[PDF] A brief history of the butterfly effect in nonlinear dynamics

39566_7AJP000425.pdf Seagulls,butterßies,andgrasshoppers:Abriefhistoryofthebutterßy effectinnonlineardynamics

RobertC.Hilborn

a) DepartmentofPhysics,AmherstCollege,Amherst,Massachusetts01002 ?Received10September2003;accepted24October2003? ThebutterßyeffecthasbecomeapopularmetaphorforsensitivedependenceoninitialconditionsÑ thehallmarkofchaoticbehavior.Idescribehow,where,andwhenthistermwasconceivedinthe

1970s.Surprisingly,thebutterßymetaphorwaspredatedbymorethan70yearsbythegrasshopper

effect.©2004AmericanAssociationofPhysicsTeachers. ?DOI:10.1119/1.1636492?

I.INTRODUCTION

Thetermbutterßyeffecthascapturedforthepublicand muchofthescientiÞccommunitytheessenceofchaoticbe- haviorindynamicalsystems:sensitivedependenceoninitial conditions.Forachaoticsystem,eventhesmallestchangein initialconditions,due,forexample,totheßappingofabut- terßyÕswings,mayleadtodramaticchangesinthebehavior ofthesystem.Fromthissimpleideafollowtheunpredict- abilityofmanydeterministicsystemsundercertaincondi- tionsandthecomplexityofspatialpatternsinturbulence,to mentionjusttwoimportantconsequences.Whatistheorigin ofthiscolorfulmetaphor?Theanswerturnsouttobecom- plex.PartofthestoryistoldinE.N.LorenzÕsbook, 1 The

EssenceofChaos.Iplayedaroleintrackingdownthehis-

toryofthetermbutterßyeffect,andinthispaperIdiscuss someofthedetailsofthishistory.Ialsorelateamorerecent discovery:thegrasshoppereffectprecededthebutterßyef- fectbymorethan70years. ThetermbutterßyeffectisusuallyattributedtoE.N.Lo- renz.Infact,inhisearlywritingonsensitivedependenceon initialconditionsanditseffectonthepredictabilityofdy- namicalsystems,Lorenz 2 usedaseagullmetaphor:ÔÔWhen theinstabilityofuniformßowwithrespecttoinÞnitesimal perturbationswasÞrstsuggestedasanexplanationforthe presenceofcyclonesandanticyclonesintheatmosphere,the ideawasnotuniversallyaccepted.Onemeteorologistre- markedthatifthetheorywerecorrect,oneßapofaseagullÕs wingswouldbeenoughtoalterthecourseoftheweather forever.Thecontroversyhasnotyetbeensettled,butthe mostrecentevidenceseemstofavorthegulls?p.431?.'Õ

Theseagullmetaphor,however,wastobeshort-lived.In

thetitleofatalkgivenbyLorenzatthe139thmeetingofthe

AmericanAssociationfortheAdvancementofScience

?AAAS?inDecember,1972,thebutterßymadeitsÞrstap- pearance:ÔÔDoestheßapofabutterßyÕswingsinBrazilset offatornadoinTexas?ÕÕInthistalk,Lorenz 1,3 raisedthe fundamentalissue:ÔÔThequestionwhichreallyinterestsusis whetherthey?thebutterßies?candoeventhisÑwhether,for example,twoparticularweathersituationsdifferingbyas littleastheimmediateinßuenceofasinglebutterßywill generallyaftersufÞcienttimeevolveintotwosituationsdif- feringbyasmuchasthepresenceofatornado.Inmore technicallanguage,isthebehavioroftheatmosphereun- stablewithrespecttoperturbationsofsmallamplitude?ÕÕ

II.WHENCETHEBUTTERFLY?

Howandwhydidtheseagullchangetoabutterßy?Letus

beginwithtwopossibleexplanations.

First,ithasbeensuggested

4 thatthebutterßymetaphor arosefromtheresemblanceoftheattractoroftheLorenz equations,whenthevariableZ,inthestandardformofthe

Lorenzsystemofequations,

5 isplottedagainstX.?SeeFig.1 forsuchaplot.?However,inhis1963paper, 2

Lorenzpre-

sentedplotsofZagainstYandXagainstY,andtheseplots donotmuchresembleabutterßy. 6

NoÞguresaccompanied

LorenzÕs1972talk.

1

Moreover,probablynooneplottedZ

againstXfortheLorenzmodeluntilafterthemid-seventies. 4

Asecondsuggestionwasmadetomeinthesummerof

1989byDavidS.Hall,anAmherstCollegephysicsmajorat

thetime,whopointedoutthata1952RayBradburystory, 7 ÔÔAsoundofthunder,ÕÕtellsthetaleoftimetravelerswho,in theirtriptothepast,accidentallykillabutterßy,andreturn totheirpresenttoÞndhistorychanged.Infactpartsofthis storyreadmuchlikesegmentsofcurrentbooksonnonlinear dynamics:ÔÔCrushingcertainplantscouldaddupinÞnitesi- mally.Alittleerrorherewouldmultiplyinsixtymillion years,alloutofproportion.Ofcoursemaybeourtheoryis wrong.MaybetimecanÕtbechangedbyus.Ormaybeitcan bechangedonlyinlittlesubtleways.Adeadmousehere makesaninsectimbalancethere,apopulationdisproportion later,abadharvestfurtheron,adepression,massstarvations, and,Þnallyachangeinsocialtemperamentinfar-ßung countries.Somethingmuchmoresubtle,likethat?p.61?.'Õ

However,Lorenz

8 informedmethathehadnotbeenaware oftheBradburystorybeforeIbroughtittohisattention.

