Chaos: A Very Short Introduction

Very Short Introductions are for anyone wanting a stimulatingand accessible way in to a new subject. They are written by experts, and havebeen published in more than 25 languages worldwide.The series began in 1995, and now represents a wide variety of topicsin history, philosophy, religion, science, and the humanities. Over the nextfew years it will grow to a library of around 200 volumes – a Very ShortIntroduction to everything from ancient Egypt and Indian philosophy toconceptual art and cosmology.Very Short Introductions available now:ANARCHISM Colin WardANCIENT EGYPT Ian ShawANCIENT PHILOSOPHYJulia AnnasANCIENT WARFAREHarry SidebottomANGLICANISM Mark ChapmanTHE ANGLO-SAXON AGEJohn BlairANIMAL RIGHTS David DeGraziaARCHAEOLOGY Paul BahnARCHITECTUREAndrew BallantyneARISTOTLE Jonathan BarnesART HISTORY Dana ArnoldART THEORY Cynthia FreelandTHE HISTORY OFASTRONOMY Michael HoskinAtheism Julian BagginiAugustine Henry ChadwickBARTHES Jonathan CullerTHE BIBLE John RichesTHE BRAIN Michael O’SheaBRITISH POLITICSAnthony WrightBuddha Michael CarrithersBUDDHISM Damien KeownBUDDHIST ETHICSDamien KeownCAPITALISM James FulcherTHE CELTS Barry CunliffeCHAOS Leonard SmithCHOICE THEORYMichael AllinghamCHRISTIAN ART Beth WilliamsonCHRISTIANITY Linda WoodheadCLASSICS Mary Beard andJohn HendersonCLAUSEWITZ Michael HowardTHE COLD WAR Robert McMahonCONSCIOUSNESS Susan BlackmoreCONTEMPORARY ARTJulian StallabrassContinental PhilosophySimon CritchleyCOSMOLOGY Peter ColesTHE CRUSADESChristopher TyermanCRYPTOGRAPHYFred Piper and Sean MurphyDADA AND SURREALISMDavid HopkinsDarwin Jonathan HowardTHE DEAD SEA SCROLLSTimothy LimDemocracy Bernard CrickDESCARTES Tom SorellDESIGN John HeskettDINOSAURS David NormanDREAMING J. Allan HobsonDRUGS Leslie IversenTHE EARTH Martin RedfernECONOMICS Partha DasguptaEGYPTIAN MYTH Geraldine Pinch

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Leonard A. SmithCHAOSA Very Short Introduction1

3Great Clarendon Street, Oxford o x 2 6 d pOxford University Press is a department of the University of Oxford.It furthers the University’s objective of excellence in research, scholarship,and education by publishing worldwide inOxford New YorkAuckland Cape Town Dar es Salaam Hong Kong KarachiKuala Lumpur Madrid Melbourne Mexico City NairobiNew Delhi Shanghai Taipei TorontoWith offices inArgentina Austria Brazil Chile Czech Republic France GreeceGuatemala Hungary Italy Japan Poland Portugal SingaporeSouth Korea Switzerland Thailand Turkey Ukraine VietnamOxford is a registered trade mark of Oxford University Pressin the UK and in certain other countriesPublished in the United Statesby Oxford University Press Inc., New York Leonard A. Smith 2007The moral rights of the author have been assertedDatabase right Oxford University Press (maker)First published as a Very Short Introduction 2007All rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,without the prior permission in writing of Oxford University Press,or as expressly permitted by law, or under terms agreed with the appropriatereprographics rights organizations. Enquiries concerning reproductionoutside the scope of the above should be sent to the Rights Department,Oxford University Press, at the address aboveYou must not circulate this book in any other binding or coverand you must impose this same condition on any acquirerBritish Library Cataloguing in Publication DataData availableLibrary of Congress Cataloging in Publication DataData availableTypeset by RefineCatch Ltd, Bungay, SuffolkPrinted in Great Britain byAshford Colour Press Ltd, Gosport, Hampshire978–0–19–285378–31 3 5 7 9 10 8 6 4 2

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

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ContentsAcknowledgements xiPrefacexiiList of illustrations xv123The emergence of chaos 1Exponential growth, nonlinearity, common sense 22Chaos in context: determinism, randomness,and noise456789101133Chaos in mathematical models 58Fractals, strange attractors, and dimension(s)76Quantifying the dynamics of uncertainty 87Real numbers, real observations, and computers 104Sorry, wrong number: statistics and chaos112Predictability: does chaos constrain our forecasts?Applied chaos: can we see through our models?Philosophy in chaos 154Glossary 163123132

