Table Of ContentInstitute for Nonlinear Science
Springer Science+Business Media, LLC
Institute for Nonlinear Science
Henry D.1. Abarbanel Analysis of Chaotic Time Series (1996)
Leon Glass, Peter Hunter, Andrew McCullogh (Eds.) Theory of Heart:
Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac
Function (1991)
Mark Millonas (Ed.) Fluctuations and Order: The New Synthesis (1996)
Linda E. Reichl The Transition to Chaos in Conservative Classical Systems:
Quantum ManiJestations (1992)
Henry D.I. Abarbanel
Analysis of Observed
Chaotic Data
With 140 Illustrations
Springer
Henry D.I. Abarbanel
Professor of Physics
Director, Institute for Nonlinear Science
University of California - San Diego
La JoHa, CA 92093-0402, USA
Editorial Board
Institute for Nonlinear Science, University of California - San Diego
Henry D.I. Abarbanel, Physics (Scripps Institution of Oceanography)
Morteza Gharib, Applied Mechanics and Engineering Sciences
Michael E. Gilpin, Biology
Walter HeHer, Economics
Katja Lindenberg, Chemistry
Manuel Rotenberg, Electrical and Computer Engineering
John D. Simon, Chemistry
Library of Congress Cataloging-in-Publication Data
Abarbanel, H.D.1.
Analysis of observed chaotic data / Henry D.1. Abarbanel.
p. cm. - (Institute for nonlinear science)
Includes bibliographical references and index.
ISBN 978-0-387-98372-1 ISBN 978-1-4612-0763-4 (eBook)
DOI 10.1007/978-1-4612-0763-4
1. Chaotic behavior in systems. 1. Title. II. Series: Institute
for nonlinear science (Series)
Ql72.5.C45A23 1995
501' .1385-dc20 95-18641
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© 1996 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 1996
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ISBN 978-0-387-98372-1 SPIN 10830839
Preface
When I encountered the idea of chaotic behavior in deterministic dynami
cal systems, it gave me both great pause and great relief. The origin of the
great relief was work I had done earlier on renormalization group properties
of homogeneous, isotropic fluid turbulence. At the time I worked on that,
it was customary to ascribe the apparently stochastic nature of turbulent
flows to some kind of stochastic driving of the fluid at large scales. It was
simply not imagined that with purely deterministic driving the fluid could
be turbulent from its own chaotic motion. I recall a colleague remarking
that there was something fundamentally unsettling about requiring a fluid
to be driven stochastically to have even the semblance of complex motion
in the velocity and pressure fields. I certainly agreed with him, but neither
of us were able to provide any other reasonable suggestion for the observed,
apparently stochastic motions of the turbulent fluid. So it was with relief
that chaos in nonlinear systems, namely, complex evolution, indistinguish
able from stochastic motions using standard tools such as Fourier analysis,
appeared in my bag of physics notions. It enabled me to have a physi
cally reasonable conceptual framework in which to expect deterministic,
yet stochastic looking, motions.
The great pause came from not knowing what to make of chaos in non
linear systems. By this I mean that when I learn about a new phenomenon
in Physics, I try to provide myself a framework in which to ask what I
have learned about the physical system in which it is observed. In the case
of the BeS theory of superconductivity, just to stray a bit, I recall un
derstanding immediately that it represented a deep insight into collective
effects in quantum systems leading to remarkable macroscopic properties.
vi Preface
In the case of chaos, I was simply bewildered why nature chose such an ap
parently unwelcome outcome for the evolution of physical systems. From
the outset it was clear, especially for a physicist, from the elegant fluid
dynamical experiments of colleagues like Jerry Gollub and Harry Swinney,
that chaos was a real property of physical systems, and thus I could not
ignore it as an interesting, but irrelevant, mathematical construct. Yet I
could not give meaning to why a system would not toe the line and behave
as I had learned in my courses on classical physics, namely, in nice regular
ways. Perhaps I was ready to accept a superposition of nice regular ways,
namely, quasi-periodic behavior composed of a collection of incommensu
rate sinusoids. I was not ready for evolution which was nonperiodic and
thus possessed of a continuous, broad Fourier spectrum. So it gave me great
pause, and this book in some manner puts forward some of the answers I
have put together to move beyond that pause.
While I may still not have a fully acceptable (metaphysical, I guess)
answer as to why physical systems choose to behave chaotically, I have
some clearer ideas on what is accomplished by their evolving chaotically. I
even have some personally satisfying conjectures as to what is achieved in
a functional sense for both physical and biological systems by the ability to
develop chaotic motion. My understanding rests on the perception that the
ability of a system in chaotic motion to explore wide regions of its phase
space, and thus utilize information about a large collection of potential
states of the system, can have distinct positive implications for what the
system can do in achieving its required functions. Regular systems moving
on a torus in some dimension explore a really limited domain of phase space.
