Department of Applied Mathematical Studies,

University of Leeds.

email:
carsten@amsta.leeds.ac.uk.

The explosive combination of a strong interest in nonlinear science and a lack of common sense in economical decisions has left me with a considerable collection of books related to nonlinear science and dynamical systems in particular. After becoming a pop-subject the field of dynamical systems was inundated with new text books, monographs, reprint collections etc. Consequently the book market is now a jungle and newcomers may want some advice before investing too much in fancy, but colourful, books. On the other hand they probably won't. Too late, because here is my $0.02's worth.

- Introduction
- The Classics
- Textbooks
- More Specialised Books
- Numerical Aspects
- Concluding Remarks
- Appendix: Contents of Books
- Appendix: Bibliography

The first I heard about nonlinear science was something about strange pictures which under magnification appeared very different; apparently these pictures could be magnified repeatedly and keep showing new interesting structures. In the university book shop I saw some of these pictures in the book by Peitgen and Richter[25]. Of course, in those days, being a student, I had plenty of money for food, rent and other inessentials. Needless to say, I bought the book.

In the following I attempt to give a superficial, but unfair, survey of some of the books in my collection. For good measure I should mention that I am unreasonably biased for dissipative systems and somewhat indifferent toward subjects such as Hamiltonian systems, complex dynamical systems, and fractals. So if you have a problem with that, well, that is your problem.

Few books in the general area of nonlinear science have risen to
the status of undisputed classics.
One that have achieved this fame is undoubtedly (and deservedly)
**Guckenheimer and Holmes**[12].
Their coverage of dynamical systems is thorough,
although brief, making it demanding to read.
Don't read this book from cover to cover
(few books are designed to be used this way).
However, if you need to look up a definition or find an example,
chances are you will find it here.
As with most of the older books this book is mainly concerned with
differential equations, but most results, eg
the centre manifold theorem, are also stated for maps.
A second edition, containing no major alterations, has appeared.
It remains good value for money.

Slightly pre-dating Guckenheimer and Holmes is
**Lichtenberg and Lieberman**[19].
Their emphasis is very different, focusing on mechanics, in particular
Hamiltonian systems.
Classical subjects such as perturbation theory are also covered.
Only the final chapter deals with dissipative dynamical systems;
the coverage is, however, quite decent even by todays standards (or possibly
because of), eg the description of Melnikov's method is not bad.

I should also mention **Thompson and Stewart**[32].
Although very different stylistically from Guckenheimer and Holmes,
it is a nice first text in dynamical systems.
As the subtitle of the book suggests, it presents a geometrical view
on dynamical systems, and gets away with it quite successfully.
It helped me understand global bifurcations, such
as crises, and in the process
taught me the importance of a good geometric intuition.
The book takes the differential equation view, but contains
two chapters on bifurcations and chaos in maps.
These chapters are non-rigorous but neatly intuitive; a good read.
Some comments on Sarkovskiis theorem is a bit, hmmm (p. 170, 1st ed.).
A second edition (supposedly available) includes a useful
glossary of geometrical dynamics.

One of my favourites is **Devaney**[8].
This deals mainly with iterated maps and
the presentation is very rigorous.
The textbook starts from scratch with virtually no prerequisites
and very rapidly arrives at an analysis of chaotic motion.
This is a winner with students (at least it was with me).
Having said this, the book is
not an easy read; some commitment and determination are required.
The second edition is divided into three main parts:
the first deals with one-dimensional dynamics;
the second treats higher dimensional theory, though not
beyond the third dimension; and
in the final part complex dynamical systems, ie primarily
Julia sets and the Mandelbrot set, are covered.

If I had to spend time on an isolated desert island with
a single book from my collection, my choice would probably be
**Abraham and Shaw**[1] (previously published in
four separate volumes at an obscene price).
This book is quite close to being ingenious.
It presents a very intuitive approach to dynamical systems,
explained through a mother lode of drawings which develop from
from simple two-dimensional trajectories to the complex
structure of stable and unstable manifolds in three-dimensional phase space.
This book also contains a large amount of historical information,
a particularly interesting part is the account of the
origin of bifurcation analysis.
However, though the style is very relaxed and non-rigorous
you shouldn't be fooled: it is not a completely straightforward read,
and it covers a lot of ground including structural stability,
explosive bifurcations and Birkhoff signatures.

New textbooks keep appearing, and in the following I shall comment on a few of these, as well as some not so new ones. The presentation is not chronological, but by subject matter covered.

