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Chaos exists in systems all around us. Even the simplest system can be subject to chaos, denying us accurate predictions of its behavior, and sometimes giving rise to astonishing structures of large-scale order. Here, Leonard Smith shows that we all have an intuitive understanding of chaotic systems. He uses accessible math and physics to explain Chaos Theory, and points to numerous examples in philosophy and literature that illuminate the problems. This book provides a complete understanding of chaotic dynamics, using examples from mathematics, physics, philosophy, and the real world, with an explanation of why chaos is important and how it differs from the idea of randomness. The author's real life applications include the weather forecast, a pendulum, a coin toss, mass transit, politics, and the role of chaos in gambling and the stock market. Chaos represents a prime opportunity for mathematical lay people to finally get a clear understanding of this fascinating concept.
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Chaos: A Very Short Introduction

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Leonard A. Smith

A Very Short Introduction



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1 3 5 7 9 10 8 6 4 2

To the memory of Dave Paul Debeer,
A real physicist, a true friend.

This page intentionally left blank


Acknowledgements xi


List of illustrations xv


The emergence of chaos 1
Exponential growth, nonlinearity, common sense 22
Chaos in context: determinism, randomness,
and noise



Chaos in mathematical models 58
Fractals, strange attractors, and dimension(s)


Quantifying the dynamics of uncertainty 87
Real numbers, real observations, and computers 104
Sorry, wrong number: statistics and chaos


Predictability: does chaos constrain our forecasts?
Applied chaos: can we see through our models?
Philosophy in chaos 154
Glossary 163



Further reading 169
Index 173


This book would not have been possible without my parents, of
course, but I owe a greater debt than most to their faith, doubt, and
hope, and to the love and patience of a, b, and c. Professionally my
greatest debt is to Ed Spiegel, a father of chaos and my thesis
Professor, mentor, and friend. I also profited immensely from
having the chance to discuss some of these ideas with Jim Berger,
Robert Bishop, David Broomhead, Neil Gordon, Julian Hunt,
Kevin Judd, Joe Keller, Ed Lorenz, Bob May, Michael Mackey,
Tim Palmer, Itamar Procaccia, Colin Sparrow, James Theiler,
John Wheeler, and Christine Ziehmann. I am happy to
acknowledge discussions with, and the support of, the Master
and Fellows of Pembroke College, Oxford. Lastly and largely, I’d
like to acknowledge my debt to my students, they know who they
are. I am never sure how to react upon overhearing an exchange
like: ‘Did you know she was Lenny’s student?’, ‘Oh, that explains
a lot.’ Sorry guys: blame Spiegel.


The ‘chaos’ introduced in the following pages reflects phenomena in
mathematics and the sciences, systems where (without cheating)
small differences in the way things are now have huge consequences
in the way things will be in the future. It would be cheating, of
course, if things just happened randomly, or if everything
continually exploded forever. This book traces out the remarkable
richness that follows from three simple constraints, which we’ll call
sensitivity, determinism, and recurrence. These constraints allow
mathematical chaos: behaviour that looks random, but is not
random. When allowed a bit of uncertainty, presumed to be the
active ingredient of forecasting, chaos has reignited a centuries-old
debate on the nature of the world.
The book is self-contained, defining these terms as they are
encountered. My aim is to show the what, where, and how of chaos;
sidestepping any topics of ‘why’ which require an advanced
mathematical background. Luckily, the description of chaos and
forecasting lends itself to a visual, geometric understanding; our
examination of chaos will take us to the coalface of predictability
without equations, revealing open questions of active scientific
research into the weather, climate, and other real-world
phenomena of interest.
Recent popular interest in the science of chaos has evolved

differently than did the explosion of interest in science a century
ago when special relativity hit a popular nerve that was to throb for
decades. Why was the public reaction to science’s embrace of
mathematical chaos different? Perhaps one distinction is that most
of us already knew that, sometimes, very small differences can have
huge effects. The concept now called ‘chaos’ has its origins both in
science fiction and in science fact. Indeed, these ideas were well
grounded in fiction before they were accepted as fact: perhaps the
public were already well versed in the implications of chaos, while
the scientists remained in denial? Great scientists and
mathematicians had sufficient courage and insight to foresee the
coming of chaos, but until recently mainstream science required a
good solution to be well behaved: fractal objects and chaotic curves
were considered not only deviant, but the sign of badly posed
questions. For a mathematician, few charges carry more shame
than the suggestion that one’s professional life has been spent on a
badly posed question. Some scientists still dislike problems whose
results are expected to be irreproducible even in theory. The
solutions that chaos requires have only become widely acceptable in
scientific circles recently, and the public enjoyed the ‘I told you so’
glee usually claimed by the ‘experts’. This also suggests why chaos,
while widely nurtured in mathematics and the sciences, took root
within applied sciences like meteorology and astronomy. The
applied sciences are driven by a desire to understand and predict
reality, a desire that overcame the niceties of whatever the formal
mathematics of the day. This required rare individuals who could
span the divide between our models of the world and the world as it
is without convoluting the two; who could distinguish the
mathematics from the reality and thereby extend the mathematics.
As in all Very Short Introductions, restrictions on space require
entire research programmes to be glossed over or omitted; I
present a few recurring themes in context, rather than a series of
shallow descriptions. My apologies to those whose work I have
omitted, and my thanks to Luciana O’Flaherty (my editor), Wendy
Parker, and Lyn Grove for help in distinguishing between what

was most interesting to me and what I might make interesting
to the reader.

How to read this introduction
While there is some mathematics in this book, there are no
equations more complicated than X = 2. Jargon is less easy to
discard. Words in bold italics you will have to come to grips with;
these are terms that are central to chaos, brief definitions of these
words can be found in the Glossary at the end of the book. Italics is
used both for emphasis and to signal jargon needed for the next
page or so, but which is unlikely to recur often throughout the book.
Any questions that haunt you would be welcome online at http:// on the discussion forum VSI Chaos. More
information on these terms can be found rapidly at Wikipedia and , and in the Further reading.

List of illustrations


The first weather map
ever published in a
newspaper, prepared
by Galton in 1875

6 A graph comparing
Fibonacci numbers and
exponential growth

© The Times/NI Syndication

2 Galton’s original sketch
of the Galton Board
3 The Times headline
following the Burns’
Day storm in 1990


© The Times/NI Syndication
Limited 1990/John Frost

4 Modern weather map
showing the Burns’ Day
storm and a two-dayahead forecast
5 The Cheat with the Ace
of Diamonds, c.1645,
by Georges de la Tour 19
Louvre, Paris. ©

7 A chaotic time series
from the Full Logistic
8 Six mathematical


9 Points collapsing onto
four attractors of the
Logistic Map

The evolution of
uncertainty under the
Yule Map


Period doubling
behaviour in the
Logistic Map



A variety of more
complicated behaviours
in the Logistic Map


bifurcation diagram and
the collapse toward
attractors in the
Logistic Map



The Lorenz attractor
and the Moore-Spiegel


The evolution of
uncertainty in the
Lorenz System

22 Predictable chaos as
seen in four iterations
of the same mouse
ensemble under the
Baker’s Map and a
Baker’s Apprentice



The Hénon attractor
and a two-dimensional
slice of the


A variety of behaviours
from the Hénon-Heilies


The Fournier Universe,
as illustrated by


Time series from the
stochastic Middle Thirds
IFS Map and the
deterministic Tripling
Tent Map

20 A close look at the
Hénon attractor,
showing fractal

Schematic diagrams
showing the action of
the Baker’s Map and a
Baker’s Apprentice

23 Card trick revealing the
limitations of digital
24 Two views of data from
Machete’s electric circuit,
suggestive of Takens’

The Not A Galton

26 An illustration of using
analogues to make a

The state space of a
climate model


Crown Copyright

28 Richardson’s dream
© F. Schuiten




29 Two-day-ahead ECMWF
ensemble forecasts of the
Burns’ Day storm

30 Four ensemble forecasts
of the Machete’s MooreSpiegel Circuit

Figures 7, 8, 9, 11, 12, 13, 19, and 20 were produced with the
assistance of Hailiang Du. Figures 24 and 30 were produced with
the assistance of Reason Machete. Figures 4 and 29 were produced
with the assistance of Martin Leutbecher with data kindly made
available by the European Centre for Medium-Range Weather
Forecasting. Figure 27 is after M. Hume et al., The UKIP02
Scientific Report, Tyndal Centre, University of East Anglia,
Norwich, UK.
The publisher and the author apologize for any errors or omissions
in the above list. If contacted they will be pleased to rectify these at
the earliest opportunity.

This page intentionally left blank

Chapter 1
The emergence of chaos

Embedded in the mud, glistening green and gold and black,
was a butterfly, very beautiful and very dead.
It fell to the floor, an exquisite thing, a small thing
that could upset balances and knock down a line of
small dominoes and then big dominoes and then
gigantic dominoes, all down the years across Time.
Ray Bradbury (1952)

Three hallmarks of mathematical chaos
The ‘butterfly effect’ has become a popular slogan of chaos. But is it
really so surprising that minor details sometimes have major
impacts? Sometimes the proverbial minor detail is taken to be the
difference between a world with some butterfly and an alternative
universe that is exactly like the first, except that the butterfly is
absent; as a result of this small difference, the worlds soon come to
differ dramatically from one another. The mathematical version of
this concept is known as sensitive dependence. Chaotic systems
not only exhibit sensitive dependence, but two other properties as
well: they are deterministic, and they are nonlinear. In this
chapter, we’ll see what these words mean and how these concepts
came into science.
Chaos is important, in part, because it helps us to cope with

unstable systems by improving our ability to describe, to
understand, perhaps even to forecast them. Indeed, one of the
myths of chaos we will debunk is that chaos makes forecasting a
useless task. In an alternative but equally popular butterfly story,
there is one world where a butterfly flaps its wings and another
world where it does not. This small difference means a tornado
appears in only one of these two worlds, linking chaos to
uncertainty and prediction: in which world are we? Chaos is the
name given to the mechanism which allows such rapid growth of
uncertainty in our mathematical models. The image of chaos
amplifying uncertainty and confounding forecasts will be a
recurring theme throughout this Introduction.


