*Chaos and Time-Series Analysis. *By Julien Clinton
Sprott.

Review by ©Frederick David Abraham, 2003

**(1) Introduction: Intended Audience**

I wanted a book to advance my own knowledge of dynamics. I got
Sprott’s book to evaluate for this purpose, anticipating it might be, as he is
a master at dynamics and has a knack for explaining things clearly and
succinctly. It was a question as to its appropriateness for my own level of
mathematical background, which I consider not very sophisticated but rather
representative of many dynamical enthusiasts in our fields of biological,
psychological, and social sciences. So I am reviewing this book as to the
appropriateness of its use as a teaching/learning instrument in these fields,
and as to its success in clarity and completeness of meeting that challenge.

A fast skim through the book showed it to be comprehensive, but
possibly a bit daunting to those of us whose mathematics is limited to basic
algebra, finite mathematics, and introductory calculus, as it includes topics
such as Jacobian matrices of partial differentials, KAM tori, Lipshitz-Hölder
exponents, Legendre transforms, and so on (things that just scare you if you
haven’t encountered them before). But closer examination of each of them makes
them seem quite tractable, and when you return after the first fast skim though
the book, you read the preface and it sets your mind at ease, and tells you
this is a book that will provide a great, complete, and manageable foundation
in chaos theory and data analysis.

In his preface, Sprott explains that the book arose from a survey
course he taught for upper-level undergraduate students, graduate students, and
“other researchers, representing a wide variety of fields in science and
engineering” As he puts it,

“This book is an
introduction to the exciting developments in chaos and related topics in
nonlinear dynamics, including the detection and quantification of chaos in
experimental data, fractals, and complex systems . . . [mentioning] most of the
important topics in nonlinear dynamics. Most of the topics are encountered
several times with increasing sophistication.” The emphasis is on concepts and
applications rather than proofs and derivations and is for “the student or
researcher who wants to learn how to use the ideas in a practical setting,
rather than the mathematically inclined reader who wants a deep theoretical
understanding.

“While many books
on chaos are purely qualitative and many others are highly mathematical, I have
tried to minimize the mathematics while still giving the essential equations in
their simplest form. I assume only an elementary knowledge of calculus. Complex
numbers, differential equations, matrices, and vector calculus are used in
places, but those tools are described as required. The level should thus be
suitable for advanced undergraduate students in all fields of science and
engineering as well as professional scientists in most disciplines.” (From the
preface.)

It delivers on all these promises. Further, it is ‘hands-on’, with
practical exercises and a programming project in each chapter. (Any language
and computer platform will do; spreadsheets or math packages such as *Maple*
or *Mathematica* may also be used if one is already capable with them. If
one is not fluent in a programming language he suggests *PowerBASIC*—DOS
or Windows versions—for its ease of learning). I found that having a dynamics
program, such as *Berkeley Madonna* (Macey, Oster, & Zahnley, 2000)
that will already solve equations with graphic displays was most useful for
additional explorations. Thus Sprott’s book is most suitable for systematic
study, but as with most textbooks, it can also serve as a useful reference work
in your library. You may also find the programs by Sportt and Rowlands, useful
supplements to the text, *Chaos Demonstrations* (1995) for examples of
several programs, and *Chaos Data Analyzer* (1995), for data analyses.

**(2) Contents**

The 15 chapters cover the following topics: Introduction,
One-dimensional maps, Nonchaotic multi-dimensional flows, Dynamical systems
theory, Lyapunov exponents, Strange attractors, Bifurcations, Hamiltonian
chaos, Time-series properties, Nonlinear prediction and noise reduction,
Fractals, Calculation of the fractal dimension, Fractal measure and
multifractals, Nonchaotic fractal sets, and Spatiotemporal chaos and
complexity. In addition, there are three fantastic appendices. The first is a
catalog of Common chaotic systems, there being 62 given, in five categories:
noninvertible maps (12), dissipative maps (11), conservative maps (6), driven
dissipative flows (8), autonomous dissipative flows (20), and conservative
flows (5), each with a graph, equations, typical parametric values and initial
conditions, Lyapunov exponents, Kaplan-Yorke dimension, correlation dimension,
and a major reference. The second appendix gives useful mathematical formulas
in ten categories: trigonometric relations, hyperbolic functions, logarithms,
complex numbers, derivatives, integrals, approximations, matrices and
determinants, roots of polynomials (including the Newton-Raphson method), and
vector calculus. And the third appendix is a list of relevant journals. The
bibliography of 715 entries covers everything from Abarbanel (1996) to Zipf
(1949). Ruelle and Grassberger are the most cited senior authors (9 each), with
Bak, Theiler, L. A. Smith, Grebogi, Arnold, Mandebrot, Lorenz, Sauer, and
Schreiber also having 5 or more citations each. The oldest citation award goes
to Huygens, 1673. There is an excellent support page with color versions of many
of the figures and much supplementary information (including answers to some of
the exercises) at http://sprott.physics.wisc.edu/chaostsa/. It is
continually updated, and contains many important links to other related pages
of both his (such as the pages for the course that spawned the book) and on
other websites.