**Wiggins**[33] has been very productive and now has
five books to his name, only three of which will be mentioned here
(I only have the first four).
Wiggins follows the tradition of Guckenheimer and Holmes in
material, but uses a more modern style.
The treatment is very rigorous, though exclusively centered around
differential equations.
The section on symbolic dynamics is short but nicely written.
The final chapter deals with global perturbation methods,
something not often considered in our educational system.
The level of difficulty is quite high (roughly post-postgraduate level)
and to solve this problem Wiggins wrote another, more
entry level, dynamical systems text[34].
This book remains one of the best
dynamical systems texts; it is very complete and yet fairly easy to read.
Certain sections are virtual copies from[33],
but never mind, as they were good in the first place.
Note that there is quite a few typos, eg Fig. 1.1.2, so don't
take everything at face value.

**Arrowsmith and Place**[2]
provide a very thorough account of most standard aspects of
dynamical systems theory.
There is a considerable amount of material on Hamiltonian systems;
the last chapter (over 90 pages) deals with area-preserving maps and
their bifurcations.
Each chapter ends with a large number of exercises (hints are given towards
the end of the book).
This is a good textbook on dynamical systems, although it is
far from elementary and biased towards Hamiltonian systems.

**Hale and Koçak**[14]
have written a clear exposition of
introductory dynamical systems theory.
It covers standard material such as
stability and local bifurcations of
flows and maps, global bifurcations are only touched upon.
The book is well-written and, dare I say, very pedagogical.
There are many good examples and illustrative figures
(though many of these are clearly produced from low resolution screen dumps).
This may well be the best introductory text considered in this review,
assuming that you can forgive the authors the last three chapters
which are too superficial to benefit students.

**Perko**[26] presents a very clear exposition of
the differential equations side of dynamical systems.
The book is very complete and almost all proofs are included,
which makes it an excellent reference book, as well as a
good textbook.

**Glendinning**[9]
has written a book centered
around differential equations. (This was reviewed in
UK Nonlinear News 1.)
Quite a bit of the book is spent on local analysis and one chapter
covers
classical subjects, such as perturbation theory.
Only the final two chapters (out of 12) deals with chaos and
global bifurcations.
The former primarily deals with one-dimensional iterated maps.
The exposition of the latter is a bit non-standard and frankly is not
my first choice.

**Gulick**[13]
is very similar in material, and partly presentation,
to Devaney; I very much prefer the original.
The best part of Gulick is the section on iterated function systems,
where Hausdorff distance, completeness, the contraction mapping theorem,
and Cauchy sequences etc are treated.

Another Devaney-look-alike is **Holmgren**[15].
Holmgren considers virtually the same basic material as Devaney,
ie one-dimensional and complex maps.
The level is more introductory than Devaney's, and many of the
more advanced subjects
considered by the latter are not mentioned.
The general presentation is very good, and there are many illustrative
figures.
There is a good section on Newton's method for cubic polynomials.

**Crownover**[6] places an emphasis on fractals, but
includes quite a bit of chaos theory as well.
The dynamical systems part contains material on both
one-dimensional maps (a traditional treatment)
and on maps in more general metric spaces. An example of one of the
more abstract mapping covered is
iterated function systems.
Some recent results relating to the definition of chaos are mentioned.
Complex dynamics are mentioned, although rather superficially.

In **Robinson**[27]
chapters two and three are Devaney-like in subject matter, except that
more concepts are considered, often in a setting of a general metric space.
This is followed by standard material on differential equations and
a smallish section on bifurcations.
Later, a large amount of non-standard material is covered,
topological entropy receives considerable attention.
In general, the text considers the abstract side of dynamical systems.
On the whole Robinson gets away with it, and although at a
very high level of difficulty it is commendable.

**Moon**[21] makes a very informal presentation
of basic nonlinear dynamics such as period-doubling cascades,
intermittency and fractals (including fractal basin boundaries).
The most technical aspect is the mentioning of Melnikov's method.
A nice aspect of the book is that it very often compares
theory with experimental results.
A new and improved version is available, but since I wasn't
particularly impressed by the first version I haven't checked it out,
aside from the usual glance in the book store.

**Schuster**[29] presents a large number
of
dynamical systems results from a non-rigorous perspective.
He covers the standard routes to chaos,
eg period-doubling cascades and intermittency.
This is the sort of book that contains a large number of pictures.
A new edition is on the market, containing new appendices.