Whispers of chaos
Warnings of chaos are everywhere, even in the nursery. The
warning that a kingdom could be lost for the want of a nail can be
traced back to the 14th century; the following version of the familiar
nursery rhyme was published in Poor Richard’s Almanack in 1758
by Benjamin Franklin:
For want of a nail the shoe was lost,
For want of a shoe the horse was lost,
and for want of a horse the rider was lost,
being overtaken and slain by the enemy,
all for the want of a horse-shoe nail.

We do not seek to explain the seed of instability with chaos, but
rather to describe the growth of uncertainty after the initial seed is
sown. In this case, explaining how it came to be that the rider was
lost due to a missing nail, not the fact that the nail had gone
missing. In fact, of course, there either was a nail or there was not.
But Poor Richard tells us that if the nail hadn’t been lost, then the
kingdom wouldn’t have been lost either. We will often explore the
properties of chaotic systems by considering the impact of slightly
different situations.

The study of chaos is common in applied sciences like astronomy,
meteorology, population biology, and economics. Sciences making
accurate observations of the world along with quantitative
predictions have provided the main players in the development of
chaos since the time of Isaac Newton. According to Newton’s Laws,
the future of the solar system is completely determined by its
current state. The 19th-century scientist Pierre Laplace elevated
this determinism to a key place in science. A world is deterministic
if its current state completely defines its future. In 1820, Laplace
conjured up an entity now known as ‘Laplace’s demon’; in doing so,
he linked determinism and the ability to predict in principle to the
very notion of success in science.
We may regard the present state of the universe as the effect of its
past and the cause of its future. An intellect which at a certain
positions of all items of which nature is composed, if this intellect
were also vast enough to submit these data to analysis, it would
embrace in a single formula the movements of the greatest bodies of
the universe and those of the tiniest atom; for such an intellect
nothing would be uncertain and the future just like the past would
be present before its eyes.

Note that Laplace had the foresight to give his demon three
properties: exact knowledge of the Laws of Nature (‘all the forces’),
the ability to take a snapshot of the exact state of the universe (‘all
the positions’), and infinite computational resources (‘an intellect
vast enough to submit these data to analysis’). For Laplace’s
demon, chaos poses no barrier to prediction. Throughout this
Introduction, we will consider the impact of removing one or more
of these gifts.
From the time of Newton until the close of the 19th century, most
scientists were also meteorologists. Chaos and meteorology are
closely linked by the meteorologists’ interest in the role uncertainty
plays in weather forecasts. Benjamin Franklin’s interest in

The emergence of chaos

moment would know all forces that set nature in motion, and all


meteorology extended far beyond his famous experiment of flying
a kite in a thunderstorm. He is credited with noting the general
movement of the weather from west towards the east and testing
this theory by writing letters from Philadelphia to cities further
east. Although the letters took longer to arrive than the weather,
these are arguably early weather forecasts. Laplace himself
discovered the law describing the decrease of atmospheric pressure
with height. He also made fundamental contributions to the theory
of errors: when we make an observation, the measurement is never
exact in a mathematical sense, so there is always some uncertainty
as to the ‘True’ value. Scientists often say that any uncertainty in an
observation is due to noise, without really defining exactly
what the noise is, other than that which obscures our vision of
whatever we are trying to measure, be it the length of a table, the
number of rabbits in a garden, or the midday temperature.
Noise gives rise to observational uncertainty, chaos helps us to
understand how small uncertainties can become large
uncertainties, once we have a model for the noise. Some of the
insights gleaned from chaos lie in clarifying the role(s) noise
plays in the dynamics of uncertainty in the quantitative
sciences. Noise has become much more interesting, as the study
of chaos forces us to look again at what we might mean by the
concept of a ‘True’ value.
Twenty years after Laplace’s book on probability theory appeared,
Edgar Allan Poe provided an early reference to what we would now
call chaos in the atmosphere. He noted that merely moving our
hands would affect the atmosphere all the way around the planet.
Poe then went on to echo Laplace, stating that the mathematicians
of the Earth could compute the progress of this hand-waving
‘impulse’, as it spread out and forever altered the state of the
atmosphere. Of course, it is up to us whether or not we choose to
wave our hands: free will offers another source of seeds that chaos
might nurture.
In 1831, between the publication of Laplace’s science and Poe’s

all manner of insects, vultures, infinite billions of life forms are
thrown into chaos and destruction . . . Step on a mouse and you
leave your print, like a Grand Canyon, across Eternity. Queen
Elizabeth might never be born, Washington might not cross the
Delaware, there might never be a United States at all. So be careful.
Stay on the Path. Never step off!

Needless to say, someone does step off the Path, crushing to
death a beautiful little green and black butterfly. We can only
consider these ‘what if’ experiments within the fictions of
mathematics or literature, since we have access to only one
realization of reality.
The origins of the term ‘butterfly effect’ are appropriately shrouded

The emergence of chaos

fiction, Captain Robert Fitzroy took the young Charles Darwin on
his voyage of discovery. The observations made on this voyage led
Darwin to his theory of natural selection. Evolution and chaos have
more in common than one might think. First, when it comes to
language, both ‘evolution’ and ‘chaos’ are used simultaneously to
refer both to phenomena to be explained and to the theories that are
supposed to do the explaining. This often leads to confusion
between the description and the object described (as in ‘confusing
the map with the territory’). Throughout this Introduction we will
see that confusing our mathematical models with the reality they
aim to describe muddles the discussion of both. Second, looking
more deeply, it may be that some ecosystems evolve as if they were
chaotic systems, as it may well be the case that small differences in
the environment have immense impacts. And evolution has
contributed to the discussion of chaos as well. This chapter’s
opening quote comes from Ray Bradbury’s ‘A Sound Like Thunder’,
in which time-travelling big game hunters accidentally kill a
butterfly, and find the future a different place when they return to it.
The characters in the story imagine the impact of killing a mouse,
its death cascading through generations of lost mice, foxes, and
lions, and:

in mystery. Bradbury’s 1952 story predates a series of scientific
papers on chaos published in the early 1960s. The meteorologist Ed
Lorenz once invoked sea gulls’ wings as the agent of change,
although the title of that seminar was not his own. And one of his
early computer-generated pictures of a chaotic system does
resemble a butterfly. But whatever the incarnation of the ‘small
difference’, whether it be a missing horse shoe nail, a butterfly, a sea
gull, or most recently, a mosquito ‘squished’ by Homer Simpson, the
idea that small differences can have huge effects is not new.
Although silent regarding the origin of the small difference, chaos
provides a description for its rapid amplification to kingdomshattering proportions, and thus is closely tied to forecasting and


The first weather forecasts
Like every ship’s captain of the time, Fitzroy had a deep interest in
the weather. He developed a barometer which was easier to use
onboard ship, and it is hard to overestimate the value of a
barometer to a captain lacking access to satellite images and radio
reports. Major storms are associated with low atmospheric
pressure; by providing a quantitative measurement of the
pressure, and thus how fast it is changing, a barometer can give
life-saving information on what is likely to be over the horizon.
Later in life, Fitzroy became the first head of what would become
the UK Meteorological Office and exploited the newly deployed
telegraph to gather observations and issue summaries of the
current state of the weather across Britain. The telegraph allowed
weather information to outrun the weather itself for the first time.
Working with LeVerrier of France, who became famous for using
Newton’s Laws to discover two new planets, Fitzroy contributed to
the first international efforts at real-time weather forecasting.
These forecasts were severely criticized by Darwin’s cousin,
statistician Francis Galton, who himself published the first
weather chart in the London Times in 1875, reproduced in
Figure 1.

1. The first weather chart ever published in a newspaper. Prepared by
Francis Galton, it appeared in the London Times on 31 March 1875


If uncertainty due to errors of observation provides the seed that
chaos nurtures, then understanding such uncertainty can help us
better cope with chaos. Like Laplace, Galton was interested in the
‘theory of errors’ in the widest sense. To illustrate the ubiquitous
‘bell-shaped curve’ which so often seems to reflect measurement
errors, Galton created the ‘quincunx’, which is now called a Galton
Board; the most common version is shown on the left side of Figure
2. By pouring lead shot into the quincunx, Galton simulated a
random system in which each piece of shot has a 50:50 chance of
going to either side of every ‘nail’ that it meets, giving rise to a bellshaped distribution of lead. Note there is more here than the oneoff flap of a butterfly wing: the paths of two nearby pieces of lead
may stay together or diverge at each level. We shall return to Galton
Boards in Chapter 9, but we will use random numbers from the
bell-shaped curve as a model for noise many times before then. The
bell-shape can be seen at the bottom of the Galton Board on the left
of Figure 2, and we will find a smoother version towards the top of
Figure 10.
The study of chaos yields new insight into why weather forecasts
remain unreliable after almost two centuries. Is it due to our
missing minor details in today’s weather which then have major
impacts on tomorrow’s weather? Or is it because our methods,
while better than Fitzroy’s, remain imperfect? Poe’s early
atmospheric incarnation of the butterfly effect is complete with the
idea that science could, if perfect, predict everything physical. Yet
the fact that sensitive dependence would make detailed forecasts of
the weather difficult, and perhaps even limit the scope of physics,
has been recognized within both science and fiction for some time.
In 1874, the physicist James Clerk Maxwell noted that a sense of
proportion tended to accompany success in a science:
This is only true when small variations in the initial circumstances
produce only small variations in the final state of the system. In a
great many physical phenomena this condition is satisfied; but there
are other cases in which a small initial variation may produce a very

The emergence of chaos

2. Galton’s 1889 schematic drawings of what are now called ‘Galton

great change in the final state of the system, as when the
displacement of the ‘points’ causes a railway train to run into
another instead of keeping its proper course.