** **

**(3)
****Brief explorations of the chapters**

The ** Introduction** (chapter 1) is fairly brief but
reveals both some major strategies as well as some subtle ones employed in the
book. It mentions several examples of chaotic astronomical, physical, and
nonphysical systems, usually quite briefly, many with photographs and diagrams.
For a few he gives introductory equations, such as a second-order differential
equation for the driven pendulum. These include some parametric values which
yield chaos, and for one, its Poincaré sections. There is also a section on
electrical circuits, which are of value whether one is intrinsically interested
in electronic circuits or not. They have been tools for investigating dynamical
systems since van der Pol in the 1920s. Since some can be easily and
inexpensively constructed, they provide nice demonstrations for lecturing and
for comparing analog and computer results. I found not only a color enlargement
of one of the analog devices which made it easier to understand the circuit,
and this led to questions about further details for the circuit found on
another page (those in the book provide enough for one knowledgeable in basic
circuits but I needed a little more help).

Another example of supplementary information on the website, was an
expansion of biographical notes on Poincaré. A nice aspect of the book is the
inclusion brief footnotes on individuals, such as *predicting* the weather is easy, but predicting it *correctly*
is not”; a *torus* described as resembling an American, not a British
doughnut). After this very brief, skeletal introduction you are challenged with
14 exercises and a programming project with the logistic equation (I first did
that with an ancient dialect of Basic back in the mid 80s on an Apple II; Paul
Rapp also had done it and suggested to me adding a statement to produce tones,
which is a nice enhancement, discussed later in Sprott’s book as *sonification*,
a method for hearing bifurcations and attractor patterns). Programming the
logistic equation prepares you for the next chapter on ** one dimensional
maps** (chapter 2) where the logistic equation is the main object of
study, and where the programming exercise is taken up with an exploration of
its bifurcation diagram. Similarly, the nice thing about the electronic
circuits that Sprott shows is that their output can be displayed on an
oscilloscope and a cheap speaker. But the first exercise. Whew! “Derive a set
of four first-order ordinary differential equations whose solution would give
the three-body trajectory of Figure 1. Well the figure reduces the possibilities
to the simplest ones usually considered, but I suspect this problem, over which
many have spent much time since Poincaré first stumbled on it, was offered to
get one to wrestle with the problem rather than get to a solution. I found
myself trying to cheat already. Not having Sprott’s demonstration program
currently running on my computer (

** Nonchaotic multi-dimensional flows** (chapter 3) moves
on to the continuous, multivariate case where time moves continuously rather
than in discrete steps. It starts with a simple first-order, explicit, linear
differential equation for population growth and decay. For those of us who have
hassled over the meaning and history of some terms in dynamics, he gives most
synonyms and discussion of some of these etymological issues, here and
throughout the book. He uses this growth/decay model to provide the distinction
between maps and flows, which he summarizes in a table. Continuing this
comparison, he next takes up the logistic differential equation, (Verhulst,
1845). Then he proceeds to some multi-dimensional models: circular motion, the
simple harmonic oscillator (to be taken up again in chapter 8 on Hamiltonian
systems), the driven harmonic oscillator (from the introductory chapter), the
damped harmonic oscillator, the driven damped harmonic oscillator, and the van
der Pol equation. Under the driven harmonic oscillator, he takes up an
important topic, namely that of converting a system of nonautonomous
differential equations into an autonomous system, making it amenable to
solution for representation in state space. Such clear explanations are not
easily found elsewhere. The chapter finishes with an explanation of the various
numerical methods of solving equations, the Euler, Leap-frog, Runge-Kutte
second-order and fourth-order methods, and their advantages and disadvantages.
This information is essential for anyone studying geometric properties of
trajectories, attractors, and portraits[1]
(the set of all possible trajectories that a system of equations can generate
although the portrait is usually shown with just a few representative
trajectories and critical features of the portrait such as limit sets, saddles,
and separatrices).