Whatactuallyhappenedinthetransformationofthesea

gulltothebutterßy?AccordingtoLorenz, 6 hewasoutofthe countryinthefallof1972,andPhilipMerilees,convenerof theAAASsessionatwhichLorenzwastogivehistalk, concoctedthetitleofthetalkinLorenzÕsabsence.Merilees respondedtomyinquiryaboutthebutterßyÕsoriginswiththe following: 9

ÔÔAsIrecallthecircumstances,Iwasworkingas

ExecutiveScientistfortheUCAR?UniversityCorporation forAtmosphericResearch?GARP?GlobalAtmosphericRe- searchProgram?CouncilunderthelateWaltRoberts.Iwas onleavefromMcGillandwasgiventhejoboforganizingan

AAASsessiononGARP.Oneofthefundamentalissuesin

GARPwasthepredictabilityoftheweather,inparticularthe sensitivedependenceoninitialconditions.Walttoldmethat itwasimportanttotrytocome?up?withintriguingtitlesfor thetopicsinthesessionbecausetherewasmuchcompetition fortheattentionofparticipants.IhadfollowedEdLorenzÕs workverycloselyandwasawareoftheseagullmetaphor, butIthoughtthebutterßymightbemoreappealing.Inaddi- tion,Itriedforsomealliteration;butterßyÑBrazil, tornadoÑTexas.IsupposeseagullinSenegalmighthave

425425Am.J.Phys.72?4?,April2004http://aapt.org/ajp©2004AmericanAssociationofPhysicsTeachers

workedaswell.IcanÕtrecallbeinginßuencedbysomeone elseÕsuseofthebutterßyinthiscontext,butitwasnearly18 yearsago?ÕÕ AfewdaysbeforereceivingMerileesÕletter,Ireceiveda telephonecallfromDouglasLilly,whoseofÞceattheNa- tionalCenterforAtmosphericResearchin1972wasnear Merilees.Lillyrecalledtossingaroundideasfortalktitles withMerileesin1972andthoughtthatitwashe?Lilly?who suggestedtheuseofbutterßyinLorenzÕstitle.Buthecould notrecallaspeciÞcinßuencethatledtotheadoptionofthe butterßy.Hedidsuggestthatthe1941novelStormby

GeorgeR.Stewart

10 mighthavemotivatedhisthinking.A carefulreadingofthenovel?anintriguingstoryofmeteo- rologiststrackingaPaciÞccoaststorm?revealednobutter- ßies.ButIdidÞndthebasicideaofsensitivedependenceon initialconditionsexpressedinthefollowingway:ÔÔHe?the juniormeteorologist?thoughtofhisoldprofessorÕssaying:A

Chinaman?sic?sneezinginShen-simaysetmentoshovel-

ingsnowinNewYorkCity?p.44?.'Õ

Withthisexchangeoflettersandtelephonecalls,myin-

vestigationsintotheoriginofthebutterßymetaphorreached adeadend.Thebutterßyenterednonlineardynamicsinthe conversationsbetweenMerileesandLillyin1972atNCAR. Apparently,itwasbornastheresultofsubtlenonlinearin- teractionsamongMerilees,Lilly,andLillyÕsrecollectionsof StewartÕsStormcausing,darewesay,abifurcationfromthe seagullmetaphortothatofthebutterßy.Buttheprecise initialconditionsofthesedivergingtrajectoriesarenowbe- yondourpowersofobservation.

Oneminorhistoricalproblemremains:Howdidthebut-

terßymetaphorturnintothebutterßyeffect?Lorenzhimself neverusedthephrasebutterßyeffect.Thetermappearsin

SchusterÕs1984textDeterministicChaos

11 andinthenow famousChaosbyJamesGleick. 4

Infact,Gleick

4 wrotethat ÔÔtheButterßyEffectÑthenotionthatabutterßystirringthe airinPekingcantransformstormsystemsnextmonthin NewYork?p.8?.'ÕGleickÕsstatementisanintriguingcom- binationofphrasesfromLorenzÕstitleandStewartÕsStorm.