This example is again atypical of chaos in that it is ‘one-off’
sensitivity, but it does serve to distinguish sensitivity and
uncertainty: this sensitivity is no threat as long as there is no
uncertainty in the position of the points, or in which train is on
which track. Consider pouring a glass of water near a ridge in the


Rocky Mountains. On one side of this continental divide the water
finds its way into the Colorado River and to the Pacific Ocean, on
the other side the Mississippi River and eventually the Atlantic
Ocean. Moving the glass one way or the other illustrates
sensitivity: a small change in the position of the glass means a
particular molecule of water ends up in a different ocean. Our
uncertainty in the position of the glass might restrict our ability to
predict which ocean that molecule of water will end up in, but only
if that uncertainty crosses the line of the continental divide. Of
course, if we were really trying to do this, we would have to
question whether any such mathematical line actually divided
continents, as well as the other adventures the molecule of water
might have which could prevent it reaching the ocean. Usually,
chaos involves much more than a single one-off ‘tripping point’; it
tends to more closely resemble a water molecule that repeatedly
evaporates and falls in a region where there are continental divides
all over the place.
Nonlinearity is defined by what it is not (it is not linear). This kind
of definition invites confusion: how would one go about defining a
biology of non-elephants? The basic idea to hold in mind now is
that a nonlinear system will show a disproportionate response: the
impact of adding a second straw to a camel’s back could be much
bigger (or much smaller) than the impact of the first straw. Linear
systems always respond proportionately. Nonlinear systems need
not, giving nonlinearity a critical role in the origin of sensitive

The Burns’ Day storm
But Mousie, thou art no thy lane,
In proving foresight may be vain:
The best-laid schemes o mice an men
Gang aft agley,
An lea’e us nought but grief an pain,
For promis’d joy!

Still thou art blest, compar’d wi me!
The present only toucheth thee:
But och! I backward cast my e’e,
On prospects drear!
An forward, tho I canna see,
I guess an fear!
Robert Burns, ‘To A Mouse’ (1785)

In contrast, the Great Storm of 1987 is famous for a BBC television
meteorologist’s broadcast the night before, telling people not to
worry about rumours from France that a hurricane was about to
strike England. Both storms, in fact, managed gusts of over
100 miles per hour, and the Burns’ Day storm caused much
greater loss of life; yet 20 years after the event, the Great Storm of
1987 is much more often discussed, perhaps exactly because the
Burns’ Day storm was well forecast. The story leading up to this
forecast beautifully illustrates a different way that chaos in our

The emergence of chaos

Burns’ poem praises the mouse for its ability to live only in the
present, not knowing the pain of unfulfilled expectations nor the
dread of uncertainty in what is yet to pass. And Burns was writing
in the 18th century, when mice and men laid their plans with little
assistance from computing machines. While foresight may be pain,
meteorologists struggle to foresee tomorrow’s likely weather every
day. Sometimes it works. In 1990, on the anniversary of Burns’
birth, a major storm ripped through northern Europe, including the
British Isles, causing significant property damage and loss of life.
The centre of the storm passed over Burns’ home town in Scotland,
and it became known as the Burns’ Day storm. A weather chart
reflecting the storm at noon on 25 January is shown in the top
panel of Figure 4 (page 14). Ninety-seven people died in northern
Europe, about half of this number in Britain, making it the highest
death toll of any storm in 40 years; about 3 million trees were blown
down, and total insurance costs reached £2 billion. Yet the Burns’
Day storm has not joined the rogues’ gallery of famously failed
forecasts: it was well forecast by the Met Office.


models can impact our lives without invoking alternate worlds,
some with and some without butterflies.
In the early morning of 24 January 1990, two ships in the
mid-Atlantic sent routine meteorological observations from
positions that happened to straddle the centre of what would
become the Burns’ Day storm. The forecast models run with these
observations give a fine forecast of the storm. Running the model
again after the event showed that when these observations are
omitted, the model predicts a weaker storm in the wrong place.
Because the Burns’ Day storm struck during the day, the failure to
provide forewarning would have had a huge impact on loss of life,
so here we have an example where a few observations, had they
not been made, would have changed the forecast and hence the
course of human events. Of course, an ocean weather ship is
harder to misplace than a horse shoe nail. There is more to this
story, and to see its relevance we need to look into how weather
models ‘work’.
Operational weather forecasting is a remarkable phenomenon in
and of itself. Every day, observations are taken in the most remote
locations possible, and then communicated and shared among
national meteorological offices around the globe. Many different
nations use this data to run their computer models. Sometimes an
observation is subject to plain old mistakes, like putting the
temperature in the box for wind speed, or a typo, or a glitch in
transition. To keep these mistakes from corrupting the forecast,
incoming observations are subject to quality control: observations
that disagree with what the model is expecting (given its last
forecast) can be rejected, especially if there are no independent,
nearby observations to lend support to them. It is a well-laid plan.
Of course, there are rarely any ‘nearby’ observations of any sort in
the middle of the Atlantic, and the ship observations showed the
development of a storm that the model had not predicted would be
there, so the computer’s automatic quality control program simply
rejected these observations.

3. Headline from The Times the day after the Burns’ Day storm

4. A modern weather chart reflecting the Burns’ Day storm as seen
through a weather model (top) and a two-day-ahead forecast targeting
the same time showing a fairly pleasant day (bottom)

Luckily, the computer was overruled. An intervention forecaster
was on duty and realized that these observations were of great
value. His job was to intervene when the computer did something
obviously silly, as computers are prone to do. In this case, he tricked
the computer into accepting the observations. Whether or not to
take this action is a judgement call: there was no way to know at the
time which action would yield a better forecast. The computer was
‘tricked’, the observation was used. The storm was forecast, and
lives were saved.


The emergence of chaos

There are two take-home messages here: the first is that when our
models are chaotic then small changes in our observations can have
large impacts on the quality of our foresight. An accountant looking
to reduce costs and computing the typical benefit of one particular
observation from any particular weather station is likely to vastly
underestimate the value of a future report from one of those
weather stations that falls at the right place at the right time, and
similarly the value of the intervention forecaster, who often has to
do nothing, literally. The second is that the Burns’ Day forecast
illustrates something a bit different from the butterfly effect.
Mathematical models allow us to worry about what the real future
will bring not by considering possible worlds, of which there may be
only one, but by contrasting different simulations of our model, of
which there can be as many as we can afford. As Burns might
appreciate, science gives us new ways to guess and new things to
fear. The butterfly effect contrasts different worlds: one world with
the nail and another world without that nail. The Burns effect
places the focus firmly on us and our attempts to make rational
decisions in the real world given only collections of different
simulations under various imperfect models. The failure to
distinguish between reality and our models, between observations
and mathematics, arguably between an empirical fact and scientific
fiction, is the root of much confusion regarding chaos both by the
public and among scientists. It was research into nonlinearity and
chaos that clarified yet again how import this distinction remains.
In Chapter 10, we will return to take a deeper look at how today’s

weather forecasters would have used insights from their
understanding of chaos when making a forecast for this event.
We have now touched on the three properties found in chaotic
mathematical systems: chaotic systems are nonlinear, they are
deterministic, and they are unstable in that they display sensitivity
to initial condition. In the chapters that follow we will constrain
them further, but our real interests lie not only in the mathematics
of chaos, but also in what it can tell us about the real world.

Chaos and the real world: predictability and a
21st-century demon
There is no more greater an error in science, than to believe that just
because some mathematical calculation has been completed, some
aspect of Nature is certain.

Alfred North Whitehead (1953)

What implications does chaos hold for our everyday lives? Chaos
impacts the ways and means of weather forecasting, which affect us
directly through the weather, and indirectly through economic
consequences both of the weather and of the forecasts themselves.
Chaos also plays a role in questions of climate change and our
ability to foresee the strength and impacts of global warming. While
there are many other things that we forecast, weather and climate
can be used to represent short-range forecasting and long-range
modelling, respectively. ‘When is the next solar eclipse?’ would be a
weather-like question in astronomy, while ‘Is the solar system
stable?’ would be a climate-like question. In finance, when to buy
100 shares of a given stock is a weather-like question, while a
climate-like question might address whether to invest in the stock
market or real estate.
Chaos has also had a major impact on the sciences, forcing a close
re-examination of what scientists mean by the words ‘error’ and
‘uncertainty’ and how these meanings change when applied to our

world and our models. As Whitehead noted, it is dangerous to
interpret our mathematical models as if they somehow governed
the real world. Arguably, the most interesting impacts of chaos
are not really new, but the mathematical developments of the last
50 years have cast many old questions into a new light. For
instance, what impact would uncertainty have on a 21st-century
incarnation of Laplace’s demon which could not escape
observational noise?