That may be perfectly fine for some actions, but such limited behavior is
rather restrictive, and if the systems want to get to another state by action
of external forces or equivalently a changed environment, it will be most
difficult to do that when the basic motion is confined to an integrable
subspace of the full system phase space. Biological systems that do not
take advantage of chaos in their state spaces may not be able to participate
in evolution to the extent we understand that as adaptation to changing
environments. Of course, we are unlikely to see such systems as they are
now extinct. Similarly, the range of alternatives available to which one
can control a system is larger if chaotic motion, seen as a wide ranging
exploration of state space, is allowed.
Perhaps that sounds more like a credo than an understanding, but keep
ing in mind the possibilities that wide domains of phase space are opened
up by chaotic motion focuses one's attention on the main theme of this
book: geometry in phase space provides answers to a large body
of questions as to how one can extract useful physical informa
tion from observations of chaos. The thrust of this book is precisely to
provide the physicist and other persons encountering chaotic time series,
a set of tools for establishing a framework in which clear questions about
the source of the chaotic signal can be asked and answered. An alternative
Preface vii
title for the book could have been "The Chaotic Mechanic's Tool Kit: An
Owner's Manual."
The book starts on a tone unfamiliar in nonlinear dynamics, namely,
with an extended example from manufacturing and milling machines. This
example is developed with experimental observations from the laboratory
of Professor Frank Moon at Cornell. It is meant to place in context the main
goal of this work: to provide tools for scientists and engineers who wish to
extract useful content from observations of quantities important in their
work but for whom time dependence is irregular in appearance. With data
from one set of observations we draw conclusions which are unsupported
at first but will rest on the material in the remainder of the book. The
discussion of this example will be completed much later in the book.
After this I move right into the analysis of signals observed from chaotic
motion. Ideas of phase space and its reconstruction from single observed
quantities are introduced. Then details are given on time delay phase space
reconstruction. Following this I introduce and discuss the notion of invari
ants of the chaotic motion, namely, those things not sensitive to changes
in initial condition.
These preliminaries are then used to discuss model making and predic
tion within the context of chaos. This is not now, and may never be, a
subject which we can consider complete in a mathematical sense. This is
an area in which it is quite easy to do very well with rather simple tools,
and I shall outline how one accomplishes prediction within the limits of the
predictability of the source. Indeed it is a most pleasing aspect of the anal
ysis in this whole book that the limits on predictability of the dynamical
system can be systematically extracted from observations of the system.
Following model making I address the issue of signal separation in chaotic
systems: given a contaminated chaotic signal, how can we separate the
contamination from the chaos. Thinking of this as signal separation, rather
than noise reduction which some would term it, allows us to go beyond just
removing the contamination, for it may be that the contamination itself is
precisely what we wish to capture. Next I give some discussion of various
methods which have been explored for controlling chaotic motions either
back to regular motions or to chaotic motions somehow more desirable than
the one handed to us.
Synchronization of chaotic systems is taken up next. This is in many ways
related to control. In controlling chaos we seek to move the free running
system we are given into more regular or more chaotic motions by adding
dynamics to the original system in the form of time varying parameters or
external driving of other forms. In synchronization we seek subspaces of
the coupled system space in which a special kind of motion which relates
the coupled systems takes place.
I will illustrate the ideas as they are introduced with detailed analysis of
computer model calculations and with analysis of data from experiments.
Typically the model will either be the three degree of freedom Lorenz model
viii Preface
from 1963, the two-dimensional Henon map, the Rossler attractor [Roe76],
or the two-dimensional Ikeda map describing a laser beam in a ring cavity.
The experimental data will be taken from observations of nonlinear circuits
by Tom Carroll and Lou Pecora of the Naval Research Laboratory or ex
periments by N. F. Rul'kov at the Institute for Nonlinear Science. At the
end of the book will be a chapter discussing a variety of other examples
of the application of the chaos toolkit which is the bottom line output of
the discussion. The topics which will be covered in this chapter will give
us the luxury of allowing us to move from analysis of a sequence of ASCII
numbers to attempts to use the tools to uncover innovative understand
ing of the source of the time series. Among the topics we will discuss are
(a) chaotic motions of coherent vortex structures in a turbulent boundary
layer, (b) chaotic fluctuations of a solid state laser. (In particular I will
show how the tools allow us to establish that some of these fluctuations
are due to classical chaos and others have their origin in intrinsic quantum
fluctuations of the laser processes.), and (c) chaotic motions of the volume
of the Great Salt Lake.
In the case of the laser intensity fluctuations I will also discuss a dynam
ical model which ties together the conclusions drawn from our analysis of
a single observed quantity: the intensity of infrared light.