** Dynamical systems theory** (chapter 4) is the heart of the book.
Saying that it covers two-dimensional equilibria, stability, the damped
harmonic oscillator (again), saddle points, area contraction and expansion,
nonchaotic three-dimensional attractors, chaotic dissipative flows (including
several well known ones such as those by Lorenz, Rössler, van der Pol, Ueda,
some even simpler ones—some 19 are summarized in a table, and some jerk systems)
would hardly reveal the importance of the concepts included. It finishes with
‘shadowing’ on the relationship between a true trajectory (theoretical or
experimental) and the computed one. These provide the foundations for chaotic
dynamical system theories. Here you get the no-intersection in two dimensions
theorem (Poincaré-Bendixson theorem, Hirsch & Smale, 1974) which “is a
cornerstone of dynamical systems theory”. Here also is where the use of of
eigenvalues (characteristic exponents and multipliers) and the use of the
determinant of the Jacobian are clearly developed. You need never have heard of
these before. Previously I have Googled for such information. If I may be
permitted (and even if I am not), I cannot do better to characterize the role of
this chapter than to quote its first two paragraphs:

“We have seen
examples of dynamical systems for iterated maps and continuous flows. Maps are
simpler to analyze numerically and have a rich variety of dynamical behaviors,
even in one dimension. By contrast, except in one dimension the solutions
cannot do anything more complicated than grow or decay to an equilibrium point.
Even in two dimensions, the most complicated behavior is growth or decay to a
periodic limit cycle.

“In this chapter
we will develop a more general theory of dynamical systems and extend the ideas
to three dimensions where the flows can exhibit chaos. Although it is often
difficult to calculate the trajectory, much can be gained from identifying the
equilibrium points and examining the flow in their vicinity. Since the flow is
usually smooth near these equilibria, we can make linear approximation and use
these ideas developed in the previous chapter. From this knowledge, we can
construct a good qualitative picture of how the flow must behave throughout the
entire state space. The material in this chapter is slightly more formal than
usual and makes some use of complex numbers and matrix algebra.”

** Lyapunov exponents** (chapter 5) are important for depicting
the converging and diverging properties of a chaotic attractor. These are
closely related to the eigenvalues upon which they depend but important
differences are noted and are summarized in a table. There are important
distinctions and relationships between local and global Lyapunov exponents.
Examples for many systems are given and also numerical methods for their
computation. The concepts of a 3-torus, hyperchaos, and non-integer dimension
emerge from consideration of these systems.

The Kaplan-Yorke (Lyapunov) dimension is defined by interpolating
between the number of the largest Lyapunov exponents summing to a positive
number, its *topological dimension*—the attractor would expand (i.e., not
exist) if unfolded in the dimensions represented by those exponents—and the
minimum number for the sum to be negative—the attractor would contract (i.e.,
exist) if unfolded in the dimensions represented by those exponents. The
non-integer result of this interpolation gives a measure of the dimensionality
of the attractor. “The dimension of the attractor is the lower limit on the
number of variables required to model its dynamics.” While related, the
Liapunov[2]
exponents measure behavior of a system over time; the dimension measures the
complexity of the attractor. For a chaotic system of differential equations, at
least one exponent must be positive (for the divergence necessary for chaos),
one at least one must be zero, and at least one must be negative to insure the
sum is negative and that the volume is contracting (Abarbanel, 1996, p. 27).

** Strange attractors** (chapter 6) begins with 12 properties
such as limit set, invariance, stability, sensitivity to (divergence from)
nearby initial conditions, and yes, aesthetics. He develops the idea of the
probability of chaos with a dynamical system (the proportion of parameter space
yielding chaos) and search techniques for finding values yielding chaos. Sprott
is well known for developing this approach (see his earlier book or his web
site). He is also known for programs that display such attractors in 3D (a
plane with color for the third dimension). His development of such programs to
display them in various formats and with their statistical properties not only
illuminates these properties, but makes them available for use in research such
as in his studies of the aesthetics of the attractors (many URLs are given). I
remember about 1998 that a friend, Elliot Middleton, informed me of such a
program on Sprott’s site which I immediately downloaded, recognizing its
potential for doing a study of perception as a function of attractor dimension.
I was mesmerized for hours by the beauty of the attractors, and since have used
the program in psychophysical research. The chapter also includes a discussion
of the routes to chaos, basin boundaries, fractal basin boundaries, and
structural stability.

