Stability and Chaos in Celestial Mechanics
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Alessandra Celletti
Stability and Chaos in Celestial Mechanics
Published in association with
Praxis Publishing Chichester, UK
Professor Alessandra Celletti Department of Mathematics University of Rome ``Tor Vergata'' Rome Italy
SPRINGER±PRAXIS BOOKS IN ASTRONOMY AND PLANETARY SCIENCES SUBJECT ADVISORY EDITORS: Philippe Blondel, C.Geol., F.G.S., Ph.D., M.Sc., Senior Scientist, Department of Physics, University of Bath, UK; John Mason, M.B.E., B.Sc., M.Sc., Ph.D.
ISBN 978-3-540-85145-5 Springer-Verlag Berlin Heidelberg New York Springer is part of Springer-Science + Business Media (springer.com) Library of Congress Control Number: 2009936867 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. # Praxis Publishing Ltd, Chichester, UK, 2010 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speci®c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Jim Wilkie Project copy editor: Mike Shardlow Author-generated LaTex, processed by EDV-Beratung Herweg, Germany Printed in Germany on acid-free paper
To the memory of my mother Mirella, my best teacher and friend, and in fond remembrance of my grandmother Emilia. Without them I would never have had the privilege of studying the sky.
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV 1
Order and chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Continuous and discrete systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Conservative and dissipative systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The attractors and basins of attraction . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The standard map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The dissipative standard map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 H´enon’s mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 4 6 7 9 12 15 17
2
Numerical dynamical met
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