Feldman's ** Chaos and Fractals: An Elementary Introduction** attempts
to teach the mathematics of dynamical systems and fractals starting at a
genuinely elementary level, assuming only basic algebra. It is designed
either as "a college course for students who are not maths or science
majors and have not necessarily taken calculus" or as a mathematics course
for high school students, taught instead of or as a precursor to calculus.

So Feldman begins with the basic idea of a function and different ways of viewing functions and works up from there. He doesn't even assume, to take one example, comfort with logarithms: "If you do not like logarithms and prefer to solve equations like Eq. (16.9) by guessing and checking, that is OK." (With a marginal note: "Logarithms are incredibly useful. If you had a bad experience with logarithms in the past, as seems to be the case for many students, I encourage you to try again to form a good relationship with them.")

Despite the low level of assumed knowledge and a relatively discursive
exposition, Feldman manages to get a remarkably long way, both in
conveying an understanding of the concepts of chaotic dynamics and fractal
geometry and in introducing new mathematical ideas. On chaos he covers,
among other topics, iterated functions, discrete dynamical systems
and the logistic function, sensitive dependence to initial conditions,
Lyapunov exponents, the shadowing lemma, bifurcation diagrams and the
universality of the period-doubling route to chaos, and renormalization.
With fractals he introduces similarity and box-counting dimensions,
random fractals and the chaos game, the collage theorem, and power laws.
And the last third of *Chaos and Fractals* covers Julia and Mandelbrot
sets and higher-dimensional systems such as the Hénon map and the Lorentz
attractor. Many of these are only touched on, but it's impressive to see
such advanced topics treated at all within this framework — as Feldman
writes, "there are surprisingly few non-technical discussions of the
phenomenon of universality".

A considerable amount of general mathematics is introduced in support of this. That includes relatively simple ideas such as time series, plots, histograms, final state diagrams, and binary numbers. But it also includes at least an introduction to ergodicity and statistical stability, normal distributions and the central limit theorem, power laws and long tails, different kinds of dimension, countable and uncountable infinities, complex numbers, cellular automata, basic calculus and differential equations!

Feldman also includes some excellent material on the scientific and philosophical and historical context, looking at such things as the difference between popular and scientific understandings of "chaos", the historical relationship of dynamics to ideas of causality and the Scientific Revolution, the difficulty of giving a single definition of "fractal", the difference between mathematical and real fractals, and so forth. This is the kind of material one might find in a popular science book and will certainly help with the motivation, but here it remains as background to the central exposition of the mathematics.

Anyone with university mathematics, or even just a good grasp of calculus,
may find *An Elementary Introduction* a bit too elementary (I skipped
over almost all the mathematical exposition myself). I would have loved
it when I was twelve, but it is hard to judge how well it would work
for ordinary school students or mathematics-averse university students:
I think it would take really exceptional teaching to get someone scared
of logarithms all the way through it. But it offers at least the
possibility of a radically different trajectory for school teaching,
providing a motivated pathway to a lot of fascinating mathematics not
normally considered accessible at that level. (The lack of solutions
to the exercises might make it a little less useful for self-study.)

** Fractals and Chaos: An Illustrated Course** (1997) should be accessible
to science undergraduates and a broad range of scientists: its focus is
on applications to the physical sciences and it is aimed at those who
want to be able to actually

*use*fractal geometry and chaotic dynamics. It assumes acquaintance with some mathematics, but no details: so the basic idea of limits but no involved calculus, the concepts of Fourier transforms and differential equations without any ability to manipulate them, and so forth. And it makes no attempt at rigour or formalism, with the emphasis instead on numerical examples and visualisation.

Addison has a knack for focusing on the central ideas of a topic, with just the essential details; he makes no attempt to teach any mathematics for its own sake and tackles a relatively restricted range of topics compared to the other two books. The emphasis is on how the key concepts can be used in science, both experimental and observational, and the result is exceptionally well motivated, at least for the scientifically-minded.

*An Illustrated Course* begins with a tour of some classical fractals and
an introduction to the concept of similarity dimension. This proceeds
directly to random fractals and box-counting and divider dimensions
(with "a brief overview of the Hausdorff dimension for completeness of
the text"), introducing Richardson plots and glancing at multifractals.
Then there's a chapter on Brownian motion, both regular and fractional,
which looks at, among other things, diffusive processes, Lévy flights,
and different coloured noises.

A chapter on discrete dynamical systems uses the logistic map, the Hénon map, and the Mandelbrot set as key examples. A chapter on chaotic oscillations touches on the Duffing oscillator, the Lorenz model, Rössler systems, and (briefly) spatially extended systems: the transition from laminar to turbulent flow (Taylor-Couette flow) and different routes to chaos/turbulence. And a final chapter "Characterizing Chaos" offers practical advice on methods for distinguishing and characterizing systems that exhibit chaos: visual inspection, frequency components, Lyapunov exponents, various kinds of dimension estimates, attractor reconstruction from time series, and the effects of noise.