I consider ** Bifurcations** (chapter 7) the most important
property of nonlinear systems. Sprott considers them important “because they
provide strong evidence of determinism in otherwise seemingly random systems,
especially if the parameters can be repeatedly changed back and forth across
the” bifurcation point. I consider them important from a slightly different
perspective, that of a system that can explain differing patterns of
experimental data whose connection may not have been previously noticed and for
which different models might have been suggested. Both perspectives represent
the same parsimonious point of view. In addition to the usual topics and forms
of bifurcation, such as folds, transcritical, pitchfork, flip, Hopf, Neimark,
and blue sky (R. H. Abraham & C.D. Shaw, 1992; R. H. Abraham & Stewart,
1986, and on the cover of Thompson & Stewart, 1986), it also includes
homoclinic and heteroclinic bifurcations, examination of Lyapunov spectra as a
function of control parameters especially with bifurcations that involve
transient chaos, and crises.

Conservative systems (ideal systems that do not dissipate energy)
include familiar examples like ideal frictionless pendula for which Hamiltonian
equations may be used for analysis (** Hamiltonian chaos**, chapter
8). It also includes simplectic maps, which are important for systems where
“the numerical methods may not precisely conserve the invariants”. I liked this
section because it also explains the mathematical relationship between the flow
(in n dimensions) and the map (in n-1 dimensions) to approximate its Poincaré
section conservatively.

Since dynamical systems evolve in time, ** Time-series properties**
(chapter 9) are the heart of the analysis of data obtained from them. Often
combining traditional linear methods with nonlinear methods helps to illuminate
a dynamical process. Also because there are some methodological similarities of
some aspects of the analyses, knowing linear methods helps to understand the
nonlinear methods. Sprott provides a quick review of traditional linear methods
(including topics of stationarity, detrending, noise, autocorrelation, Fourier
analyses). Comparison of a noise signal with one produced by a one-dimensional
map illustrates the use of surrogate data (Monte Carlo methods) and return maps
to determine if your data contain deterministic as well as stochastic
information. The final part of the chapter introduces time-delay embeddings
used for attractor reconstruction and for “determining the dimension of an
attractor”. The computer project for this chapter involves running an
autocorrelation function on data generated by the Lorenz equations.

“One of the most important applications of
time-series analysis is *prediction* (or *forecasting*)”, that is ** Nonlinear
prediction and noise reduction** (chapter 10). Myself, I am more
interested in models than prediction, but these turn to be fundamentally
related, and thus this chapter is exceptionally exciting. “In some cases,
prediction entails developing a global dynamical model for the data, which may
illuminate the underlying mechanisms.” Therefore, comparison of models with
data lies at the heart of both prediction and evaluation of models. After
showing how the linear methods of autoregression cannot help with prediction of
chaos, and also showing the practical limitations of nonlinear methods,
including both methods that use equations and those that use comparison of
nearby trajectories, Sprott then shows how a method called

Prediction depends on evaluating divergence of nearby trajectories,
which, for data not modelled by equations, means measuring the rate of
divergence and summarizing them in Lyapunov exponents. Sprott explains several
methods of doing this and compares their relative advantages and drawbacks.

The subject of embedding from the end of the previous chapter (9) is
examined further in this chapter with the discussion of *false nearest
neighbors*, a method for estimating the optimal Cartesian embedding
dimension by systematically increasing the embedding dimension until separation
of nearby points no longer occurs (Kennel et al, 1992; Abarbanel et al., 1993).
While traditionally the main use has been to get a best view of the attractor,
to help determine when other measures of fractal dimensionality have saturated
(become asymptotic), and to estimate the likely number of variables involved in
the dynamical system under investigation, Sprott points out other uses, such as
evaluating if sufficient points are in a neighborhood to support prediction.
Stewart has described the extension of the technique to multivariate data[3]
(Stewart, 1996; also presented in Abraham, 1997). The related subject of
recurrence plots is also taken up, along with a derivative (actually
integrated) space-time plot method, developed by Smith (1992) and Provenzale *et
al.*, (1992). Sprott evaluates various computational algorithms suggesting
Schreiber’s (1995) as “a good compromise between simplicity and efficiency”.

