Analysis of Chaotic Data and The Exploitation of Chaotic Systems

New York: John Wiley & Sons

Blueberry Brain Institute & Chaos Cooperative

http://www.pacweb.com/blueberry/

This book is important for several reasons. First of all, it uses the word "chaos" in the title on a book that is expectionally competent and sophisticated. This should help legitimize the word when many of us have been concerned that its popularization may have destroyed its academic usefulness. We seem to schizophrenically want both the respectability of disciplinary academics and the metaphoric creativity that transcends the usual academic market place. This book well affirms the former role of dynamics as an important scientific strategy.

Second. While the development of theoretical dynamical models is considerably advanced, the design and analysis of experiments in our fields has been insufficiently developed (Abraham, 1997). This book and the classic review paper by Abarbanel, Brown, Sidorowich, and Tsimring (1993; now expanded into a fine book: Abarbanel, 1996) provide some very sophisticated techniques that should do much to develop this embryonic field.

Third. Many of us in the psychological, social, and life sciences have a mature mathematical conceptual outlook but not the time or training to be really proficient in mathematics to the extent we might wish. Many of us need mathematical colleagues for some assistance with our efforts in dynamics. This book can help us considerably. While this book's mathematics may be a stretch for some of us, its exposition is so clear that we can follow the basic concepts and improve our mathematics at the same time.

This book has three main parts, 15 chapters, and reprints of 42 classic papers.

Part I, labeled "Background" is a clear and concise primer on some basics of dynamics (pp. 2-62). The first chapter is a review of dynamical systems, attractors, and chaos. In this brief 12-page chapter you get an excellent summary of the basics, and clear destinctions between an invertible map (Hénon) and a noninvertible (logistic) map, between attractors from conservative Hamiltonian systems obeying Liouville's Theorem and attractors from nonconservative systems (this distinction involves evaluating volume change in an attractor using a Jacobian determinant). The box-counting definition of fractal dimensions is given, and then a definition of chaos (it specifies sensitive dependence on initial conditions; debated by some) using Moon & Holmes' (1979) beam apparatus as an example. It also shows how the second-order nonautonomous equation of Duffing can be converted into autonomous first order differential equations, and ends by providing a definition of Lyapunov exponents. I mention these as examples of terms that can seem a bit formidable at first but become crystal clear in their use by these authors. For those of us who might benefit from it, the book may provide motivation for review and further learning.

The remaining four chapters of this introductory primer focus on issues of measuring properties of chaotic attractors: dimension, symbolic dynamics, Lyapunov exponents and entropy, and the theory of embedding.

Part II, "Analysis of Data from Chaotic Systems" is comprised of four chapters of 3-6 reprints each (pp. 65-203). Each chapter starts with a review of basic principles by the editors, with a description of what each paper contributes to the subject. These papers include the great classics that we have all seen quoted over and over. Few of us have had the convenience of having them on our shelves, an opportunity provided by this book.

For example, in chapter 6, *The Practice of Embedding*, the paper
by Packard, Crutchfield, Farmer, & Shaw (1980) 'Geometry from a time
series', is the foremost introduction to the time-lagged (delay-coordinate)
technique of reconstructing an attractor from a single time series. Broomhead
& King's (1986) 'Extracting qualitative dynamics from experimental
data', added filtering using singular value decomposition to the technique.
Kennel, Brown, & Abarbanel's (1992) 'Determining embedding dimension
for phase-space reconstruction using a geometrical construction', added
the false-neighbors improvements to diminish the number of dimensions needed
for emedding and assisted in distinguishing deterministic from stochastic
components. Kaplan & Glass' (1992) 'Direct test for determinism in
a time series' adds another technique for evaluating deterministic and
stochastic contributions by comparing an averaged vectorfield with that
from a randomized vectorfield. I liked the editor's pointing out these
techniques and their chapter 5 (from Part I) on the theory of embedding
were appropriate for simultaneous multiple variables as well as time-delayed
variables (Abraham, 1997; Sauer, Yorke, & Casdagli, 1991; Stewart,
1996).