Further reading 169Index 173

AcknowledgementsThis book would not have been possible without my parents, ofcourse, but I owe a greater debt than most to their faith, doubt, andhope, and to the love and patience of a, b, and c. Professionally mygreatest debt is to Ed Spiegel, a father of chaos and my thesisProfessor, mentor, and friend. I also profited immensely fromhaving 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 toacknowledge discussions with, and the support of, the Masterand Fellows of Pembroke College, Oxford. Lastly and largely, I’dlike to acknowledge my debt to my students, they know who theyare. I am never sure how to react upon overhearing an exchangelike: ‘Did you know she was Lenny’s student?’, ‘Oh, that explainsa lot.’ Sorry guys: blame Spiegel.

PrefaceThe ‘chaos’ introduced in the following pages reflects phenomena inmathematics and the sciences, systems where (without cheating)small differences in the way things are now have huge consequencesin the way things will be in the future. It would be cheating, ofcourse, if things just happened randomly, or if everythingcontinually exploded forever. This book traces out the remarkablerichness that follows from three simple constraints, which we’ll callsensitivity, determinism, and recurrence. These constraints allowmathematical chaos: behaviour that looks random, but is notrandom. When allowed a bit of uncertainty, presumed to be theactive ingredient of forecasting, chaos has reignited a centuries-olddebate on the nature of the world.The book is self-contained, defining these terms as they areencountered. My aim is to show the what, where, and how of chaos;sidestepping any topics of ‘why’ which require an advancedmathematical background. Luckily, the description of chaos andforecasting lends itself to a visual, geometric understanding; ourexamination of chaos will take us to the coalface of predictabilitywithout equations, revealing open questions of active scientificresearch into the weather, climate, and other real-worldphenomena of interest.Recent popular interest in the science of chaos has evolved

differently than did the explosion of interest in science a centuryago when special relativity hit a popular nerve that was to throb fordecades. Why was the public reaction to science’s embrace ofmathematical chaos different? Perhaps one distinction is that mostof us already knew that, sometimes, very small differences can havehuge effects. The concept now called ‘chaos’ has its origins both inscience fiction and in science fact. Indeed, these ideas were wellgrounded in fiction before they were accepted as fact: perhaps thepublic were already well versed in the implications of chaos, whilethe scientists remained in denial? Great scientists andmathematicians had sufficient courage and insight to foresee thecoming of chaos, but until recently mainstream science required agood solution to be well behaved: fractal objects and chaotic curveswere considered not only deviant, but the sign of badly posedquestions. For a mathematician, few charges carry more shamethan the suggestion that one’s professional life has been spent on abadly posed question. Some scientists still dislike problems whoseresults are expected to be irreproducible even in theory. Thesolutions that chaos requires have only become widely acceptable inscientific 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 rootwithin applied sciences like meteorology and astronomy. Theapplied sciences are driven by a desire to understand and predictreality, a desire that overcame the niceties of whatever the formalmathematics of the day. This required rare individuals who couldspan the divide between our models of the world and the world as itis without convoluting the two; who could distinguish themathematics from the reality and thereby extend the mathematics.As in all Very Short Introductions, restrictions on space requireentire research programmes to be glossed over or omitted; Ipresent a few recurring themes in context, rather than a series ofshallow descriptions. My apologies to those whose work I haveomitted, and my thanks to Luciana O’Flaherty (my editor), WendyParker, and Lyn Grove for help in distinguishing between what

was most interesting to me and what I might make interestingto the reader.How to read this introductionWhile there is some mathematics in this book, there are noequations more complicated than X 2. Jargon is less easy todiscard. Words in bold italics you will have to come to grips with;these are terms that are central to chaos, brief definitions of thesewords can be found in the Glossary at the end of the book. Italics isused both for emphasis and to signal jargon needed for the nextpage or so, but which is unlikely to recur often throughout the book.Any questions that haunt you would be welcome online at on the discussion forum VSI Chaos. Moreinformation on these terms can be found rapidly at Wikipedia and , and in the Further reading.