*Principal component analysis* (also known by several synonyms including
*singular value decomposition* and *Karhunen-Loéve decomposition*) is
useful for noise reduction, in estimating dimension, and in building model
equations with polynomials.

Among *artificial neural network predictors*, *single-layer
feed-forward networks* are mentioned for their computational and conceptual
ease and for the large literature on optimization and training. Two methods are
mentioned: *multi-dimensional Newton-Raphson* (repeat the calculation of
the error until it “stops changing or you lose patience”) and a simplified
variant, *simulated annealing*. I might mention a really nice and
innovational combination of the methods of artificial neural nets (but using
multi-layer feedback) with the dynamical analysis of real neural nets in
molluskan brains in the work of Mpitsos (2000).

Chapter 10 thus covered a lot of topics in a rather short space, and
thus while providing a great introduction to them was sometimes rather brief,
and may require additional reading or the use of Sprott’s web page and other
links and references. I found that sometimes following indices to the matrix
algebra required a bit of such effort, but that is related to the atrophy of my
rudimentary matrix algebraic skills acquired quite some time ago.

Sprott takes up ** Fractals** (chapter 11) from a broader
view than simply that of its “geometric manifestation of chaotic dynamics”.
“Dynamical systems are only one way to produce fractals.” Some of these include

A few of the methods are presented for the ** Calculation of the
fractal dimension **(chapter 12). The Kaplan-Yorke dimension works when
the equations are known. Estimating the Lyapunov spectrum for empirical
time-series is difficult, so other methods, of which there are many, may be
used. These include the

What do you do if your correlation dimension isn’t converging
nicely? Are your various fractal dimensions disparate? Is your fractal not very
homogeneous? Is that what is getting you down? You need ** Fractal measure**
for your

** Nonchaotic fractal sets **(chapter 14) is a discussion of

** Spatiotemporal chaos and complexity** is the final
chapter (15). Spatiotemporal chaos means that chaos is exhibited over spatial
as well as temporal dimensions. Complexity refers to broad class of subjects
not unified by any theory, but many of them depend on dynamical system
concepts. The term includes not only chaos, fractals, and neural networks, and
artificial life, but also complex dynamical systems which in turn includes
cellular automata, lattices, and self-organization. Cellular automata, for
example, are networks of dynamical systems where nearby neighbors are coupled
(share variables). Some special systems that have had widespread deployment in
many sciences include

His final concluding remarks are of special interest to me, as they
reflect the way I used to finish off many of my articles, issues also
emphasized by Christine Hardy (1998), and they deal with issues of free will
and responsibilities, social, ecological, and for Sprott, aesthetical, for a
more beautiful world.

** **

**(4)
****Conclusion and recommendations**

It should be clear by now that I consider this book an indispensable
addition to my bookshelf. I intend to use it to get a good foundation in almost
all aspects of dynamical systems theory, especially, of course, chaos theory
and data analysis. For those with the minimum recommended mathematical
background (elementary calculus and some matrix algebra), some of the review
might still have sounded a bit forbidding, but as Sprott promises, the
explanations and support material will fill in the necessary updating of your
mathematics. In most instances the explanations were clear, and covered almost
all related aspects of each subject. It was very impressive. Occasionally I
found myself struggling with keeping the meanings of indices straight with the
matrix algebraic expressions, but much less so than in any other similar book
that I have encountered. The later chapters sometimes tried to compress so many
topics into them, that the compression demanded some supplementation from the
web sources or other books. I suspect
that if you do the exercises and programming projects, that you will find them
very challenging and time consuming, but very rewarding. The extensive and
evolving website back-up makes the book unique and even more valuable. It stays
up to date as the field evolves. The book is thus perfect for self-instruction,
or for use as a classroom textbook, and of course, as a reference work for
workers in any field of science.