Chapter 7, *Dimension Calculations*, includes: Albano, Muench,
Schwartz, Mees, & Rapp's (1988) 'Singular-value decomposition and the
Grassberger-Procaccia algorithm'. It provides results of the Broomhead
& King procedure for measuring the correlation dimension. Eckmann &
Ruelle's (1992) 'Fundamental limitations for estimating dimensions and
Lyapunov exponents in dynamical systems' and Ding, Grebogi, Ott, Sauer,
& Yorke's (1993) 'Plateau onset for correlation dimension: When does
it occur?' both address issues of inadequate data. Theiler, Eubank, Longtin,
Galdrakian, & Farmer's (1992) 'Testing for nonlinearity in time series:
The method of surrogate data' is known to our field because they used EEG
among their test time series and so it is frequently quoted in our disciplines.
Econometric and epidemiological data have also been so tested. They introduced
Monte Carlo techniques for establishing a null hypothesis and statistical
testing of attractor invariants. The Brandstater & Swinney (1987) 'Strange
attractors in weakly turbulent Couette-Taylor flow' examined portraits,
power spectra, circle maps, and dimension estimates to evaluate bifurcations
and problems of data requirements. The final reprint of this chapter, Guckenheimer
& Buzyna's (1983) 'Dimension measurements for geostrophic turbulence'
is also a study of bifurcation in turbulence. Their study used multiple
simultaneous time series. The editor's note, "It is our feeling that simultaneous
measurements will often give superior results [to the delay-coordinate
procedure], and should be used, if available (p. 106).

Chapter 8, *Calculation of Lyapunov Exponents*, contains three
papers which take up the evolution of these computations since Wolf, Swift,
Swinney, & Vastano's (1985) heuristic method for estimating the largest
exponents. The idea is to convolute an *m*-frame of orthogonal vectors
over a trajectory to get at local linear dynamics. The first paper here,
Eckmann, Kamphorst, Ruelle, & Ciliberto's (1986) 'Liapunov exponents
from time series' is an application of this method to synthetic and experimental
data. It is one of the first to provide computation of several characteristic
exponents. The next paper, Bryant, Brown, & Abarbanel's (1990) 'Lyapunov
exponents from observed time series' adds the use of hypothetical higher
degree polynomial models during the convolution. See Brown, Bryant, &
Abarbanel (1991) for more details of this technique which provides more
reliable estimates of the entire spectrum, including negative Lyapuov exponents.
The bad news is that the process is seriously degraded by even small amounts
of noise. The last paper in this chapter, Parlitz' (1992) 'Identification
of true and spurious Lyapunov exponents from time series' addresses the
issue of getting spurious Lyapunov's from these techniques and suggests
trolling with time reversals which should reverse the sign of true Lyapunov
exponents. The good news is the suggestion that regularization and smoothing
data contaminated with noise can overcome some of the degredative effects
of noise.

The final chapter of this section contains four papers on *Periodic
Orbits and Symbolic Dynamics*. These papers provide attractor reconstructions,
return and Poincaré maps, and parameter maps from the analysis of
various physical systems.

Part III (pp. 206-395) deals with the practical problems of "Prediction,
Filtering, Control and Communication in Chaotic Systems". The editors note
that after the initial development of the concept of reconstruction of
the attractor (Taken's embedding theorem, 1981), estimating the correlation
dimension (Grassberger & Procaccia, 1983), reconstruction of the local
linear dynamics (Eckmann & Ruelle, 1985), and the approximation of
Lyapunov exponents (Sano & Sawada, 1985), the way was paved for the
development of ideas of prediction and nonlinear forecasting. The first
chapter (10), *Prediction*, contains four principal papers in this
development: Farmer & Sidorowich (1987) who "discuss the scaling of
prediction error for several artificial and experimental time series",
Casdagli (1989) who "compares local linear approximations to the dynamics
with global approximations, in the form of radial basis functions", Sugihara
& May (1990) who use local linear short term techniques and discuss
difficult issues of measurement error, and Sauer (1993) "using singlular
value decomposition to restrict the dynamics to the tangent plane of the
attractor, and using Fourier interpolation to counteract undersampling
difficulties." Most of these use known systems to calibrate the techniques
in addition to testing them on experimental data.

Chapter 11 on *Noise Reduction* is important because distinguishing
noise from deterministic chaos is difficult due to the continuous nature
of chaotic power spectra. Three papers by Kostgelich & Yorke (1988),
Grassberger et al. (1993), and Hammel (1990) are included.