In his 1927 Gifford Lectures, Sir Arthur Eddington went to the
heart of the problem of chaos: some things are trivial to predict,
especially if they have to do with mathematics itself, while other
things seem predictable, sometimes:
A total eclipse of the sun, visible in Cornwall is prophesied for
11 August 1999 . . . I might venture to predict that 2 + 2 will be
equal to 4 even in 1999 . . . The prediction of the weather this time
next year . . . is not likely to ever become practicable . . . We should
require extremely detailed knowledge of present conditions, since
a small local deviation can exert an ever-expanding influence.
We must examine the state of the sun . . . be forewarned of volcanic
eruptions, . . . , a coal strike . . . , a lighted match idly thrown
away . . .

The emergence of chaos

Consider an intelligence that knew all the laws of nature precisely
and had good, but imperfect, observations of an isolated chaotic
system over an arbitrarily long time. Such an agent – even if
sufficiently vast to subject all this data to computationally exact
analysis – could not determine the current state of the system and
thus the present, as well as the future, would remain uncertain in
her eyes. While our agent could not predict the future exactly, the
future would hold no real surprises for her, as she could see what
could and what could not happen, and would know the probability
of any future event: the predictability of the world she could see.
Uncertainty of the present will translate into well-quantified
uncertainty in the future, if her model is perfect.

Our best models of the solar system are chaotic, and our best
models of the weather appear to be chaotic: yet why was Eddington
confident in 1928 that the 1999 solar eclipse would occur? And
equally confident that no weather forecast a year in advance would
ever be accurate? In Chapter 10 we will see how modern weather
forecasting techniques designed to better cope with chaos helped
me to see that solar eclipse.


When paradigms collide: chaos and controversy
One of the things that has made working in chaos interesting over
the last 20 years has been the friction generated when different
ways of looking at the world converge on the same set of
observations. Chaos has given rise to a certain amount of
controversy. The studies that gave birth to chaos have
revolutionized not only the way professional weather forecasters
forecast but even what a forecast consists of. These new ideas often
run counter to traditional statistical modelling methods, and still
produce both heat and light on how best to model the real world.
This battle is broken into skirmishes by the nature of the field and
our level of understanding in the particular system of which a
question is asked, be it the population of voles in Scandinavia,
a mathematical calculation to quantify chaos, the number of
spots on the Sun’s surface, the price of oil delivered next month,
tomorrow’s maximum temperature, or the date of the last ever solar
The skirmishes are interesting, but chaos offers deeper insights
even when both sides are fighting for traditional advantage, say, the
‘best’ model. Here studies of chaos have redefined the high ground:
today we are forced to reconsider new definitions for what
constitutes the best model, or even a ‘good’ model. Arguably, we
must give up the idea of approaching Truth, or at least define a
wholly new way of measuring our distance from it. The study of
chaos motivates us to establish utility without any hope of achieving
perfection, and to give up many obvious home truths of forecasting,

like the naı̈ve idea that a good forecast consists of a prediction that
is close to the target. This did not appear naı̈ve before we
understood the implications of chaos.

La Tour’s realistic vision of science in the real world

5. The Cheat with the Ace of Diamonds, by Georges de la Tour, painted
about 1645

The emergence of chaos

To close this chapter, we illustrate how chaos can force us to
reconsider what constitutes a good model, and revise our beliefs as
to what is ultimately responsible for our forecast failures. This
impact is felt by scientists and mathematicians alike, but the
reconsideration will vary depending on the individual’s point of
view and the empirical system under study. The situation is nicely
personified in Figure 5, a French baroque painting by Georges de la
Tour showing a card game from the 17th century. La Tour was
arguably a realist with a sense of humour. He was fond of fortune
telling and games of chance, especially those in which chance
played a somewhat lesser role than the participants happened to
believe. In theory, chaos can play exactly this role. We will interpret


this painting to show a mathematician, a physicist, a statistician,
and a philosopher engaged in an exercise of skill, dexterity, insight,
and computational prowess; this is arguably a description for doing
science, but the task at hand here is a game of poker. Exactly who is
who in the painting will remain open, as we will return to these
personifications of natural science throughout the book. The
insights chaos yields vary with the perspective of the viewer, but a
few observations are in order.
The impeccably groomed young man on the right is engaged in
careful calculations, no doubt a probability forecast of some nature;
he is currently in possession of a handsome collection of gold coins
on the table. The dealer plays a critical role, without her there is no
game to be played; she provides the very language within which we
communicate, yet she seems to be in nonverbal communication
with the handmaiden. The role of the handmaiden is less clear; she
is perhaps tangential, but then again the provision of wine will
influence the game, and she herself may feature as a distraction.
The roguish character in ramshackle dress with bows untied is
clearly concerned with the real world, not mere appearances in
some model of it; his left hand is extracting one of several aces of
diamonds from his belt, which he is about to introduce into the
game. What then do the ‘probabilities’ calculated by the young man
count for, if, in fact, he is not playing the game his mathematical
model describes? And how deep is the insight of our rogue? His
glance is directed to us, suggesting that he knows we can see his
actions, perhaps even that he realizes that he is in a painting?
The story of chaos is important because it enables us to see the
world from the perspective of each of these players. Are we merely
developing the mathematical language with which the game is
played? Are we risking economic ruin by over-interpreting some
potentially useful model while losing sight of the fact that it, like all
models, is imperfect? Are we only observing the big picture, not
entering the game directly but sometimes providing an interesting
distraction? Or are we manipulating those things we can change,

acknowledging the risks of model inadequacy, and perhaps even our
own limitations, due to being within the system? To answer these
questions we must first examine several of the many jargons of
science in order to be able to see how chaos emerged from the noise
of traditional linear statistics to vie for roles both in understanding
and in predicting complicated real-world systems. Before the
nonlinear dynamics of chaos were widely recognized within science,
these questions fell primarily in the domain of the philosophers;
today they reach out via our mathematical models to physical
scientists and working forecasters, changing the statistics of
decision support and even impacting politicians and policy makers.

The emergence of chaos


Chapter 2
Exponential growth,
nonlinearity, common sense

One of the most pervasive myths about chaotic systems is that they
are impossible to predict. To expose the fallacy of this myth, we
must understand how uncertainty in a forecast grows as we predict
further and further into the future. In this chapter we investigate
the origin and meaning of exponential growth, since on average a
small uncertainty will grow exponentially fast in a chaotic system.
There is a sense in which this phenomenon really does imply a
‘faster’ growth of uncertainty than that found in our traditional
ideas of how error and uncertainty grow as we forecast further into
the future. Nevertheless, chaos can be easy to predict, sometimes.

Chess, rice, and Leonardo’s rabbits:
exponential growth
An oft-told story about the origin of the game of chess illustrates
nicely the speed of exponential growth. The story goes that a king of
ancient Persia was so pleased when first presented with the game
that he wanted to reward the game’s creator, Sissa Ben Dahir. A
chess board has 64 squares arranged in an 8 by 8 pattern; for his
reward, Ben Dahir requested what seemed a quite modest sum
of rice determined using the new chess board: one grain of rice
was to be put on the first square of the board, two to be put on
the second, four for the third, eight for the fourth, and so on,
doubling the number on each square until the 64th was reached. A

mathematician will often call any rule for generating one number
from another one a mathematical map, so we’ll refer to this simple
rule (‘double the current value to generate the next value’) as the
Rice Map.
Before working out just how much rice Ben Dahir has asked for, let
us consider the case of linear growth where we have one grain on
the first square, two on the second square, three on the third, and so
on until we need 64 for the last square. In this case we have a total
of 64 + 63 + 62 + . . . + 3 + 2 + 1, or around 1,000 grains. Just for
comparison, a 1 kilogram bag of rice contains a few tens of
thousands of grains.

By comparing the amount of rice on a given square in the case of
linear growth with the amount of rice on the same square in the
case of exponential growth, we quickly see that exponential is much
faster than linear growth: on the fourth square we already have
twice as many grains in the exponential case as in the linear case
(8 in the first, only 4 in the second), and by the eighth square, at the
end of the first row, the exponential case has 16 times more!
Soon thereafter we have the astronomical numbers.

Exponential growth, nonlinearity, common sense

The Rice Map requires one grain for the first square, then two for
the second, four for the third, then 8, 16, 32, 64, and 128 for the last
square of the first row. On the third square of the second row, we
pass 1,000 and before the end of the second row there is a square
which exhausts our bag of rice. To fill the next square alone will
require another entire bag, the following square two bags, and so
on. Some square in the third row will require a volume of rice
comparable to a small house, and we will have enough rice to fill the
Royal Albert Hall well before the end of the fifth row. Finally, the
64th square alone will require billions and billions, or to be exact,
263 (= 9, 223, 372, 036, 854, 775, 808) grains, for a total of
18,446,744,073,709,551,615 grains. That is a non-trivial quantity of
rice! It is something like the entire world’s rice production over two
millennia. Exponential growth quickly grows out of all proportion.


