Table Of ContentThe Theory of Chaotic Attractors
Springer Science+Business Media, LLC
Brian R. Hunt
Judy A. Kennedy
Tien-Yien Li
Helena E. Nusse
Editors
The Theory of Chaotic Attractors
Springer
Brian R. Hunt Judy A. Kennedy
Institute for Physical Science and Technology Department of Mathematics
University of Maryland University of Delaware
College Park, MD 20742 Newark, Delaware 19716
USA USA
[email protected] [email protected]
Tien-Yien Li Helena E. Nusse
Department of Mathematics Department of Econometrics
Michigan State University University of Groningen
East Lansing, Michigan 48824 NL-9700 A V Groningen
USA The Netherlands
[email protected] [email protected]
Mathematics Subject Classification (2000): 37045, 37A40, 37C40, 37Exx, 28005, 01A75
Library of Congress Cataloging-in-Publication Data
The theory of chaotic attractors / [edited by] Brian R. Hunt, Judy A. Kennedy, Tien-Yien Li, Helena E. Nusse.
p. cm.
Includes bibliographical references
ISBN 978-1-4419-2330-1 ISBN 978-0-387-21830-4 (eBook)
DOI 10.1007/978-0-387-21830-4
1. Chaotic behavior in systems. 1. Nusse, Helena Engelina, 1952- II. Hunt, Brian R.
QI72.5.C45T44 2003
033'.857--dc21 2003053003
ISBN 0-387-40349-3 Printed on acid-free paper.
© 2004 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2004
Softcover reprint ofthe hardcover Ist edition 2004
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This volume is dedicated to James A. Yorke in commemoration of
his 60th birthday.
Preface
The editors felt that the time was right for a book on an important topic, the
history and development of the notions of chaotic attractors and their "natu
ral" invariant measures. We wanted to bring together a coherent collection of
readable, interesting, outstanding papers for detailed study and comparison.
We hope that this book will allow serious graduate students to hold seminars
to study how the research in this field developed. Limitation of space forced us
painfully to exclude many excellent, relevant papers, and the resulting choice
reflects the interests of the editors.
Since James Alan Yorke was born August 3, 1941, we chose to have this
book commemorate his sixtieth birthday, honoring his research in this field.
The editors are four of his collaborators.
We would particularly like to thank Achi Dosanjh (senior editor math
ematics), Elizabeth Young (assistant editor mathematics), Joel Ariaratnam
(mathematics editorial), and Yong-Soon Hwang (book production editor) from
Springer Verlag in New York for their efforts in publishing this book.
College Park, MD Brian R. Hunt
June 2003 Judy A . Kennedy
Tien-Yien Li
Helena E. Nusse
Contents
Preface ....................................................................................................................... vii
Introduction.. .. .. .. .. .. . ...... .. ... ...... ....... .. . .. .. .... ... .. .... .. .. ... .. .... . . . .. .. .. . .. ... .. .. . .. ........ ... . ... .. . . 1
Deterministic Nonperiodic Flow
E.N. Lorenz................ ..... ..... ...... ....... .......... ........................................... ...... ... .......... 25
On Invariant Measures for Expanding Differentiable Mappings
K. Krzyzewski and W. Szlenk................................................................................... 37
On the Existence of Invariant Measures for Piecewise Monotonic
Transformations
A. Lasota and J.A. Yorke .......................................................................................... 47
The Ergodic Theory of Axiom A Flows
R. Bowen and D. Ruelle............................................................................................ 55
Period Three Implies Chaos
T.-y' Li and J.A. yorke.............................................................................................. 77
Simple Mathematical Models with Very Complicated Dynamics
R.M. May ................................................................................................................... 85
A Two-Dimensional Mapping with a Strange Attractor
M. Henon ................................................................................................................... 94
Strange Attractors and Chaotic Motions of Dynamical Systems
E. Ott ......................................................................................................................... 103
Ergodic Properties of Invariant Measures for Piecewise Monotonic
Transformations
F. Hofbauer and G. Keller ........................................................................................ 120
The Dimension of Chaotic Attractors
D. J. Farmer, E. Ott, and J.A. Yorke ....................................................................... 142
x Contents
Measuring the Strangeness of Strange Attractors
P. Crassberger and 1. Procaccia ................................................................................ 170
Invariant Measures and Variational Principle for Lozi Applications
M. Rychlik ................................................................................................................. 190
Ergodic Properties of The Lozi Mappings
P. Collet and Y Levy................................................................................................. 222
On the Concept of Attractor
J. Milnor .................................................................................................................... 243
Bowen-Ruelle Measures for Certain Piecewise Hyperbolic Maps
L.-S. Young ................................................................................................................ 265
Ergodic Theory of Chaos and Strange Attractors
J.-P. Eckmann and D. Ruelle ................................................................................... 273
Another Proof of Jakobson's Theorem and Related Results
M.R. Rychlik ............................................................................. , ............................... 313
Unstable Periodic Orbits and the Dimensions of Multifractal Chaotic Attractors
C. Crebogi, E. Ott, and J.A. Yorke ................................................................... '" ...... 335
Absolutely Continuous Invariant Measures for Piecewise Expanding C2
Transformation in RN
P. Cora and A. Boyarsky .......................................................................................... 349
Sinai-Bowen-Ruelle Measures for Certain Henon Maps
M. Benedicks and L.-S. young .................................................................................. 364
On the Approximation of Complicated Dynamical Behavior
M. Dellnitz and O. Junge .......................................................................................... 400
Absolutely Continuous Invariant Measures for Piecewise Real-Analytic
Expanding Maps on the Plane
M. Tsujii .................................................................................................................... 425
SRB Measures for Partially Hyperbolic Systems Whose Central Direction
is Mostly Expanding
J.P. Alves, C. Bonatti, and M. Viana ....................................................................... 443
SLYRB Measures: Natural Invariant Measures for Chaotic Systems
HR. Hunt, J.A. Kennedy, T.-Y Li, and H.E. Nusse ............................................... 491
Credits ........................................................................................................................ 513
Introduction
Part I: Who is James Yorke?
