Dynamical Systems and Cosmology
Dynamical systems theory is especially well-suited for determining the possible asymptotic states (at both early and late times) of cosmological models, particularly when the governing equations are a finite system of autonomous ordinary differential equa
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ASTROPHYSICS AND SPACE SCIENCE LIBRARY VOLUME 291
EDITORIAL BOARD Chainnan W.B. BURTON, National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A. ([email protected]); University of Leiden, The Netherlands ([email protected])
Executive Committee J. M. E. KUIJPERS, Faculty of Science, Nijmegen, The Netherlands E. P. J. VAN DEN HEUVEL, Astronomical Institute, University of Amsterdam, The Netherlands H. VAN DER LAAN, Astronomical Institute, University of Utrecht, The Netherlands MEMBERS 1. APPENZELLER, Landesstemwarte Heidelberg-Konigstuhl, Gennany J. N. BAHCALL, The Institute for Advanced Study, Princeton, U.S.A. F. BERTOLA, Universita di Padova, Italy J. P. CASSINELLI, University of Wisconsin, Madison, U.S.A. C. J. CESARSKY, Centre d'Etudes de Saclay, Gif-sur-Yvette Cedex, France O. ENGVOLD, Institute of Theoretical Astrophysics, University of Oslo, Norway R. McCRAY, University of Colorado, JILA, Boulder, U.S.A. P. G. MURDIN, Institute of Astronomy, Cambridge, U.K. F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India K. SATO, School of Science, The University of Tokyo, Japan F. H. SHU, University of California, Berkeley, U.S.A. B. V. SOMOV, Astronomical Institute, Moscow State University, Russia R. A. SUNYAEV, Space Research Institute, Moscow, Russia Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan S. TREMAINE, CITA, Princeton University, U.S.A. N. O. WEISS, University of Cambridge, U.K.
DYNAMICAL SYSTEMS AND COSMOLOGY by A.A. COLEY Dalhousie University, Halifax, Canada
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6329-8 ISBN 978-94-017-0327-7 (eBook) DOI 10.1007/978-94-017-0327-7
Printed on acid-free paper
All Rights Reserved © Springer Science+Business Media Dordrecht 2003 Originally published by Kluwer Academic Publishers in 2003. Softcover reprint of the hardcover I st edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
Contents
I.
II.
Introduction
1
A. Self-similarity
3
The Theory of Dynamical Systems
7
A. Linear Autonomous Differential Equations
8
1.
Topological Equivalence
13
2.
Linear Stability
14
B. Non-Linear Differential Equations 1.
Liapunov Theory
17
2.
Linearization and the Hartman-Grobman Theorem
19
C. Periodic Orbits and The Poincare-Bendixson Theorem in the Plane
21
D. More General Non-Linear Behaviour
23
1.
III.
IV.
15
Higher Dimensions
23
Spatially Homogeneous Models
27
A. Definitions and Kinematical Quantities
28
B. Asymptotic States of Perfect Fluid Bianchi Models
30
C. More Recent Wo
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