Of course, we hid the values of some parameters in the example
above: we could have made the linear growth faster by adding
not one additional grain for each square, but instead, say,
1,000 additional grains. This parameter, the number of
additional grains, defines the constant of proportionality between
the number of the square and the number of grains on that
square, and gives us the slope of the linear relationship between
them. There is also a parameter in the exponential case: on
each step we increased the number of grains by a factor of two,
but it could have been a factor of three, or a factor of one and a
One of the surprising things about exponential growth is that
whatever the values of these parameters, there will come a time at
which exponential growth surpasses any linear growth, and will
soon thereafter dwarf linear growth, no matter how fast the linear
growth is. Our ultimate interest is not in rice on a chess board, but
in the dynamics of uncertainty in time. Not just the growth of a
population, but the growth of our uncertainty in a forecast of the
future size of that population. In the forecasting context, there will
come a time at which an exponentially growing uncertainty which is
very small today will surpass a linearly growing uncertainty which is
today much larger. And the same thing happens when contrasting
exponential growth with growth proportional to the square of time,
or to the cube of time, or to time raised to any power (in symbols:
steady exponential growth will eventually surpass the growth
proportional to t2 or t3 or tn for any value of n.). It is for this reason
among others that exponential growth is mathematically
distinguished, and taken to provide a benchmark for defining
chaos. It has also contributed to the widespread but fundamentally
mistaken impression that chaotic systems are hopelessly
unpredictable. Ben Dahir’s chess board illustrates that there is a
deep sense in which exponential growth is faster than linear growth.
To place this in the context of forecasting, we move forward a few
hundred years in time and a few hundred miles northwest, from
Persia to Italy.

So what does this ‘population dynamic’ look like? In the first month
we have one immature pair, in the second month we have one
mature pair, in the third month we have one mature pair and a new
immature pair, in the fourth month we have two mature pairs and
one immature pair, in the fifth month we have three mature pairs
and two immature.
If we count up all the pairs each month, the numbers are 1, 1, 2, 3, 5,
8, 13, 21 . . . . Leonardo noted that the next number in the series is
always the sum of the previous two numbers (1 + 1 = 2, 2 + 1 = 3,
3 + 2 = 5, . . . ) which makes sense, as the previous number is the
number we had last month (in our model all rabbits survive no
matter how many there are), and the penultimate number is the
number of mature pairs (and thus the number of new pairs arriving
this month).
Now it gets a bit tedious to write ‘and in the sixth month we have
12 pairs of rabbits’, so scientists often use a short-hand X for the
number of pairs of rabbits and X6 to denote the number of pairs in
month six. And since the series 1, 1, 2, 3, 5, 8, . . . reflects how the
population of rabbits evolves in time, this series and others like it
are called time series. The Rabbit Map is defined by the rule:

Exponential growth, nonlinearity, common sense

At the beginning of the 13th century, Leonardo of Pisa posed a
question of population dynamics: given a newborn pair of rabbits in
a large, lush, walled garden, how many pairs of rabbits will we have
in one year if their nature is for each mature pair to breed and
produce a new pair every month, and newborn rabbits mature in
their second month? In the first month we have one juvenile pair. In
the second month this pair matures and breeds to produce a new
pair in the third month. So in the third month, we have one mature
pair and one newborn pair. In the fourth month we once again have
one new born pair from the original pair of rabbits and now two
mature pairs for a total of three pairs. In the fifth month, two new
pairs are born (one from each mature pair), and we have three
mature pairs for a total of five pairs. And so on.

Add the previous value of X to the current value of X, and take
the sum as the new value of X.


The numbers in the series 1, 1, 2, 3, 5, 8, 13, 21, 34 . . . are
called Fibonacci numbers (Fibonacci was a nickname of
Leonardo of Pisa), and they arise again and again in nature: in the
structure of sunflowers, pine cones, and pineapples. They are of
interest here because they illustrate exponential growth in time,
almost. The crosses in Figure 6 are Fibonacci’s points – the
rabbit population as a function of time – while the solid line
reflects two raised to the power λt, or in symbols 2λt , where t is
the time in months and λ is our first exponent. Exponents which
multiply time in the superscript are a useful way of quantifying
uniform exponential growth. In this case, λ is equal to the
logarithm of a number called the golden mean, a very special
number which is discussed in the Very Short Introduction to

6. The series of crosses showing the number of pairs of rabbits each
month (Fibonacci numbers); the smooth curve they lie near is the
related exponential growth

So how can Leonardo’s rabbits help us to get a feel for the growth of
forecast uncertainty? Like all observations, counting the number
of rabbits in a garden is subject to error; as we saw in Chapter 1,
observational uncertainties are said to be caused by noise. Imagine
that Leonardo failed to notice a pair of mature rabbits also in the
garden in the first month; in that case, the number of pairs
actually in the garden would have been 2, 3, 5, 8, 13, . . . The error in
the original forecast (1, 1, 2, 3, 5, 8 . . . ) would be the difference
between the Truth and that forecast, namely: 1, 2, 3, 5 . . . (again, the
Fibonacci series). In month 12, this error has reached a very
noticeable 146 pairs of rabbits! A small error in the initial number
of rabbits results in a very large error in the forecast. In fact, the
error is growing exponentially in time. This has many implications.
Consider the impact of the exponential error growth on the
uncertainty of our forecasts. Let us again contrast linear growth and
exponential growth. Let’s assume that, for a price, we can reduce
the uncertainty in the initial observation that we use in generating

Exponential growth, nonlinearity, common sense

The first thing to notice about Figure 6 is that the points lie close to
the curve. The exponential curve is special in mathematics because
it reflects a function whose increase is proportional to its current
value. The larger it gets, the faster it grows. It makes sense that
something like this function would describe the dynamics of
Leonardo’s rabbit population since the number of rabbits next
month is more or less proportional to the number of rabbits this
month. The second thing to notice about the figure is that the points
do not lie on the curve. The curve is a good model for Fibonacci’s
Rabbit Map, but it is not perfect: at the end of each month the
number of rabbits is always a whole number and, while the curve
may be close to the correct whole number, it is not exactly equal to
it. As the months go by and the population grows, the curve gets
closer and closer to each Fibonacci number, but it never reaches
them. This concept of getting closer and closer but never quite
arriving is one that will come up again and again in this


our forecast. If the error growth is linear, and we reduce our
initial uncertainty by a factor of ten, then we can forecast the
system ten times longer before our uncertainty exceeds the same
threshold. If we reduce the initial uncertainty by a factor of 1,000,
then we can get forecasts of the same quality 1,000 times longer.
This is an advantage of linear models. Or, more accurately, this is
an apparent advantage of studying only linear systems. By
contrast, if the model is nonlinear and the uncertainty grows
exponentially, then we may reduce our initial uncertainty by a
factor of ten yet only be able to forecast twice as long with the
same accuracy. In that case, assuming the exponential growth in
uncertainty is uniform in time, reducing the uncertainty by a
factor of 1,000 will only increase our forecast range at the same
accuracy by a factor of eight. Now reducing the uncertainty in a
measurement is rarely free (we have to hire someone else to count
the rabbits a second time), and large reductions of uncertainty
can be expensive, so when uncertainty grows exponentially fast,
the cost sky-rockets. Attempting to achieve our forecast goals by
reducing uncertainty in initial conditions can be tremendously
Luckily, there is an alternative that allows us to accept the simple
fact that we can never be certain that any observation has not been
corrupted by noise. In the case of rabbits or grains of rice, it seems
there really is a fact of the matter, a whole number that reflects the
correct answer. If we reduce the uncertainty in this initial condition
to zero then we can predict without error. But can we ever really be
certain of the initial condition? Might there not be another bunny
hiding in the noise? While our best guess is that there is one pair in
the garden, there might be two, or three, or more (or perhaps zero).
When we are uncertain of the initial condition, we can examine the
diversity of forecasts under our model by making an ensemble of
forecasts: one forecast started from each initial condition we think
plausible. So one member of the ensemble will start with X equal to
one, another ensemble member will start with X equals two, and so
on. How should we divide our limited resources between computing

more ensemble members and making better observations of the
current number of rabbits in the garden?

But what if we were measuring something that is not a whole
number, like temperature, or the position of a planet? And is
temperature in an imperfect weather model exactly the same thing
as temperature in the real world? It was these questions that
initially interested our philosopher in chaos. First, we should
consider the more pressing question of why rabbits have not taken
over the world in the 9,000 months since 1202?

Stretching, folding, and the growth of uncertainty
The study of chaos lends credence to the meteorological maxim that
no forecast is complete without a useful estimate of forecast
uncertainty: if we know our initial condition is uncertain then we
are not only interested in the prediction per se, but equally in
learning what the likely forecast error will be. Forecast error for any

Exponential growth, nonlinearity, common sense

In the Rabbit Map, differences between the forecasts of different
members of the ensemble will grow exponentially fast, but with an
ensemble forecast we can see just how different they are and use
this as a measure of our uncertainty in the number of rabbits we
expect at any given time. In addition, if we carefully count the
number of rabbits after a few months, we can all but rule out some
of the individual ensemble members. Each of these ensemble
members was started from some estimate of the number of rabbits
that were in the garden originally, so ruling an ensemble member
out in effect gives us more information about the original number of
rabbits. Of course, this information need only prove accurate if our
model is literally perfect, meaning, in this case, that our Rabbit Map
captures the reproductive behaviour and longevity of our rabbits
exactly. But if our model is perfect, then we can use future
observations to learn about the past; this process is called noise
reduction. If it turns out that our model is not perfect, then we may
end up with incoherent results.