Each chapter includes an excellent and extensive (though unfortunately now dated) guide to further reading, pointing at further mathematics but mostly surveying the literature on applications in various areas of chemistry, physics, biology, medicine, geology, engineering and so forth. Each chapter also includes some exercises, with full solutions provided in an appendix. Many of these are qualitative "think about this" questions; others are suggestions for computational explorations, both numeric and visual.

*Fractals and Chaos: An Illustrated Course* is well designed for
self-study, making it a great practical resource for those working in
the physical sciences or engineering as well as for students.

Despite its size and length, ** Chaos and Fractals: New Frontiers of
Science** (2004, first edition 1992) is also highly accessible. It is
relatively discursive and easy to read, with each chapter telling a
coherent story, and it highlights the key concepts and ideas, examining
a few models in detail and using worked numerical examples as well as
visualisations and illustrations (including twelve pages of colour
halftones). The assumed knowledge is greater than Feldman, and the
target audience might be early undergraduates studying mathematics or
the computational and physical sciences.

There is a little background history and philosophy, notably in a
preface by Feigenbaum and an introduction, and occasional discussion of
applications, such as using the Hilbert curve for dithering, but *New
Frontiers of Science* remains focused on the mathematics. Full formal
rigour is eschewed, but there are derivations and outlines of proofs,
and the extra space available is used both to broaden the coverage and
to delve into some topics in depth.

There is no attempt to be comprehensive — it is not the "authoritative
general reference" one of the back-cover blurbs proclaims — or to
present current research, but *New Frontiers of Science* treats all
the major topics, at least touching on the central ideas and concepts,
and goes into quite some detail with some. The treatment of Julia sets,
for example, looks at field lines and equipotentials, computation using
a chaos game algorithm, the structure of disconnected "Cantor" Julia
sets, and even quaternion Julia sets. And there's more detail here on
the fine structure of the Mandelbrot set, following work by Douady and
Hubbard and by Tan Lei, than in more rigorous books on fractal geometry.

Peitgen et al. begin with feedback processes and iterators, a look at some classical fractals and the notion of self-similarity, and the idea of limits. Only then do they introduce concepts of dimension. Returning to iterated functions, they introduce the idea of multiple copy machine reduction and explain how images can be encoded using iterated function systems. This leads naturally to chaos game algorithms, ways of growing fractals from simple generators using recursion, and the use of randomness in constructing fractals, notably with Brownian motion and landscapes. They also look at cellular automata, Pascal's Triangle, and some number theory.

Turning to dynamics, *New Frontiers of Science* uses some simple
transformations to introduce sensitivity to initial conditions, mixing
and periodic orbits, and ergodicity, before proceeding to the quadratic
iterator, period-doubling, the Feigenbaum point, intermittency, and paths
to chaos. Moving to higher dimensions, there is treatment of strange
attractors, continuous systems, the Rössler and Lorentz attractors,
Lyapunov exponents and the dimensions of attractors, and reconstruction
of attractors from time series. And the book ends with the fractal
boundaries of basins of attraction, Julia sets, and the Mandelbrot set
as a catalogue of Julia sets.

Some general mathematics is introduced as an aid to understanding — to help give a feel for limits, for example, there's an explanation of methods for approximating pi and the square-root of two — and there are explorations of some number theoretical results that seem a little peripheral. Otherwise mathematics is introduced only as necessary: towards the end there are brief introductions to complex numbers and differential equations, for example.

Some more technical material, amounting to around a third of the book, is separated out from the main text, pushed to the inside instead of the outside of the page and in a different font. This mostly involves filling in details, offering a different perspective, or using slightly more sophisticated mathematics, and can be skipped without breaking the flow of the presentation. Examples of topics treated in this fashion include Hausdorff dimension and how it differs from box-counting dimension, the physical interpretation of the original Lorentz attractor, and various number-theoretic results involving binomial coefficients.

*New Frontiers of Science* makes an excellent entry to the broader
mathematics of fractals and chaos, especially for students who are
curious about the details as well as the core concepts but don't want to
get bogged down in formal mathematics. It does however lack exercises,
which may make it less useful for some kinds of self-study.

Collectively these three books illustrate just how good modern textbooks are. They are all, given their different goals and audiences, pedagogically excellent, clearly refined by practical teaching experience. They make good use of diagrams and illustrations and of concrete, worked examples, and they keep a focus on the key ideas they are trying to convey.

Note: For anyone after more rigorous mathematics I recommend Falconer's
*Fractal Geometry: Mathematical Foundations and Applications* and
Barnsley's *Fractals Everywhere*, both of which have recently appeared
in new editions. More introductory works, but ones which still attempt
to present some actual mathematics, include Falconer's recent *Fractals:
A Very Short Introduction* and Lauwerier's older *Fractals: Endlessly
Repeated Geometrical Figures*.

February 2016

**External links:**-
*Chaos and Fractals: An Elementary Introduction*

- buy from Amazon.com or Amazon.co.uk

- information from David P. Feldman

*Fractals and Chaos: An Illustrated Course*

- buy from Amazon.com or Amazon.co.uk

*Chaos and Fractals: New Frontiers of Science*

- buy from Amazon.com or Amazon.co.uk

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