A newer book following the same tradition is
**Ott**[22].
Ott covers many, relatively new, subjects
that are often not mentioned in dynamical systems texts:
control of chaos, crisis phenomena, nonattracting chaotic sets,
multifractals, quantum chaos.
Some of these sections are on the short side.
Well-written and amply illustrated (sorry, no colour pics).
It is available in a reasonably cheap paperback version,
just what our students need.
Good value for money and an excellent first text
if mathematics is not of the essence.

The first half of **Marek and Schreiber**[20] presents
an informal description of dynamical systems theory, including
fractals and chaos theory.
A section on numerical methods is included, and an appendix
contains Fortran code for continuation.
In the second half a large number of examples are provided,
often with some bifurcation analysis involved.
Most of the examples are chemical systems,
and many experimental results are mentioned.
Each chapter contains a large number of references.

**Bai-lin**[3] presents an unusual, but interesting,
description of dynamical systems, with primary focus on symbolic dynamics.
This book contains a great deal
of valuable information on symbolic dynamics.
In the second part of the book more standard subjects are treated, eg
dimensions, entropies, Lyapunov exponents and multifractals.
The first part is definitely the best.
A plus point is that practical numerical aspects are discussed several times
in the book, but do not expect state-of-the-art methods.
The book is not intended as a textbook, but even so the
presentation and organisation is lacking.
I have not seen many of the subjects treated in this book in books
before.

**Iooss and Joseph**[16] treat stability and bifurcations
of solutions to differential equations.
The exposition is very detailed and complete, but
is too technical for my taste; it contains very few good examples.
As a reference book on local bifurcations it may serve a purpose.
The vey short final chapter outlines stability and bifurcations
in conservative systems.
Why the title of this text contains the word `elementary' is
beyond me; the text is very demanding, for example, on page one
the general evolution problem is stated in nonlinear operator form.
A much more comprehensive treatment of bifurcation theory
from the point of view of singularity theory can be found in
**Golubitsky and Schaeffer**[10] and
**Golubitsky, Stewart and Schaeffer**[11].

**Kuznetsov**[18]
(see Alan Champney's review in
UK Nonlinear News 2)
has written a quite comprehensive text on bifurcation theory with some
emphasis on practical aspects.
This book contains
a wealth of valuable information about global and
codimension-two bifurcations.
Another positive aspect is the final chapter which addresses
numerical analysis of dynamical systems;
I would have liked this chapter to be even longer, but never mind
as it is the best found in a standard text (so far).
The book contains some of the best illustrations I have seen for a long time,
with the possible exception of Abraham and Shaw,
but that is a somewhat high standard.
It is an excellent reference book that you should have access to, but
it should not be read cover to cover.

**Ruelle**[28].
This book is in the French tradition, need I say more?
The treatment of just about everything in this book is very brief,
however, it does cover a lot of ground.
Only consult this book if you know the subject already.

If you should ever find yourself with a lack
of purpose in life, then try to understand every
detail in **Collet and Eckmann**[5].
That should keep you busy for a while.
Collet and Eckmann focus entirely on the dynamics of
one-dimensional mappings, apart from the first part which is merely
motivation.
The two main parts of the book treat kneading theory for
individual iterated maps and for families of maps.
Ergodic theory is often ignored or mentioned superficially in dynamical
systems texts, here it is addressed in some detail.
If you find reading this book easy going you are ready for
**de Melo and van Strien**[7].
This is `the`
authoritative source on one-dimensional dynamical systems.
This covers the works:
combinatorial, topological, ergodic as well as smooth theory.
Everything is covered from the (relatively) simple circle diffeomorphisms
to renormalisation theory.
Taking on this book may well earn you the Purple Heart.
Excellent for references, but reading requires a solid background in maths.
**Block and Coppel**[4] is a slightly earlier book dealing
with similar material.
This book also deals exclusively with one-dimensional dynamics,
but from a topological viewpoint, containing no information on ergodic theory.
The exposition is very complete including almost all proofs.

**Palis and Takens**[23] present a wealth of results on
homoclinic bifurcations, including some of the more recent results
on homoclinic bifurcations and strange attractors.
Beware, this is a pure mathematics book and is extremely challenging.

In **Sparrow**[31] you will find a quite good exposition
of one of the most classical dynamical systems: The Lorenz equations.
Although focused on an extensive case study, you may still learn
much of the general theory from this book;
stability, Lyapunov functions, global bifurcations and symbolic dynamics
are all covered.
Don't let it be your first book on dynamical systems though.

