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Once again KAM theory is committed in the context of nearly integrable Hamiltonian systems. While elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form <EM>n-parameter families. Hence, without the need for untypical conditions or external parameters, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system. The text moves gradually from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations have to be replaced by Cantor sets. Planar singularities and their versal unfoldings are an important ingredient that helps to explain the underlying dynamics in a transparent way.
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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1893 Heinz Hanßmann Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems Results and Examples ABC Author Heinz Hanßmann Mathematisch Instituut Universiteit Utrecht Postbus 80010 3508 TA Utrecht The Netherlands Library of Congress Control Number: 2006931766 Mathematics Subject Classification (2000): 37J20, 37J40, 34C30, 34D30, 37C15, 37G05, 37G10, 37J15, 37J35, 58K05, 58K70, 70E20, 70H08, 70H33, 70K30, 70K43 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-38894-x Springer Berlin Heidelberg New York ISBN-13 978-3-540-38894-4 Springer Berlin Heidelberg New York DOI 10.1007/3-540-38894-x 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 ° 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. Typesetting by the author and SPi using a Springer LATEX package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper SPIN: 11841708 VA41/3100/SPi 543210 to my parents Preface Life is in color, But black and white is more realistic. Samuel Fuller The present notes are devoted to the study of bifurcations of invariant tori in Hamiltonian systems. Hamiltonian dynamical systems can be used to model frictionless mechanics, in particular celestial mechanics. We are concerned with the nearly integrable context, where Kolmogorov–Arnol’d– Moser (KAM) theory shows that most motions are quasi-periodic whence the (invariant) closure is a torus. An interesting aspect is that we may encounter torus bifurcations of high co-dimension in a single given Hamiltonian system. Historically, bifurcation theory has first been developed for dissipative dynamical systems, where bifurcations occur only under variation of external parameters. Bifurcations of equilibria and periodic orbits The structure of any dynamical system is organized by its invariant subsets, the equilibria, periodic orbits, invariant tori and the stable and unstable manifolds of all these. Invariant subsets form the framework of the dynamics, and one is interested in the properties that are persistent under small perturbations. The most si