III.GRASSHOPPERSWEREFIRST

Thenotionofsensitivedependenceoninitialconditions

haslongbeenapartoftheloreofmeteorologyasStewartÕs Stormindicates.Irecentlylearnedthatthebutterßymetaphor was,infact,predatedbynearly70yearsbywhatweshould callthegrasshoppereffect.ThelateAlMcLennanofLehigh University,knowingofmyinterestinnonlineardynamics, hadalertedmein2002toareview 12 ofPierreDuhemÕs

Traite

" E " le " mentairedeMe " chaniquefonde " esurlaThermody- namique?1897?.Thereviewwaswrittenin1898byW.S. Franklin,thenaprofessorofphysicsatLehigh.Discussing thesensitivityoftheatmospheretosmallperturbations, FranklinwrotethatÔÔLongrangedetailedweatherprediction isthereforeimpossible,andtheonlydetailedprediction whichispossibleistheinferenceoftheultimatetrendand characterofastormfromobservationsofitsearlystages; andtheaccuracyofthispredictionissubjecttothecondition thattheßightofagrasshopperinMontanamayturnastorm asidefromPhiladelphiatoNewYork?ÕÕ Itseemsasifthenotionofsensitivedependenceoninitial conditions,thehallmarkofchaos,hasbeenintheair?soto speak?forsometimeandthatinsectshavebeenthecreatures ofchoiceforvividmetaphorsfortheseeffects.

ACKNOWLEDGMENTS

IthankE.N.Lorenz,P.Merilees,D.Lilly,A.McLennan,

andD.S.Hallfortheirassistanceintrackingdownthehis- toryofthebutterßyeffect.

Noteaddedinproof.ProfessorLorenzrecentlyalertedme

toapublicationinwhichaÔÔbutterßyÕÕappears,predatinghis

1972AAASpaper.ThebutterßymetaphoroccursinJoseph

Smagorinsky,ÔÔProblemsandpromisesofdeterministicex- tendedrangeforecasting,ÕÕBull.Am.Meteor.Soc.50,286Ð

311?1969?.ProfessorLorenztoldmethathehadreadthe

SmagorinskypaperwhenitÞrstappeared,buthadnotre- memberedthebutterßy?onpage289?untilrecentlyashe waspreparingatalkforanApril2003symposiumhonoring ProfessorSmagorinskyÕsreceiptoftheBenjaminFranklin MedalinEarthScience.Therelevantsentencereads,ÔÔOr, wouldtheßutterofabutterßyÕswingsultimatelyamplifyto thepointwherethenumericalsimulationdepartsfromreal- ity,sothattherewillcomeatimewhentheymustberan- domlyrelatedtoeachother?ÕÕItispossiblethatthisappear- anceofthebutterßysubconsciouslyinßuencedMerilees, whoalsoreadtheSmagorinskypaperin1969,whenhelater createdthetitleforLorenzÕstalk. a?

Electronicmail:rchilborn@amherst.edu

1 EdwardN.Lorenz,TheEssenceofChaos?UniversityofWashington

Press,Seattle,1994?.

2 EdwardN.Lorenz,ÔÔThepredictabilityofhydrodynamicßow,ÕÕTrans.

N.YAcad.Sci.25,409Ð432?1963?.

3 TheChaosAvant-Garde,MemoriesoftheEarlyDaysofChaosTheory, editedbyRalphAbrahamandYoshisukeUeda?WorldScientiÞc,Sin- gapore,2000?.Seepp.91Ð92. 4 JamesGleick,Chaos?Viking,NewYork,1987?.Gleickwrites:THELO- RENZATTRACTOR?onfacingpage?.Thismagicalimage,resemblingan owlÕsmaskorbutterßyÕswings,becameanemblemfortheearlyexplorers ofchaos.Thisishistoricallyincorrectbecausetheimagewasnotproduced assuchuntilthelate1970Õs. 5 RobertC.Hilborn,ChaosandNonlinearDynamics?OxfordUniversity

Press,Oxford,2000?,2nded.

Fig.1.AphasespaceprojectionontotheXZplaneofatrajectoryofthe Lorenzmodel.?ZistheordinateandXtheabscissa.?IthankE.N.Lorenz forprovidingtheÞgure.

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6 EdwardN.Lorenz,lettertoOliverM.Ashford,27December,1989. 7 RayBradbury,ÔÔAsoundofthunder,ÕÕinRisforRocket?Bantam,New

York,1962?.

8 EdwardN.Lorenz,letter,1May,1990,toRobertC.Hilborn. 9 PhilipMerilees,letter,28August,1990,toRobertC.Hilborn. 10 GeorgeR.Stewart,Storm?RandomHouse,NewYork,1941?.Inatele- phoneconversation,Lorenztoldmethatafterreadinganearlydraftofthis paper,herecalledtheChinamanmetaphorfromhavingreadtheStewart novelandthatthisnovelwasinßuentialinhisdecisiontogointometeo- rology. 11 H.G.Schuster,DeterministicChaos?VCH,NewYork,1984?. 12

W.S.Franklin,ReviewofP.Duhem,Traite"E

" le"mentairedeMe"chanique fonde"esurlaThermodynamique?Paris,1897?,Vols.1and2,inPhys.Rev.

6,170Ð175?1898?.

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