The last **Wiggins**
book to receive a mention in this exclusive review
is his book on chaotic transport[35].
In this Wiggins
takes the view that in many dynamical systems the transient
behaviour is as important, if not more so, than the steady state behaviour.
The text is very specialised and consequently not casual reading.
Quite a lot on Hamiltonian systems and examples from fluid mechanics (sigh!).

The before mentioned books all suffer from the complete lack of any description of numerical aspects of dynamical systems. Luckily, a few books have appeared that deal with numerical methods for dynamical systems.

**Kubícek and Marek**[17] was one of the first texts
to consider numerical bifurcation analysis.
Their treatment is not exactly modern, but quite detailed and a large
amount of
Fortran code can be found in appendices.
Many of their examples are drawn from chemistry (surprise, surprise) and
many bifurcation diagrams are presented.

**Parker and Chua**[24] present a wealth of algorithms
for the standard tasks faced when carrying out a numerical study of
a dynamical system.
Among the tools described are determination of invariant tori,
continuation methods, and
methods for calculating stable and unstable manifolds.
Pseudo-code is included, so that programming in
languages such as C, Fortran, C++ or Pascal is relatively straightforward;
however, beware, there are many errors in the code
(this may be corrected in the second edition; in the first edition
even the horseshoes are drawn incorrectly).

The second edition of **Seydel**[30]
is a very careful and extremely detailed text,
to be recommended to anyone truly interested in the numerical
analysis of dynamical systems.
It is aimed at bifurcation analysis, and does not cover
invariant manifolds etc.
Following Parker and Chua, Seydel has included
a section on chaos theory.
It not quite successful, and should have been left out; just as
Parker and Chua should have left out certain superficially covered subjects.
Nevertheless, the latter two books are definitely handy to have within reach.

A few questions naturally arise when attempting a review such as this: are there a sufficient number of quality textbooks books around?; are there any subjects that are under represented or neglected?; what is needed in future textbooks? There are a number of excellent textbooks in dynamical systems, several of the above mentioned texts fall into this category. The choice of a particular one depends entirely on the purpose. Considering that the vast majority of dynamical systems activity lies in practical applications, the numerical analysis of dynamical systems is shamefully neglected in most texts. This is a great pity, as there are interesting theoretical aspects to cover here as well; strange as it may sound, treating numerics does not necessarily imply thousands of lines of Fortran code. The final question mentioned above cannot be answered in a simple way, as it depends on future developments. It could, however, be the topic of an interesting discussion. Related to this is the question of how to teach the subject, at undergraduate, as well as postgraduate, level. UK Nonlinear News would be a natural place to take up such a discussion.

The following table is intended to give the reader an overview of the contents of the various books mentioned above. A plus indicates that the subject is considered in the book. If a subject is mentioned in a footnote or in an insultingly short section I have not included it.

The following abbreviations have been used:

GB: global bifurcations;

SD: symbolic dynamics;

NA: numerical analysis of dynamical systems;

HD: Hamiltonian dynamics;

CD: complex dynamical systems;

F: Fractals.