Exponential growth: an example from
Miss Nagel’s third grade class
A few months ago, I received an email written by an old
friend of mine from elementary school. It contained another
email that had originated from a third grader in North
Carolina whose class was studying geography. It requested
that everyone who read the email send a reply to the school
stating where they lived, and the class would locate that place
on a school globe. It also requested that each reader pass on
the email to ten friends.
I did not forward the message to anyone, but I did write an
email to Miss Nagel’s class stating that I was in Oxford,


England. I also suggested that they tell their mathematics
teacher about their experiment and use it as an example to
illustrate exponential growth: if they sent the message to ten
people, and the next day each of them sent it to ten more
people, that would be 100 on day three, 1,000 on day four,
and more emails than there are email addresses within a
week or so. In a real system, exponential growth cannot go
on forever: eventually we run out of rice, or garden space, or
new email addresses. It is often the resources that limit
growth: even a lush garden provides only a finite amount
of rabbit food. There are limits to growth which bound
populations, if not our models of populations.
I never found out whether Miss Nagel’s class learned their
lesson in exponential growth. The only answer I ever
received was an automated reply stating that the school’s
email in-box had exceeded its quota and had been closed.


real system should not grow without limit; even if we start with a
small error like one grain or one rabbit, the forecast error will not
grow arbitrarily large (unless we have a very naı̈ve forecaster), but
will saturate near some limiting value, as would the population
itself. Our mathematician has a way to avoid ludicrously large
forecast errors (other than naı̈veté), namely by making the initial
uncertainty infinitesimally small – smaller than any number you
can think of, yet greater than zero. Such an uncertainty will stay
infinitesimally small for all time, even if it grows exponentially fast.

Whenever our model goes into never-never land (suggesting values
where no data have ever gone before), then something is likely to
give, unless something in our model has already broken. Often, as
our uncertainty grows too large, it starts to fold back on itself.
Imagine kneading dough, or a toffee machine continuously
stretching and folding toffee. An imaginary line of toffee connecting
two very nearby grains of sugar will grow longer and longer as these
two grains separate under the action of the machine, but before it
becomes bigger than the machine itself, this line will be folded back
into itself, forming a horrible tangle. The distance between the
grains of sugar will stop growing, even as the string of toffee
connecting them continues to grow longer and longer, becoming a
more and more complicated tangle. The toffee machine gives us a
way to envision limits to the growth of prediction error whenever
our model is perfect. In this case, the error is the growing distance

Exponential growth, nonlinearity, common sense

Physical factors, like the total amount of rabbit food in the garden
or the amount of disk space on an email system, limit growth in
practice. The limits are intuitive even if we do not know exactly
what causes them: I think I have lost my keys in the car park; of
course they might be several miles from there, but it is exceedingly
unlikely that they are farther away than the moon. I do not need to
understand or believe the laws of gravity to appreciate this.
Similarly, weather forecasters are rarely more than 100 degrees off,
even for a forecast one year in advance! Even inadequate models
can usually be constrained so that their forecast errors are bounded.


between the True state and our best guess of that state: any
exponential growth of error would correspond only to the rapid
initial growth of the string of toffee. But if our forecasts are not
going to zoom away towards infinity (the toffee must stay in the
machine, only a finite number of rabbits will fit in the garden, and
the like), then eventually the line connecting Truth and our forecast
will be folded over on itself. There is simply nowhere else for it to
grow into. In many ways, identifying the movement of a grain of
sugar in the toffee machine with the evolution of the state of a
chaotic system in three dimensions is a useful way to visualize
chaotic motion.
We want to require a sense of containment for chaos, since it is
hardly surprising that it is difficult to predict things that are flying
apart to infinity, but we do not want to impose so strict a condition
as requiring a forecast to never exceed some limited value, no
matter how big that value might be. As a compromise, we require
the system to come back to the vicinity of its current state at some
point in the future, and to do so again and again. It can take as long
as it wants to come back, and we can define coming back to mean
returning closer to the current point than we have ever seen it
return before. If this happens, then the trajectory is said to be
recurrent. The toffee again provides an analogy: if the motion was
chaotic and we wait long enough, our two grains of sugar will again
come back close together, and each will pass close to where it was at
the beginning of the experiment, assuming no one turns off the
machine in the meantime.


Chapter 3
Chaos in context:
determinism, randomness,
and noise

All linear systems resemble one another, each nonlinear system is
nonlinear in its own way.
After Tolstoy’s Anna Karenina

Dynamical systems
Chaos is a property of dynamical systems. And a dynamical system
is nothing more than a source of changing observations: Fibonacci’s
imaginary garden with its rabbits, the Earth’s atmosphere as
reflected by a thermometer at London’s Heathrow airport, the
economy as observed through the price of IBM stock, a computer
program simulating the orbit of the moon and printing out the
date and location of each future solar eclipse.
There are at least three different kinds of dynamical systems. Chaos
is most easily defined in mathematical dynamical systems. These
systems consist of a rule: you put a number in and you get a new
number out, which you put back in, to get yet a newer number out,
which you put back in. And so on. This process is called iteration.
The number of rabbits each month in Fibonacci’s imaginary garden
is a perfect example of a time series from this kind of system. A
second type of dynamical system is found in the empirical world of
the physicist, the biologist, or the stock market trader. Here, our
sequence of observations consists of noisy measurements of reality,


which are fundamentally different from the noise-free numbers of
the Rabbit Map. In these physical dynamical systems – the Earth’s
atmosphere and Scandinavia’s vole population, for example –
numbers represent the state, whereas in the Rabbit Map they were
the state. To avoid needless confusion, it is useful to distinguish a
third case when a digital computer performs the arithmetic
specified by a mathematical dynamical system; we will call this a
computer simulation – computer programs that produce TV
weather forecasts are a common example. It is important to
remember that these are different kinds of systems and that each is
a different beast: our best equations for the weather differ from our
best computer models based on those equations, and both of these
systems differ from the real thing the Earth’s atmosphere itself.
Confusingly, the numbers from each of our three types of systems
are called time series, and we must constantly struggle to keep in
mind the distinction between what these are time series of: a
number of imaginary rabbits, the True temperature at the airport (if
such a thing exists), a measurement representing that temperature,
and a computer simulation of that temperature.
The extent to which these differences are important depends on
what we aim to do. Like la Tour’s card players, scientists,
mathematicians, statisticians, and philosophers each have different
talents and aims. The physicist may aim to describe the
observations with a mathematical model, perhaps testing the
model by using it to predict future observations. Our physicist is
willing to sacrifice mathematical tractability for physical relevance.
Mathematicians like to prove things that are true for a wide range
of systems, but they value proof so highly that they often do not
care how widely they must restrict that range to have it; one
should almost always be wary whenever a mathematician is
heard to say ‘almost every’. Our physicist must be careful not to
forget this and confuse mathematical utility with physical
relevance; physical intuitions should not be biased by the properties
of ‘well-understood’ systems designed only for their mathematical

Mathematical dynamical systems and attractors
We commonly find four different types of behaviour in time series.
They can (i) grind to a halt and more or less repeat the same fixed
number over and over again, (ii) bounce around in a closed loop like
a broken record, periodically repeating the same pattern: exactly
the same series of numbers over and over, (iii) move in a loop that
has more than one period and so does not quite repeat exactly but
comes close, like the moment of high tide drifting through the time
of day, or (iv) forever jump about wildly, or perhaps even calmly,
displaying no obvious pattern. The fourth type looks random, yet
looks can be deceiving. Chaos can look random but it is not random.
In fact, as we have learned to see better, chaos often does not even
look all that random to us anymore. In the next few pages we will
introduce several more maps, though perhaps without the rice or
rabbits. We need these maps in order to generate interesting
artefacts for our tour in search of the various types of behaviour just
noted. Some of these maps were generated by mathematicians for
this very purpose, although our physicist might argue, with reason,
that a given map was derived by simplifying physical laws. In truth,
the maps are simple enough to have each come about in several
different ways.

Chaos in context: determinism, randomness, and noise

Our statistician is interested in describing interesting statistics
from the time series of real observations and in studying the
properties of dynamical systems that generate time series which
look like the observations, always taking care to make as few
assumptions as possible. Finally, our philosopher questions the
relationships among the underlying physical system that we claim
generated the observations, the observations themselves, and the
mathematical models or statistical techniques that we created to
analyse them. For example, she is interested in what we can know
about the relationship between the temperature we measure and
the true temperature (if such a thing exists), and in whether the
limits on our knowledge are merely practical difficulties we might
resolve or limits in principle that we can never overcome.


Before we can produce a time series by iterating a map, we need
some number to start with. This first number is called an initial
condition, an initial state that we define, discover, or arrange for
our system to be. As in Chapter 2, we adopt the symbol X as shorthand for a state of our system. The collection of all possible states X
is called the state space. For Fibonacci’s imaginary rabbits, this
would be the set of all whole numbers. Suppose our time series is
from a model of the average number of insects per square mile at
mid-summer each year. In that case, X is just a number and the
state space, being the collection of all possible states, is then a line.
It sometimes takes more than one number to define the state, and if
so X will have more than one component. In predator-prey models,
for instance, the populations of both are required and X has two
components: it is a vector. When X is a vector containing both the
number of voles (prey) and the number of weasels (predators) on
the first of January each year, then the state space will be a twodimensional surface – a plane – that contains all pairs of numbers.
If X has three components (say, voles, weasels, and annual
snowfall), then the state space is a three-dimensional space
containing all triplets of numbers. Of course, there is no reason to
stop at three components; although the pictures become more
challenging to draw in higher dimensions, modern weather models
have over 10,000,000 components. For a mathematical system,
X can even be a continuous field, like the height of the surface
of the ocean or the temperature at every point on the surface of
the Earth. However, our observations of physical systems will
never be more complicated than a vector, and since we will only
measure a finite number of things, our observations will always be
finite-dimensional vectors. For the time being, we will consider the
case in which X is a simple number, such as one-half.
Recalling that a mathematical map is just a rule that transforms one
set of values into the next set of values, you can define the
Quadrupling Map by the rule:
Multiply X by four to form the new value of X.