These items represent a broad spectrum of scientific interests and phe
nomena. They will be discussed in some detail as to their scientific impor
tance, and then I will provide detailed information on what the tools we
have developed tell us about these diverse scientific questions. The point of
the further examples is to (a) illustrate the power of the methods central to
this book, (b) to indicate specific scientific interpretations of aspects of the
dynamics of the sources of the various signals, and (c) to place in proper
context the practical use of the tools we have so that the reader can see that
they do not provide unaided algorithmic paths to correct interpretations of
data-humans playa critical role which must be kept in mind.
In the closing chapter we discuss a more abstract topic, namely, the max
imum accuracy with which one can expect to ever estimate a dynamical
variable of interest when it is chaotic and observed in the presence of exter
nal additive noise. This provides a generalization of the usual statement of
the implications of the Cramer-Rao bound [Tre68], and ties together some
observations in earlier sections, especially surrounding signal separation
using manifold decomposition methods.
Despite my original inclinations, I have chosen to forgo the customary
introduction to nonlinear dynamics found in most books which have ap
peared in recent years on topics related to dynamics. At this time there
are many excellent introductions and advanced monographs on these top
ics [Ott93, Dra92 , TS86, Mo092, GH83, LL91, Arn78], and the reader is
well advised to refer to them when issues of details about strange attractors
or other matters arise. I violate this in the Appendix where I discuss two
topics which I deem of some importance: (1) the role of information theory
Preface IX
in nonlinear systems, and (2) the notion of strange at tractors or chaotic be
havior arising through a series of instabilities of nonlinear systems. Indeed,
the view that motion on a strange attractor is unstable everywhere in state
space, yet bounded and possessed of structure in the state space, underlies
much of what this book proposes one use in the analysis of such systems.
In the case of the Lorenz attractor, seeing some detail about how these
instabilities develop and arise from stressing the system to make it perform
some task, the transport of heat by convection in the case of the Lorenz
model, is worth noting. To make up for the details of these topics I have also
included a glossary of terms commonly appearing in this book. This is not
a substitute for working through many of the important details to be found
in the books I mentioned, but perhaps it will be helpful to the reader just
becoming well acquainted with those details to have a handy peek into the
meaning of the main items underlying the analysis here. A few sentences
describing a strange attractor, for example, can hardly substitute for the
reader's working out its properties through examining some examples nu
merically and reading some of the mathematical results known about such
objects, but perhaps the glossary will serve some benefit nonetheless.
It is my hope that the reader will be interested in trying out the methods
described in this book on his/her own data and perhaps further developing
the analysis tools found here. The algorithms described below are available
in a mixture of FORTRAN and C programs. The algorithms have been
ported to many computing environments, and a working version running
under OSF /Motif on SUN Workstations and other similar UNIX Work
stations is available for purchase. Please contact the author. This book is
probably the best 'manual' for those programs, though something of a more
formal 'here's how to do it' set of directions accompany the software.
The collections of results which make up this book could not have been
created by my efforts alone despite the partial hubris required to put them
into print above my name alone. I had extensive assistance in putting to
gether the picture painted in this book. Primary among my colleagues
have been the graduate students and postdoctoral fellows with whom I
have worked over the past several years. Specifically I wish to acknowledge
Matt Kennel, Clif Liu, Misha Sushchik, Jesse Goldberg, Garrett Lisi, John
J. ("SID") Sidorowich, Reggie Brown, Pierre Marteau, Nikolai Rul'kov,
and Paul Bryant. More senior colleagues have also played a critical role,
and among them I especially thank Lev Tsimring, Doug Mook, Ed Lorenz,
Misha Rabinovich, Cory Myers, and David Ruelle. Ted Frison has sup
ported the effort to put together this book in ways he is too modest to
acknowledge. Tom Carroll, Lou Pecora, U pmanu Lall, and Raj Roy have
been both friends and invaluable colleagues in providing and interpreting
data. Without them this book might seem an amusing set of conjectures;
they have made its needed contact with reality.
I have received continuing financial support for the research on which
this book could be based. Oscar Manley at the Department of Energy,
x Preface
Mikhael Ciftan at the Army Research Office, and Michael Shlesinger at the
Office of Naval Research have steadfastly given their encouragement to the
work here and enabled it to proceed with financial stability. Working with
Doug Mook and Cory Meyers of the Lockheed/Sanders Corporation was
critical as the algorithms devised here were brought to bear on real world
problems.
Over the years I have wondered why authors give such substantial thanks
to their families for the support they received while composing scientific
volumes. This is no longer a mystery to me as I have so richly benefited
from the generosity of my wife Beth Levine and our remarkable daughters
Brett and Sara. It is customary to think that one could not have done it
without them, but I suspect I have done it so much with them that they
could have done it for me. My love and thanks.
October 4, 1995 Henry D. 1. A bar banel
Del Mar, California