List of illustrations1The first weather mapever published in anewspaper, preparedby Galton in 18756 A graph comparingFibonacci numbers andexponential growth267 The Times/NI SyndicationLimited2 Galton’s original sketchof the Galton Board93 The Times headlinefollowing the Burns’Day storm in 199013 The Times/NI SyndicationLimited 1990/John FrostNewspapers4 Modern weather mapshowing the Burns’ Daystorm and a two-dayahead forecast145 The Cheat with the Aceof Diamonds, c.1645,by Georges de la Tour 19Louvre, Paris. A chaotic time seriesfrom the Full LogisticMap398 Six mathematicalmaps409 Points collapsing ontofour attractors of theLogistic Map4810The evolution ofuncertainty under theYule Map5211Period doublingbehaviour in theLogistic Map1261A variety of morecomplicated behavioursin the Logistic Map62

13Three-dimensionalbifurcation diagram andthe collapse towardattractors in theLogistic Map632114The Lorenz attractorand the Moore-Spiegelattractor6715The evolution ofuncertainty in theLorenz System22 Predictable chaos asseen in four iterationsof the same mouseensemble under theBaker’s Map and aBaker’s ApprenticeMap1001668The Hénon attractorand a two-dimensionalslice of theMoore-Spiegelattractor7017A variety of behavioursfrom the Hénon-HeiliesSystem7218The Fournier Universe,as illustrated byFournier7819Time series from thestochastic Middle ThirdsIFS Map and thedeterministic TriplingTent Map8220 A close look at theHénon attractor,showing fractalstructureSchematic diagramsshowing the action ofthe Baker’s Map and aBaker’s ApprenticeMap9823 Card trick revealing thelimitations of digitalcomputers10824 Two views of data fromMachete’s electric circuit,suggestive of Takens’Theorem11825The Not A GaltonBoard26 An illustration of usinganalogues to make aforecast13427The state space of aclimate model136Crown Copyright28 Richardson’s dream F. Schuiten84128137

29 Two-day-ahead ECMWFensemble forecasts of theBurns’ Day storm14030 Four ensemble forecastsof the Machete’s MooreSpiegel Circuit150Figures 7, 8, 9, 11, 12, 13, 19, and 20 were produced with theassistance of Hailiang Du. Figures 24 and 30 were produced withthe assistance of Reason Machete. Figures 4 and 29 were producedwith the assistance of Martin Leutbecher with data kindly madeavailable by the European Centre for Medium-Range WeatherForecasting. Figure 27 is after M. Hume et al., The UKIP02Scientific Report, Tyndal Centre, University of East Anglia,Norwich, UK.The publisher and the author apologize for any errors or omissionsin the above list. If contacted they will be pleased to rectify these atthe earliest opportunity.

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Chapter 1The emergence of chaosEmbedded 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 thingthat could upset balances and knock down a line ofsmall dominoes and then big dominoes and thengigantic dominoes, all down the years across Time.Ray Bradbury (1952)Three hallmarks of mathematical chaosThe ‘butterfly effect’ has become a popular slogan of chaos. But is itreally so surprising that minor details sometimes have majorimpacts? Sometimes the proverbial minor detail is taken to be thedifference between a world with some butterfly and an alternativeuniverse that is exactly like the first, except that the butterfly isabsent; as a result of this small difference, the worlds soon come todiffer dramatically from one another. The mathematical version ofthis concept is known as sensitive dependence. Chaotic systemsnot only exhibit sensitive dependence, but two other properties aswell: they are deterministic, and they are nonlinear. In thischapter, we’ll see what these words mean and how these conceptscame into science.Chaos is important, in part, because it helps us to cope with1

unstable systems by improving our ability to describe, tounderstand, perhaps even to forecast them. Indeed, one of themyths of chaos we will debunk is that chaos makes forecasting auseless task. In an alternative but equally popular butterfly story,there is one world where a butterfly flaps its wings and anotherworld where it does not. This small difference means a tornadoappears in only one of these two worlds, linking chaos touncertainty and prediction: in which world are we? Chaos is thename given to the mechanism which allows such rapid growth ofuncertainty in our mathematical models. The image of chaosamplifying uncertainty and confounding forecasts will be arecurring theme throughout this Introduction.ChaosWhispers of chaosWarnings of chaos are everywhere, even in the nursery. Thewarning that a kingdom could be lost for the want of a nail can betraced back to the 14th century; the following version of the familiarnursery rhyme was published in Poor Richard’s Almanack in 1758by 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, butrather to describe the growth of uncertainty after the initial seed issown. In this case, explaining how it came to be that the rider waslost due to a missing nail, not the fact that the nail had gonemissing. 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 thekingdom wouldn’t have been lost either. We will often explore theproperties of chaotic systems by considering the impact of slightlydifferent situations.2