Given an initial condition, like X equals one-half, this mathematical
dynamical system produces a time series of values of X, in this case
½ × 4 = 2, 2 × 4 = 8, 8 × 4 = 32 . . . and the time series is 0.5, 2, 8, 32,
128, 512, 2048 . . . And so on. This series just gets bigger and
bigger and, dynamically speaking, that is not so interesting. If a
time series of X grows without limit like this one does, we call it
unbounded. In order to get a dynamical system where X is bounded,
we’ll take a second example, the Quartering Map:
Take X divided by four as the new X

In the Full Logistic Map, time series from almost every X bounces
around irregularly between zero and one forever:

Chaos in context: determinism, randomness, and noise

Starting at X = ½ yields the time series 1/8, 1/32, 1/128, . . . . At first
sight, this is not very exciting since X rapidly shrinks towards zero.
But in fact, the Quartering Map has been carefully designed to
illustrate special mathematical properties. The origin – the state
X = 0 – is a fixed point: if we start there we will never leave, since
zero divided by four is again zero. The origin is also our first
attractor; under the Quartering Map the origin is the inevitable if
unreachable destination: if we start with some other value of X, we
never actually make it to the attractor, although we get close as the
number of iterations increases without limit. How close? Arbitrarily
close. As close as you like. Infinitesimally close, meaning closer
than any number you can name. Name a number, any number, and
we can work out how many iterations are required after which X
will remain closer to zero than that number. Getting arbitrarily
close to an attractor as time goes on while never quite reaching it is
a common feature of many time series from nonlinear systems. The
pendulum provides a physical analogue: each swing will be smaller
than the last, an effect we blame on air resistance and friction. The
analogue of the attractor in this case is the motionless pendulum
hanging straight down. We will have more to say about attractors
after we have added a few more dynamical systems to our

Subtract X2 from X, multiply the difference by four and take the
result as the new X.


If we multiply components of state variables by other components,
things become nonlinear. What is the time series in this case if we
again start with X equals one-half? Starting with ½, X minus X2 is
¼, times four is one, so our new value is one. Continuing with X
now equal to one, we have X minus X2 is zero. But four times zero is
always zero, so we’ll get zeros forever. And our time series is 0.5, 1,
0, 0, 0 . . . This does not blow up, but it is hardly exciting; recall the
warning about ‘almost every’.
The order of the numbers in a time series is important, whether the
series reflects monthly values of Fibanocci’s rabbits or iterations of
the Full Logistic Map. Using the short-hand suggested in Chapter 2,
we will write X5 for the fifth new value of X, and X0 for the initial
state (or observation), and in general Xi for the ith value. Whether
we are iterating the map or taking observations, i is always an
integer and is often called ‘time’.
In the Full Logistic Map with X0 is equal to 0.5, X1 is equal to 1, X2
is 0, X3 is 0, X4 is 0, and Xi will be zero for all i greater than four as
well. So the origin is again a fixed point. But under the Full Logistic
Map small values of X grow (you can check this with a hand
calculator), X = 0 is unstable and so the origin is not an attractor.
A time series started near the origin is in fact unlikely to take one of
the first three options noted at the opening of this section, but to
bounce about chaotically forever.
Figure 7 shows a time series starting near X0 equals 0.876; it
represents a chaotic time series from the Full Logistic Map. But
look at it closely: does it really look completely unpredictable? It
looks like small values of X are followed by small values of X, and
that there is a tendency for the time series to linger whenever it is
near three-quarters. Our physicist would look at this series and
expect it to be predictable at least sometimes, while, after a few

calculations, our statistician might even declare it random.
Although we can see this structure, the most common statistical
tests cannot.

A menagerie of maps
The rule that defines a map can be stated either in words, or as an
equation, or in a graph. Each panel of Figure 8 defines the rule
graphically. To use the graph, find the current value of X on the
horizontal axis, and then move directly upward until you hit the
curve; the value of this point on the curve on the vertical axis is
the new value of X. The Full Logistic Map is shown graphically in
Figure 8 (b), while the Quarter Map is in panel (a).
An easy way of using the graph to see if a fixed point is unstable is to
look at the slope of the map at the fixed point: if the slope is steeper
than 45 degrees (either up or down); then the fixed point is

Chaos in context: determinism, randomness, and noise

7. A chaotic time series from the Full Logistic Map starting near
X0 equals 0.876. Note the series is visibly predictable whenever X is near
zero and three-quarters

8. Graphical presentation of the (a) Quarter Map, (b) Full Logistic Map, (c) Shift Map, (d) Tent Map, (e) Tripling Tent Map,
and (f) the Moran-Ricker Map


unstable. In the Quartering Map the slope is less than one
everywhere, while for the Full Logistic Map the slope near the
origin is greater than one. Here small but non-zero values of X grow
with each iteration but only as long as they stay sufficiently small
(the slope near ½ is zero). As we will see below, for almost every
initial condition between zero and one, the time series displays true
mathematical chaos. The Full Logistic Map is pretty simple; chaos
is pretty common.
To see if a mathematical system is deterministic merely requires
checking carefully whether carrying out the rule requires a random
number. If not, then the dynamical system is deterministic: every
time we put the same value of X in, we get the same new value of X
out. If the rule requires (really requires) a random number, then the
system is random, also called stochastic. With a stochastic system,
even if we iterate exactly the same initial condition we expect the
details of the next value of X and thus the time series to be different.
Looking back at their definitions, we see that the three maps
defined above are each deterministic; their future time series is
completely determined by the initial condition, hence the name
‘deterministic system’. Our philosopher would point out that just
knowing X is not enough, we also need to know the mathematical
system and we have to have the power to do exact calculations with
it. These were the three gifts Laplace ensured his demon possessed
200 years ago.
Our first stochastic dynamical system is the AC Map:
Divide X by four, then subtract ½ and add a random number R to
get the new X.

The AC Map is a stochastic system since applying the rule requires
access to a supply of random numbers. In fact, the rule above is
incomplete, since it does not specify how to get R. To complete the
definition we must add something like: for R on each iteration, pick
a number between zero and one in a manner that each number is
equally likely to be chosen, which implies that R will be uniformly

distributed between zero and one and that the probability of the
next value of R falling in an interval of values is proportional to the
width of that interval.

In the AC Map, each value of R is used within the map, but there is
another class of random maps – called Iterated Function Systems,
or IFS for short – which appear to use the value of R not in a
formula but to make a decision as to what to do. One example is the
Middle Thirds IFS Map, which will come in handy later when we
try to work out the properties of maps from the time series that they
generate. The Middle Thirds IFS Map is:
Take a random number R from a uniform distribution between zero
and one.
If R is less than a half, take X/3 as the new X
Otherwise take 1 – X/3 as the new X.

So now we have a few mathematical systems, and we can easily
tell if they are deterministic or stochastic. What about computer
simulations? Digital computer simulations are always
deterministic. And as we shall see in Chapter 7, the time series
from a digital computer is either on an endless loop of values
repeating itself periodically, over and over again, or it is on its
way towards such a loop. This first part of a time series in which
no value is repeated, the trajectory is evolving towards a periodic
loop but has not reached it, is called a transient. In
mathematical circles, this word is something of an insult, since

Chaos in context: determinism, randomness, and noise

What rule do we use to pick R? It could not be a deterministic rule,
since then R would not be random. Arguably, there is no finite rule
for generating values of R. This has nothing to do with needing
uniform numbers between zero and one. We’d have the same
problem if we wanted to generate random numbers which
mimicked Galton’s ‘bell-shape’ distribution. We will have to rely on
our statistician to somehow get us the random numbers we need;
hereafter we’ll just state whether they have a uniform distribution
or the bell-shaped distribution.


mathematicians prefer to work with long-lived things, not mere
transients. While mathematicians avoid transients, physical
scientists may never see anything else and, as it turns out, digital
computers cannot maintain them. The digital computers that
have proven critical in advancing our understanding of chaos
cannot, ironically, display true mathematical chaos themselves.
Neither can a digital computer generate random numbers. The
so-called random number generators on digital computers and
hand calculators are, in fact, only pseudo-random number
generators; one of the earliest of these generators was even based
on the Full Logistic Map! The difference between mathematical
chaos and computer simulations, like that between random
numbers and pseudo-random numbers, exemplifies the
difference between our mathematical systems and our computer
The maps in Figure 8 are not there by chance. Mathematicians
often construct systems in such a way that it will be relatively simple
for them to illustrate some mathematical point or allow the
application of some specific manipulation – a word they sometimes
use to obscure technical sleight of hand. The really complicated
maps – including the ones used to guide spacecraft and the ones
called ‘climate models’, and the even bigger ones used in numerical
weather prediction – are clearly constructed by physicists, not
mathematicians. But they all work the same way: a value of X goes
in and a new value X comes out. The mechanism is exactly the same
as in the simple maps defined above, even if X might have over
10,000,000 components.

Parameters and model structure
The rules that define the maps above each involve numbers other
than the state, numbers like four and one-half. These numbers are
called parameters. While X changes with time, parameters remain
fixed. It is sometimes useful to contrast the properties of time series
generated using different parameter values. So instead of

defining the map with a particular parameter value, like 4, maps
are usually defined using a symbol for the parameter, say α. We
can then contrast the behaviour of the map at α equals 4 with
that at α = 2, or α = 3.569945, for example. Greek symbols are
often used to clearly distinguish parameters from state variables.
Rewriting the Full Logistic Map with a parameter yields one of
the most famous systems of nonlinear dynamics: the Logistic
Subtract X2 from X, then multiply by α and take the result as the
new X.