I’d like to end the review with my recommendations for a basic
bookshelf. For learning dynamics, the two books I consider most important are
this book of Sprott’s, and the book that visualizes most of basic dynamics,
that of Ralph Abraham and Christopher Shaw (1992). Between them they give the
basic elements of theory and data analysis. For additional details concerning
theory, Schroeder (1991), and Thompson & Stewart (1986) are excellent, and
for data treatment, I like Abarbanel (1996), Abarbanel *et al*. (1993),
Kantz & Schreiber (1997), Rapp (1994), and Ott, Sauer, & Yorke (1994)
which has excellent introductory chapters followed by reprints of many classic
papers. I cannot say that there may not be others as good or better, but these
are what have found their way onto my bookshelf and proven most useful, and
what I use to struggle toward better understanding of dynamics. Of the many
exceptional books on fractals, all the Mandelbrot and Peitgen books are great,
but the best foundation for me has been Peitgen, Jürgens, & Saupe (1992). Give me one more
lifetime, and I might get it. Thanks, Dr. Sprott.

**(4) References**

Abarbanel, H.D.I.
(1996). *Analysis of observed chaotic data*.

Abarbanel, H.D.I.,
Brown, R., Sidorowich, J.J., & Tsimring, L.Sh. (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.

Abraham, R.H.,
& Shaw, C.D. (1992). *Dynamics: The geometry of behavior* (2^{nd}
ed.).

Abraham, R.H.,
& Stewart, H.B. (1986). A chaotic blue-sky catastrophe in forced relaxation
oscillations. *Physica 21D*, 394-400.

Hardy, C. (1998). *Networks of Meaning. *

Hirsch, M.W.,
& Smale, S. (1974). *Differential equations, dynamical systems and linear
algebra*.

Kantz, H., &
Schreiber, T. (1997). *Nonlinear time series analysis*.

Kennel, M., Brown,
R., & Abarbanel, H. (1992). Determining embedding dimension for phase-space
reconstruction using a geometrical construction. *Physical Review A* *45*,
3403-3411.

Macey, R., Oster,
G., & Zahnley, T. (2000). *Berkeley Madonna User’s Guide*, v. 8.0. UC
Berkeley: www.berkeleymadonna.com.

Mpitsos, G.J.
(2000), Attractors: Architects of Network Organization? *Brain, Behavior, and
Evolution*; *55*, 256-277.

Ott, E., Sauer,
T., & Yorke, J.A., (eds.). (1994). Coping with chaos: Analysis of chaotic
data and the exploitation of chaotic systems.[6]

Peitgen, H.-O.,
Jürgens, H., & Saupe, D. (1992). *Fractals for the classroom, Parts one
and Two*.

Rapp, P.E. (1994)
Aguide to dynamical analysis. *Integrative Physiological and Behavioral
Science*; *29*, 311-327.

Schreiber, T. (1995).
Efficient neighbor searching in nonlinear time series analysis. *International
Journal of Bifurcation and Chaos*, *5*, 349-358.

Schroeder, M.
(1991). *Fractals, chaos, power laws: Minutes from an infinite paradise*.

Smith, *Journal of the Royal Statistical Society Series B —
Methodological*., *54*, 329-352.

Sprott, J.C.
(1993). *Strange attractors: creating patterns in chaos*.

Sprott, J.C.,
& Rowlands, G. (1995). *Chaos data analyzer: the professional version*.

Sprott, J.C.,
& Rowlands, G. (1995). *Chaos demonstrations*.

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*, (80-115). Woodbury:
American Institute of Physics.

Thompson, J.M.T.,
& Stewart, H.B. (1986). *Nonlinear dynamics and chaos*.

Vaerhulst, P.F.
(1845). Récherches mathématiques sur la loi d’accrossment de la population. *Noveaux
Memoires de l’Academie Royale des Sciences et Belles-Lettres de Bruxelles*, *18*,
1-45.

[1] I omit the term ‘phase’ as a qualifier for ‘space’ and ‘portrait’ except when in the state space position of an object is a function of velocity (and other higher moments) or momentum, but common usage does not generally recognize this historical lineage as a restriction.

[2] I follow
Sprott’s transliteration, but otherwise use Liapunov as I saw it on his old
office at

[3] Here is
where the reviewer—honestly, I tried to avoid it—shamelessly inserts reference
to a paper of his own, which while not original in that it presents the work of
Kennel *et al*., 1992, and Abarbanel *et al*. 1993, is reasonably
clear; Abraham, 1997)

[4] An interesting footnote in this chapter concerns the use of a collage system (Barnsley & Hurd, 1993) which can be used to create .FIF high compression image files, first used by Microsoft enabling the whole Encarta encyclopedia to fit on one CD.

[5] Reviewed at www.blueberry-brain.org\dynamics\hardy-testimonial.htm

[6] Reviewed at www.blueberry-brain.org\dynamics\osy.htm