The study of chaos is common in applied sciences like astronomy,meteorology, population biology, and economics. Sciences makingaccurate observations of the world along with quantitativepredictions have provided the main players in the development ofchaos since the time of Isaac Newton. According to Newton’s Laws,the future of the solar system is completely determined by itscurrent state. The 19th-century scientist Pierre Laplace elevatedthis determinism to a key place in science. A world is deterministicif its current state completely defines its future. In 1820, Laplaceconjured up an entity now known as ‘Laplace’s demon’; in doing so,he linked determinism and the ability to predict in principle to thevery notion of success in science.We may regard the present state of the universe as the effect of itspast and the cause of its future. An intellect which at a certainpositions of all items of which nature is composed, if this intellectwere also vast enough to submit these data to analysis, it wouldembrace in a single formula the movements of the greatest bodies ofthe universe and those of the tiniest atom; for such an intellectnothing would be uncertain and the future just like the past wouldbe present before its eyes.Note that Laplace had the foresight to give his demon threeproperties: exact knowledge of the Laws of Nature (‘all the forces’),the ability to take a snapshot of the exact state of the universe (‘allthe positions’), and infinite computational resources (‘an intellectvast enough to submit these data to analysis’). For Laplace’sdemon, chaos poses no barrier to prediction. Throughout thisIntroduction, we will consider the impact of removing one or moreof these gifts.From the time of Newton until the close of the 19th century, mostscientists were also meteorologists. Chaos and meteorology areclosely linked by the meteorologists’ interest in the role uncertaintyplays in weather forecasts. Benjamin Franklin’s interest in3The emergence of chaosmoment would know all forces that set nature in motion, and all

Chaosmeteorology extended far beyond his famous experiment of flyinga kite in a thunderstorm. He is credited with noting the generalmovement of the weather from west towards the east and testingthis theory by writing letters from Philadelphia to cities furthereast. Although the letters took longer to arrive than the weather,these are arguably early weather forecasts. Laplace himselfdiscovered the law describing the decrease of atmospheric pressurewith height. He also made fundamental contributions to the theoryof errors: when we make an observation, the measurement is neverexact in a mathematical sense, so there is always some uncertaintyas to the ‘True’ value. Scientists often say that any uncertainty in anobservation is due to noise, without really defining exactlywhat the noise is, other than that which obscures our vision ofwhatever we are trying to measure, be it the length of a table, thenumber of rabbits in a garden, or the midday temperature.Noise gives rise to observational uncertainty, chaos helps us tounderstand how small uncertainties can become largeuncertainties, once we have a model for the noise. Some of theinsights gleaned from chaos lie in clarifying the role(s) noiseplays in the dynamics of uncertainty in the quantitativesciences. Noise has become much more interesting, as the studyof chaos forces us to look again at what we might mean by theconcept 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 nowcall chaos in the atmosphere. He noted that merely moving ourhands would affect the atmosphere all the way around the planet.Poe then went on to echo Laplace, stating that the mathematiciansof the Earth could compute the progress of this hand-waving‘impulse’, as it spread out and forever altered the state of theatmosphere. Of course, it is up to us whether or not we choose towave our hands: free will offers another source of seeds that chaosmight nurture.In 1831, between the publication of Laplace’s science and Poe’s4

all manner of insects, vultures, infinite billions of life forms arethrown into chaos and destruction . . . Step on a mouse and youleave your print, like a Grand Canyon, across Eternity. QueenElizabeth might never be born, Washington might not cross theDelaware, 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 todeath a beautiful little green and black butterfly. We can onlyconsider these ‘what if’ experiments within the fictions ofmathematics or literature, since we have access to only onerealization of reality.The origins of the term ‘butterfly effect’ are appropriately shrouded5The emergence of chaosfiction, Captain Robert Fitzroy took the young Charles Darwin onhis voyage of discovery. The observations made on this voyage ledDarwin to his theory of natural selection. Evolution and chaos havemore in common than one might think. First, when it comes tolanguage, both ‘evolution’ and ‘chaos’ are used simultaneously torefer both to phenomena to be explained and to the theories that aresupposed to do the explaining. This often leads to confusionbetween the description and the object described (as in ‘confusingthe map with the territory’). Throughout this Introduction we willsee that confusing our mathematical models with the reality theyaim to describe muddles the discussion of both. Second, lookingmore deeply, it may be that some ecosystems evolve as if they werechaotic systems, as it may well be the case that small differences inthe environment have immense impacts. And evolution hascontributed to the discussion of chaos as well. This chapter’sopening quote comes from Ray Bradbury’s ‘A Sound Like Thunder’,in which time-travelling big game hunters accidentally kill abutterfly, 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, andlions, and:

