Three-Dimensional Flows
In this book, the authors present the elements of a general theory for flows on three-dimensional compact boundaryless manifolds, encompassing flows with equilibria accumulated by regular orbits. The book aims to provide a global perspective of this theor
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A Series of Modern Surveys in Mathematics
Editorial Board G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay H. W. Lenstra, Jr., Leiden S. Müller, Bonn J. Tits, Paris D. B. Zagier, Bonn G. Ziegler, Berlin Managing Editor R. Remmert, Münster
Volume 53
Vítor Araújo Maria José Pacifico
Three-Dimensional Flows
Vítor Araújo Maria José Pacifico Instituto de Matemática Universidade Federal do Rio de Janeiro 21.945-970 Rio de Janeiro Brazil [email protected] [email protected]
ISBN 978-3-642-11413-7 e-ISBN 978-3-642-11414-4 DOI 10.1007/978-3-642-11414-4 Springer Heidelberg Dordrecht London New York Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISSN 0071-1136 Library of Congress Control Number: 2010929946 Mathematics Subject Classification (2010): 34Cxx, 34Dxx, 37C10, 37C20, 37C40, 37C70, 37D05, 37D20, 37D25, 37D30, 37D45, 37D50 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Não sou nada. Nunca serei nada. Não posso querer ser nada. À parte isso, tenho em mim todos os sonhos do mundo. Álvaro de Campos. Tabacaria.
Vítor dedica a Simone, Andreia, Manuela e Domingos. Maria José Pacifico dedica a Maria, Laura e Ricardo.
Foreword
E.N. Lorenz’s twelve pages note Deterministic nonperiodic flow, published in 1963 in the Journal of the Atmospheric Sciences [139], surely ranks among the most influential inputs from the experimental sciences in the history of 20th century Mathematics. While investigating the validity of linear regression models for weather forecast, Lorenz hit upon the observation that typical trajectories, even in very simple models, are unstable, or sensitive to small changes in the initial conditions. This observation had deep philosophical and practical implications. From a mathematician’s perspective, Lorenz’s paper set the challenge of describing and explaining on rigorous grounds the sensitivity phenomenon, and its stability under small modifications of the dynamical model. By that time, starting in the early sixties, t
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