Recently, the Science and Technology Foundation of Japan announced that
James A. Yorke had been named a winner of the 2003 Japan Prize for his work
in the field of chaos theory. This foundation has awarded Japan Prizes since
1985 under the auspices of the Japanese prime minister. It is a Japanese ver
sion of the Nobel Prize. One of the most esteemed of science and technology
prizes, Japan Prizes are given to scientists whose "original and outstanding
achievements in science and technology are recognized as having advanced
the frontiers of knowledge and served the cause of peace and prosperity for
mankind." The Science and Technology Foundation of Japan announced that
Yorke and Benoit Mandelbrot, of Yale University, will share $415,000 (50 mil
lion yen) for the "creation of universal concepts in complex systems: chaos
and fractals". The other 2003 Japan Prize was awarded for discovering the
principle behind magnetic resonance imaging (MRI). Past Japan prizes have
been awarded for achievements such as the discovery of the HIV virus, the
invention of the World Wide Web, the development of artificial intelligence,
and the eradication of smallpox. "For a scientist to be awarded the Japan
Prize is a distinction as great as any in the world," said University of Mary
land President C.D. Mote, Jr. "Jim Yorke has now been officially recognized
for his original achievements in nonlinear dynamics that have monumentally
advanced the frontiers of science and technology and served the cause of peace
and prosperity for mankind. I am so pleased for him and for the inspiration
his recognition will provide for others who by pursuing their passions can hope
to have an impact as great as Jim's." The presentation ceremony for the 2003
Japan Prize laureates is scheduled to be held in Tokyo in April 2003.
James A. Yorke came to the University of Maryland for graduate stud
ies in 1963 at the age of 22, in part because of the interdisciplinary oppor
tunities offered by the faculty of an interdisciplinary research institute now
called the Institute for Physical Science and Technology (IPST). He has spent
his entire career at the University of Maryland (College Park). From 1985
B. R. Hunt et al. (eds.), The Theory of Chaotic Attractors
© Springer Science+Business Media New York 2004
2 Introduction
through 2001 he was the director of IPST. Currently he is a Distinguished
University Professor in the Departments of Mathematics and Physics, and in
IPST at the University of Maryland. He has earned an international reputa
tion for his pioneering work in several different areas of mathematics and its
applications. The famous papers entitled "On the existence of invariant mea
sures for piecewise monotonic transformations" by A. Lasota and J.A. Yorke,
which appeared in 1973 in volume 186 of the Transactions of the American
Mathematical Society, and "Period three implies chaos" by T.-Y. Li and J.A.
Yorke, which appeared in 1975 in volume 82 of the American Mathematical
Monthly are just two examples. In particular, he has been uniquely innova
tive in his significant scientific contributions to applied topology, bifurcation
theory, numerical methods, biology (including epidemiological modeling), the
oretical nonlinear dynamics, and controlling chaos. He is coauthor of over 250
scientific papers and three books.
Yorke is a mathematician who likes to think of himself as a philosopher,
but he is also the kind of mathematician who feels compelled to put his ideas
of reality to some use, as the following examples demonstrate. In joint work
with H.W. Hethcote and A. Nold, Yorke studied the transmission dynamics
and control of gonorrhea and produced a report that persuaded the federal
government to alter its national strategies for controlling the disease. During
the gasoline crisis of the 1970s, Yorke gave testimony to the State of Maryland
arguing correctly (but unpersuasively) that the even-odd system of limiting
gasoline sales would only make lines longer. In the era of antiwar demonstra
tions, the government released a spy-plane photograph purporting to show
sparse crowds around the Washington Monument at the height of the rally.
However, Yorke proved that the photograph had actually been taken later
when the rally was breaking up (about a half-hour later), by analyzing the
monument's shadow in that photograph. Now he is working on a diverse collec
tion of projects including understanding transmission patterns of HIV / AIDS,
how to improve weather prediction, how better to determine genomes (the
DNA sequences of plants and animals), and diverse mathematics projects. A
recent example of his lively lecturing style was demonstrated during a lecture
in which he jumped on a table and stamped a grapefruit flat just to illustrate
the idea of a map from three-dimensional to two-dimensional phase space.
Those in the audience will remember that lecture.
At the institute, Yorke enjoyed unusual freedom to work on a variety of
problems and he had frequent contact with experts in a wide range of dis
ciplines. One of these experts, a fluid dynamicist, encountered Lorenz's 1963
paper "Deterministic Nonperiodic Flow" in 1972 and gave a copy to Yorke.
Lorenz's paper was the kind of scientific work that Yorke had been looking for
without even knowing it. It was a vivid physical model, a picture of a fluid in
motion, and Yorke realized that it had ideas he wanted physicists to see and
understand. To a physicist, a "legitimate" example was a differential equation
that could be written down in a simple form. When Yorke saw Lorenz's paper,
he knew that it was an example that physicists would embrace. Yorke felt that