Recall the Quartering Map, noting that after one iteration every
point between zero and one will be between zero and one-quarter.
Since all the points between zero and one-quarter are also between
zero and one, none of these points can ever escape to values greater
than one or less than zero. Dynamical systems in which, on average,
line segments (or in higher dimensions, areas or volumes) shrink
are called dissipative. Whenever a dissipative map translates a
volume of state space completely inside itself, we know immediately
that an attractor exists without knowing what it looks like.

Chaos in context: determinism, randomness, and noise

In physical models, parameters are used to represent things like
the temperature at which water boils, or the mass of the Earth,
or the speed of light, or even the speed with which ice ‘falls’ in the
upper atmosphere. Statisticians often dismiss the distinction
between the parameter and the state, while physicists tend to
give parameters special status. Applied mathematicians, as it
turns out, often force parameters towards the infinitely large or
the infinitesimally small; it is easier, for example, to study the flow
of air over an infinitely long wing. Once again, these different points
of view each make sense in context. Do we require an exact solution
to an approximate question, or an approximate answer to a
particular question? In nonlinear systems, these can be very
different things.


Whenever α is less than four we can prove that the Logistic Map
has an attractor by looking at what happens to all the points
between zero and one. The largest new value of X we can get
is the iteration of X equals one-half. (Can you see this in
Figure 8?) This largest value is α/4, and as long as α is less than
four this largest value is less than one. That means every point
between zero and one iterates to a point between zero and α/4
and is confined there forever. So the system must have an
attractor. For small values of α the point X equals zero is the
attractor, just like in the Quartering Map. But if α is greater
than one, then any value of X near zero will move away and the
attractor is elsewhere. This is an example of a non-constructive
proof: we can prove that an attractor exists but, frustratingly,
the proof does not tell us how to find it nor give any hint of its
Multiple time series of the Logistic Map for each of four different
values of α are shown in Figure 9. In each panel, we start with
512 points taken at random between zero and one. At each
step we move the entire ensemble of points forward in time.
In the first step we see that all remain greater than zero, yet
move away from X equals one never to return: we have an
attractor. In (a) we see them all collapsing onto the period one
loop; in (b) onto one of the two points in the period two loop;
in (c) onto one of the four points of the period four loop. In (d),
we can see that they are collapsing, but it is not clear what the
period is. To make the dynamics more plainly visible, one
member of our ensemble is chosen at random in the middle
of the graph, and the points on its trajectory are joined by a line
from that point forward. The period one loop (a) appears as a
straight line, while (b) and (c) show the trajectories alternating
between two or four points, respectively. While (d) first looks like
a period four loop as well, but a closer look shows that there are
many more than four options, and that while there is regularity in
the order in which the bands of points are visited, no simple
periodicity is visible.

To get a different picture of the same phenomena, we can
examine many different initial conditions and different values
for α at the same time, as shown in Figure 13 (page 63). In this
three-dimensional view, the initial states can be seen randomly
scattered on the back left of the box. At each iteration, they move
out towards you and the points collapse towards the pattern shown
in the previous two figures. The iterated initial random states are
shown after 0, 2, 8, 32, 128, and 512 iterations; it takes some time
for the transients to die away, but the familiar patterns can be seen
emerging as the states reach the front of the box.

We can see now that a dynamical system has three components: the
mathematical rule that defines how to get the next value, the
parameter values, and the current state. We can, of course, change
any of these things and see what happens, but it is useful to
distinguish what type of change we are making. Similarly, we may
have insight into the uncertainty in one of these components, and it
is in our interest to avoid accounting for uncertainty in one
component by falsely attributing it to another.
Our physicist may be looking for the ‘True’ model, or only just a
useful one. In practice there is an art of ‘tuning’ parameter values.
And while nonlinearity requires us to reconsider how we find ‘good
parameter values’, chaos will force us to re-evaluate what we mean
by ‘good’. A very small difference in the value of a parameter which
has an unnoticeable impact on the quality of a short-term forecast
can alter the shape of an attractor beyond recognition. Systems in
which this happens are called structurally unstable. Weather
forecasters need not worry about this, but climate modellers must;
as Lorenz noted in the 1960s.
A great deal of confusion has arisen from the failure to
distinguish between uncertainty in the current state, uncertainty
in the value of a parameter, and uncertainty regarding the
model structure itself. Technically, chaos is a property of a

Chaos in context: determinism, randomness, and noise

Tuning model parameters and structural stability

9. Each frame shows the evolution of 512 points, initially spread at
random between zero and one, as they move forward under the Logistic
Map. Each panel shows one of four different values of α, showing the
collapse towards (a) a fixed point, (b) a period two loop, (c) a period four
loop, and (d) chaos. The solid line starting at time 32 shows the
trajectory of one point, in order to make the path on each attractor

dynamical system with fixed equations (structure) and specified
parameter values, so the uncertainty that chaos acts on is only
the uncertainty in the initial state. In practice, these distinctions
become blurred and the situation is much more interesting, and


Statistical models of Sun spots
Chaos is only found in deterministic systems. But to understand its
impact on science we need to view it against the background of
traditional stochastic models developed over the past century.
Whenever we see something repetitive in nature, periodic motion
is one of the first hypotheses to be deployed. It can make you
famous: Halley’s comet, and the Wolf Sun spot number. In the
end, the name often sticks even when we realize that the
phenomenon is not really periodic. Wolf guessed that the Sun
went through a cycle of about 11 years at a time when he had less
than 20 years’ data. Periodicity remains a useful concept even
though it is impossible to prove a physical system is periodic
regardless of how much data we take. So are the concepts of
determinism and chaos.
The solar record showed correlations with weather, with economic
activity, with human behaviour; even 100 years ago the 11-year
cycle could be ‘seen’ in tree rings. How could we model the Sun
spots cycle? Models of a frictionless pendulum are perfectly
periodic, while the solar cycle is not. In the 1920s, the Scottish
statistician Udny Yule discovered a new model structure, realizing
how to introduce randomness into the model and get more
realistic-looking time series behaviour. He likened the observed
time series of Sun spots to those from the model of a damped
pendulum, a pendulum with friction which would have a free
period of about 11 years. If this model pendulum were ‘left alone
in a quiet room’, the resulting time series would slowly damp down
to nothing. In order to motivate his introduction of random
numbers to keep the mathematical model going, Yule extended the

analogy with a physical pendulum: ‘Unfortunately, boys with pea
shooters get into the room, and pelt the pendulum from all sides at
random.’ The resulting models became a mainstay in the
statistician’s arsenal. A linear, stochastic mainstay. We will define
the Yule Map:
Take α times X plus a random value R to be the new value of X
where R is randomly drawn from the standard bell-shaped

Yule developed a model similar to the Yule Map that behaved more
like the time series of real Sun spots. Cycles in Yule’s improved
model differ slightly from one cycle to the next due to the random
effects, the details of the pea shooters. In a chaotic model the state
of the Sun differs from one cycle to the next. What about
predictability? In any chaotic model, almost all nearby initial
states will eventually diverge, while in each of Yule’s models even far
away initial states would converge, if both experienced the same
forcing from the pea shooters. This is an interesting and rather
fundamental difference: similar states diverge under deterministic
dynamics whereas they converge under linear stochastic dynamics.
That does not necessarily make Yule’s model easier to forecast, since
we never know the details of the future random forcing, but it
changes the way that uncertainty evolves in the system, as shown in

Chaos in context: determinism, randomness, and noise

So how does this stochastic model differ from a chaotic model?
There are two differences that immediately jump out at the
mathematician: the first is that Yule’s model is stochastic – the
rule requires a random number generator, while a chaotic model
of the Sun spots would be deterministic by definition. The second
is that Yule’s model is linear. This implies more than simply that
we do not multiply components of the state together in the
definition of the map; it also implies that one can combine
solutions of the system and get other acceptable solutions, a
property called superposition. This very useful property is not
present in nonlinear systems.


10. The evolution of uncertainty under the stochastic Yule Map.
Starting as a point at the bottom of the graph, the uncertainty spreads to
the left as we move forward in time (upwards) and approaches a
constant bell-shaped distribution

Figure 10. Here an initially small uncertainty, or even an
initially zero uncertainty, at the bottom grows wider and moves
to the left with each iteration. Note that the uncertainty in the
state seems to be approaching a bell-shaped distribution, and
has more or less stabilized by the time it reaches the top of the
graph. Once the uncertainty saturates in a static state, then all
predictability is lost; this final distribution is called the ‘climate’
of the model.

Physical dynamical systems
There is no way of proving the correctness of the position of
‘determinism’ or ‘indeterminism’. Only if science were complete or
demonstrably impossible could we decide such questions.
E. Mach (1905)

The time series we want to observe now is the state of the physical
system: say, the position of our nine planets relative to the Sun, the
number of fish or grouse. As a short-hand, we will again denote the
state of the system as X, while trying to remember that there is a
fundamental difference between a model-state and the True state, if
such a thing exists. It is unclear how these concepts stand in
relation to each other; as we shall see in Chapter 11, some
philosophers have argued that the discovery of chaos implies the
real world must have special mathematical properties. Other
philosophers, perhaps sometimes the same ones, have argued that
the discovery of chaos implies mathematics does not describe the
world. Such are philosophers.
In any event, we never have access to the True state of a physical
system, even if one exists. What we do have are observations,
which we will call ‘S’ to distinguish them from the state of the
system, X. What is the difference between X and S? The unsung
hero of science: noise. Noise is the glue that bonds the
experimentalists with the theorists on those occasions when they
meet. Noise is also the grease that allows theories to slide easily
over awkward facts.

Chaos in context: determinism, randomness, and noise

There is more to the world than mathematical models. Just about
anything we want to measure in the real world, or even just think
about observing, can be taken to have come from a physical
dynamical system. It cou