of hyperbolic attractors, introducing a geometrical construction that they namedtemplateorknot-holder,and(b)resultsduetoThurston(1979–)on theclassificationof2-Ddiffeomorphismsintermsoftwomainclasses: rota- tion compatible diffeomorphisms and pseudo-Anosovdiffeomorphisms (the latter classadmits a fine structure)andthe braidcontentof the diffeomor- phism. Thurston’sresultsappearedearlierthanthetemplatedevelopment, but they were incorporated to the Physics project at a later stage. While the relationamong the mathematicaldevelopments and the programme in physics is direct and immediate, there also exist important dif- ferencesamongthem. WehavegiventhenameThe User’s Approach to Topological Methods in 3-D Dynamical Systemstotheclassification and recognitionprogrammein Physics,emphasizing that its aim is the use of the mathematical methods (emerging from Topology) in experimental situations. Unlike other programmes in chaos, the topological classifica- tion programme is still alive. In this book we intend to re-evaluate this programme. While writing this book we have come in contact with some difficult aspects concerning how to assess,prove or disprove a certain property in a system, that require a clear conception about how the programme relates to theory, experiment and numerical modeling. The readers will therefore find discussions on, and references to, epistemological matters. We have adopted as much as possible a Popperian demarcationist and fallibilistic attitude, since we are dealing with experimental science, i.e., our interest is to induce from experiments the originating properties of the underlying system in a scientifically valid way. On the contrary, we have left outside this presentation topics that are encompassed by the concept of normal scienceinthesenseofKhun(repetitionofaparadigmwithlittle variation) April10,2007 14:40 WSPC/BookTrimSizefor9inx6in wsbook Preface ix as well as some attempts which are still under development, but have not yet reachedthe level of an organizedtheory, at least in our understanding. This needsnotbe aseriousloss,thereareothersourceswherethe material canbefound. Itisourhopethatallexistingproto-attemptswillsoonreach the mature level so that they can be thoroughly assessed. Chapter 1 discusses the goals of the programme and why it is needed. Next, we dedicate the first part of the book to a presentationof the mathematical elements that constitute the basis of the programme (Chapters 2–4). Chapters 5 and 6 present the reconstruction problem, proper of the user’s approach, turning the discussionfrommostly mathematicalterms to mostlyphysical(ornaturalscience)terms. Chapter7isa guideto some pioneering works in the actual application of the methods to experimental data. Aseveryprogrammethatactuallyprogresses,thereoccursareformula- tion process while going from the dreams and illusions of the first days to ourpresent(hopefullymorerealistic)view. TheclosingwordsinChapter8 are reserved to a recollection of the conquests achieved by the programme as well as to an evaluationofthe problems that the programmefaces,having survived 20 years (about twice the survival time of failed theoretical developments, so there is a basis for keeping hopes alive) but having not reached yet a stable status. Acknowledgments Alongthe decadeswehaveworkedinthissubjectwehavemetanumberof colleagues. Many of them became friends along the way, all of them have taught us something that in one way or the other has been important for this book. Thank you all. TheLibrariansofMatematikbiblioteketaswellastheinfrastructure at Lunds Universitets Bibliotek and at the Matematikcentrum of Lunds Universitet have been of invaluable support. Thanks. One of us (MAN), having no extended family in the animal kingdom outside the homo sapiens sapiens species, thanks his many friends in different places of the world for their help in making life enjoyable. MAN gratefully acknowledges travel grants from the Swedish Veten- skapsr˚adet, from Lunds stads jubileumsfond and from Malmo¨ stads jubileumsfond. HGS thanks the continuous support of the Consejo Nacional