in mystery. Bradbury’s 1952 story predates a series of scientificpapers on chaos published in the early 1960s. The meteorologist EdLorenz once invoked sea gulls’ wings as the agent of change,although the title of that seminar was not his own. And one of hisearly computer-generated pictures of a chaotic system doesresemble a butterfly. But whatever the incarnation of the ‘smalldifference’, whether it be a missing horse shoe nail, a butterfly, a seagull, or most recently, a mosquito ‘squished’ by Homer Simpson, theidea that small differences can have huge effects is not new.Although silent regarding the origin of the small difference, chaosprovides a description for its rapid amplification to kingdomshattering proportions, and thus is closely tied to forecasting andpredictability.ChaosThe first weather forecastsLike every ship’s captain of the time, Fitzroy had a deep interest inthe weather. He developed a barometer which was easier to useonboard ship, and it is hard to overestimate the value of abarometer to a captain lacking access to satellite images and radioreports. Major storms are associated with low atmosphericpressure; by providing a quantitative measurement of thepressure, and thus how fast it is changing, a barometer can givelife-saving information on what is likely to be over the horizon.Later in life, Fitzroy became the first head of what would becomethe UK Meteorological Office and exploited the newly deployedtelegraph to gather observations and issue summaries of thecurrent state of the weather across Britain. The telegraph allowedweather information to outrun the weather itself for the first time.Working with LeVerrier of France, who became famous for usingNewton’s Laws to discover two new planets, Fitzroy contributed tothe first international efforts at real-time weather forecasting.These forecasts were severely criticized by Darwin’s cousin,statistician Francis Galton, who himself published the firstweather chart in the London Times in 1875, reproduced inFigure 1.6

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

ChaosIf uncertainty due to errors of observation provides the seed thatchaos nurtures, then understanding such uncertainty can help usbetter 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 measurementerrors, Galton created the ‘quincunx’, which is now called a GaltonBoard; the most common version is shown on the left side of Figure2. By pouring lead shot into the quincunx, Galton simulated arandom system in which each piece of shot has a 50:50 chance ofgoing 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 leadmay stay together or diverge at each level. We shall return to GaltonBoards in Chapter 9, but we will use random numbers from thebell-shaped curve as a model for noise many times before then. Thebell-shape can be seen at the bottom of the Galton Board on the leftof Figure 2, and we will find a smoother version towards the top ofFigure 10.The study of chaos yields new insight into why weather forecastsremain unreliable after almost two centuries. Is it due to ourmissing minor details in today’s weather which then have majorimpacts on tomorrow’s weather? Or is it because our methods,while better than Fitzroy’s, remain imperfect? Poe’s earlyatmospheric incarnation of the butterfly effect is complete with theidea that science could, if perfect, predict everything physical. Yetthe fact that sensitive dependence would make detailed forecasts ofthe 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 ofproportion tended to accompany success in a science:This is only true when small variations in the initial circumstancesproduce only small variations in the final state of the system. In agreat many physical phenomena this condition is satisfied; but thereare other cases in which a small initial variation may produce a very8

The emergence of chaos2. Galton’s 1889 schematic drawings of what are now called ‘GaltonBoards’great change in the final state of the system, as when thedisplacement of the ‘points’ causes a railway train to run intoanother 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 anduncertainty: this sensitivity is no threat as long as there is nouncertainty in the position of the points, or in which train is onwhich track. Consider pouring a glass of water near a ridge in the9

ChaosRocky Mountains. On one side of this continental divide the waterfinds its way into the Colorado River and to the Pacific Ocean, onthe other side the Mississippi River and eventually the AtlanticOcean. Moving the glass one way or the other illustratessensitivity: a s

been published in more than 25 languages worldwide. The series began in 1995, and now represents a wide variety of topics in history, philosophy, religion, science, and the humanities. Over the next few years it will grow to a library of around 200 volumes - a Very Short Introduction to everything from ancient Egypt and Indian philosophy to