Chapter 12, *Control: Theory of Stabilization of Unstable Orbits*
contains four papers which deal with the theory of stabilizing periodic
orbits in order to control undesirable chaotic orbits which depend on them.

Chapter 13, *Control: Experimental Stabilization of Unstable Orbits*
contains 6 papers which provide examples of such control in physical and
physiological systems such as a gravitationally buckling magnetoelastic
ribbon, thermally driven fluid convection, an electrical circuit, rabbit
cardiac tissue, a laser system, and the Belousov-Zhabotinskii reaction.
Garfinkel, Spano, Ditto, & Weiss (1992), state that chaotic systems
are "highly susceptible to control, provided that the developing chaos
can be analyzed in real time and that analysis is then used to make small
control interventions. . . . By administering electrical stimuli to the
heart at irregular times determined by chaos theory, the arrhythmia was
converted to periodic beating." Besides physiological issues of health
and disease, control theory could be considered for use in psychological
therapy, and in the control of organizations and social systems. Can we
borrow some of these techniques in our fields where noise becomes more
prominant? Time will tell.

Chapter 14, *Control: Targeting and Goal Dynamics* contains three
papers which go beyond the reduction of undesirable chaotic behavior and
tries to restrict the orbit to "a small region about some specified point
on the chaotic attractor" or to a specific dynamical goal.

Chapter 15, the final chapter, on *Synchronism and Communication*,
at first blush would seem to be the ultimate in esoteric physical systems
that, with their low noise conditions, might have the least applicability
to psychological and social systems. In living systems they can be of great
import, and do have potential applications of precision control. On the
other hand, these topics involve the aspects of psychological and social
systems, the self-organizational features, that we are most concerned with.
Thus these techniques might help in dealing with communication and harmony
within and between individuals or within and between components of social
organization. Becoming more familiar with some of these technical considerations
may yield some more explicit formulations to our metaphorical fancies as
well as some suggestions for psychological, social, and biological research.

For any of us interested in doing or reading research on chaotic systems, this book should become a cornerstone of our personal libraries and not left for an occassional visit at our institutional libraries. The clarity and elegance of the exposition is exceptional; the importance of the reprints undeniable.

Abarbanel, H. D. I., Brown, R., Sidorowich, & Tsimring, L. (1993).
The analysis of observed chaotic data in physical systems. *Reviews of
Modern Physics, 65*, 1331-1392.

Abraham, F. D. (1997). Nonlinear coherence in multivariate research:
Invariants and the reconstruction of attractors. *Nonlinear Dynamics,
Psychology, and Life Sciences, 1*, 7-33.

Brown, R., Bryant, P., & Abarbanel, H. D.. I. (1991). Computing
the Lyapunov spectrum of a dynamical system from observed time series.
*Physical Review A, 43*, 27-87.

Eckmann, J. -P., & Ruelle, D. (1985) Ergodic theory of chaos and
strange attractors. *Reviews of Modern Physics, 57*, 617-619.

Grassberger, P., & Procaccia, I. (1983). Characterisation of strange
attractors. P*hysical Review Letters, 50*, 346-369.

Moon, F. C., & Holmes, P. J. (1979). A magnetoelsastic strange attractor.
*Journal of Sound Vibration, 65*, 285.

Sano, M., & Sawada. Y. (1985). Measurement of the Lyapunov spectrum
from chaotic time series. *Physical Review Letters, 55*, 1082-1085.

Sauer, T., Yorke, J. A., & Casdagli, M. (1991). Embedology. *Journal
of Statistical Physics, 65*, 579-616.

Stewart. H. B. (1996). Chaos, dynamical structure and climate variability.
In D. Herbert (Ed.), *Chaos and the changing nature of science and medicine:
An Introduction. Conference Proceedings, 376* (pp. 80-115). New York:
American Institue of Physics.

Takens, F. (1981). Detecting strange attractors in turbulence. In D.
A. Rand & L. -S. Young (Eds), *Dynamical systems and turbulence,
Warwick, 1980. Lecture Notes in Mathematics, 898* (pp. 366-381). Berlin:
Springer-Verlag.

Wolf, A., Swift, J. B., Swinney, H., & Vastano, J. A. (1985). Determining
Lyapunov exponents from a time series. *Physica D, 16*, 285--317.