GB | SD | NA | HD | CD | F | |
---|---|---|---|---|---|---|

Lichtenberg & Lieberman | + | + | ||||

Guckenheimer & Holmes | + | + | + | + | ||

Wiggins[33] | + | + | + | + | ||

Wiggins[34] | + | + | + | + | ||

Wiggins[35] | + | + | ||||

Ruelle | + | |||||

Arrowsmith & Place | + | + | + | |||

Hale & Koçak | + | + | + | |||

Perko | + | + | ||||

Glendinning | + | + | ||||

Devaney | + | + | + | |||

Gulick | + | + | + | |||

Holmgren | + | + | ||||

Crownover | + | + | + | |||

GB | SD | NA | HD | CD | F | |

Robinson | + | + | + | + | ||

Thompson & Stewart | + | + | + | + | ||

Moon | + | + | ||||

Schuster | + | + | + | |||

Ott | + | + | ||||

Marek & Schreiber | + | + | + | + | ||

Bai-lin | + | + | + | + | ||

Iooss & Joseph | ||||||

Golubitsky & Schaeffer | ||||||

Golubitsky & Stewart & Schaeffer | ||||||

Kuznetsov | + | + | + | |||

Kubícek & Marek | + | |||||

Parker & Chua | + | + | ||||

Seydel | + |

- R.H. Abraham and C.D. Shaw,
`Dynamics: The geometry of behavior`, Addison-Wesley, Redwood City, 1992. - D.K. Arrowsmith and C.M. Place,
`An introduction to dynamical systems`, Cambridge University Press, Cambridge, 1990. - H. Bai-lin,
`Elementary symbolic dynamics and chaos in dissipative systems`, World Scientific, Singapore, 1989. - L.S. Block and W.A. Coppel,
`Dynamics in one dimension`, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. - P. Collet and J.P. Eckmann,
`Iterated maps on the interval as dynamical systems`, Birkhäuser, Basel, 1980. - R.M. Crownover,
`Introduction to fractals and chaos`, Jones and Bartlett Publishers, Boston, 1995. - W. de Melo and S. van Strien,
`One-dimensional dynamics`, Springer-Verlag, Berlin, 1993. - R.L. Devaney,
`An introduction to chaotic dynamical systems`, Benjamin/Cummings, Menlo Park, 1985. - P. Glendinning,
`Stability, instability and chaos`, Cambridge University Press, Cambridge, 1994. - M. Golubitsky and D.G. Schaeffer,
`Singularities and groups in bifurcation theory I`, Springer-Verlag, New York, 1985. - M. Golubitsky, I. Stewart, and D.G. Schaeffer,
`Singularities and groups in bifurcation theory II`, Springer-Verlag, New York, 1988. - J. Guckenheimer and P. Holmes,
`Nonlinear oscillations, dynamical systems and bifurcations of vector fields`, Springer-Verlag, New York, 1983. - D. Gulick,
`Encounters with chaos`, McGraw-Hill, New York, 1992. - J. Hale and K. Koçak,
`Dynamics and bifurcations`, Springer-Verlag, New York, 1992. - R.A. Holmgren,
`A first course in discrete dynamical systems`, Springer-Verlag, Berlin, 1994. - G. Iooss and D.D. Joseph,
`Elementary stability and bifurcation theory`, Springer-Verlag, 1980. - M. Kubícek and M. Marek,
`Computational methods in bifurcation theory and dissipative structures`, Springer-Verlag, Berlin, 1983. - Y.A. Kuznetsov,
`Elements of applied bifurcation theory`, Springer-Verlag, New York, 1995. - A.J. Lichtenberg and M.A. Lieberman,
`Regular and stochastic motion`, Springer-Verlag, New York, 1983. - M. Marek and I. Schreiber,
`Chaotic behaviour of deterministic dissipative systems`, Academia, Prague, 1991. - F.C. Moon,
`Chaotic vibrations`, John Wiley & Sons, Chichester, 1987. - E.Ott,
`Chaos in dynamical systems`, Cambridge University Press, Cambridge, 1993. - J. Palis and F. Takens,
`Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations`, Cambridge University Press, Cambridge, 1993. - T.S. Parker and L.O. Chua,
`Practical numerical algorithms for chaotic systems`, Springer-Verlag, New York, 1989. - H.O. Peitgen and P.H. Richter,
`The beauty of fractals`, Springer-Verlag, Berlin, 1986. - L.Perko,
`Differential equations and dynamical systems`, Springer-Verlag, New York, 1991. - C. Robinson,
`Dynamical systems`, CRC Press, Boca Raton, 1995. - D. Ruelle,
`Elements of differentiable dynamics and bifurcation theory`, Academic Press, San Diego, 1989. - H.G. Schuster,
`Deterministic chaos`, VCH Verlagsgesellschaft mbH, Weinheim, 1989. - R. Seydel,
`Practical bifurcation and stability analysis`, Springer-Verlag, New York, 1994. - C. Sparrow,
`The Lorenz equations: Bifurcations, chaos, and strange attractors`, Springer-Verlag, New York, 1982. - J.M.T. Thompson and H.B. Stewart,
`Nonlinear dynamics and chaos`, John Wiley & Sons, Chichester, 1986. - S. Wiggins,
`Global bifurcations and chaos`, Springer-Verlag, New York, 1988. - S. Wiggins,
`Introduction to applied nonlinear dynamical systems and chaos`, Springer-Verlag, New York, 1990. - S. Wiggins,
`Chaotic transport in dynamical systems`, Springer-Verlag, New York, 1992.

A listing of books reviewed in

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Last Updated: 31st January 1997.