thom. structural stability, catastrophe theory, and applied mathematics. the john von neumann

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Structural Stability, Catastrophe Theory, and Applied Mathematics: The John von Neumann

Lecture, 1976Author(s): Rene ThomSource: SIAM Review, Vol. 19, No. 2 (Apr., 1977), pp. 189-201Published by: Society for Industrial and Applied Mathematics

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SIAM REVIEWVol. 19, No. 2, April 1977

STRUCTURAL STABILITY, CATASTROPHE THEORY,

AND APPLIED MATHEMATICS*TheJohn onNeumannLecture, 976

RENE THOMt

Abstract.Wegive briefescriptionf atastropheheory,nd f ts pplications;omy iew,tis a fundamentallyualitative,nterpretativeheory,nd, by tself,t has no abilityo predict.Examplesregivenfnterpretationsf ingularitiesn tatisticsnd ngeophysicsplate ectonics).

It sperhaps ronical hat amnow assessing he mportancef uch deas and

theories s structuraltabilitynd catastrophe heorynfront f an audience ofappliedmathematicians.Why?Because initsvery ntentionatastrophe heoryemphasizes the qualitative aspect of empirical situations,whereas appliedmathematicssfundamentallyevotedtocomputation.

Of course, appliedmathematics annot excludequalitative hinking,s itsproblems re originally iven n ordinaryanguage-in a qualitativeway. Butmost applied mathematicianswould say-(I believe)-that "modelization" isnothing ut translatinghis ualitative roblem ntoa quantitativemodel,whichthenhas to be confronted ith xperiment. n thecontrary,atastrophe heorywouldsaythat uantitativetudies-inasmuch s they repossibleand reliable-

mayhelpindefiningocal morphologicallements singularities),romwhichglobalqualitative onstructionmaybe built.

Moreover, he truth s that do not havemyself very learpicture ftheactivityf a professionalppliedmathematician. ence it is quite possiblethatsomeofthe deas I expressheremay eemtoyoua bitout of thefield, s I neverhad theopportunityfworkingmyselfn a very uantitative asis-even inpuremathematics. ut as many people-especially in popularization rticles-haveexpressedtremendoushopes about the pragmaticpossibilities f catastrophetheory,thinkt stime o come back to amore ober ppreciationf ts mpact.

Perhapsa very mportantause of theambiguity, hendealingwith atas-trophe heory,s tsradically ovelepistomologicaltatus.You readfrequently,nthesepopular articlesabout catastrophe heory here abbreviatedC.T.), that"Catastrophetheorys a mathematical heory".The truth s thatC.T. is not amathematicalheory, uta "bodyof deas", I daresay "state ofmind".As soonas the deasdeveloped byC.T. have reached very igorousmathematicaltatus,then these ideas have been incorporatedn specific ranchesof mathematics:singularitiesf smoothmappings, tratifiedpaces, singularitiesf differentialforms, ifurcationheory,ualitative ynamics,tc.Hence, strictlypeaking, .T.isnota mathematicalheory.Ofcourse,C.T. arosefrommathematics,nd it has

led to important rogress n mathematicstself;and we may hope that this

*Thefifteenthohn onNeumannectureeliveredt he 976National eetingf he ocietyfor ndustrialndAppliedMathematics,eld tChicago,llinois, une 6-18, 976.Receivedy heeditors une , 1976.

t Institutes Hautes tudes cientifiques,ures-Sur-Yvette,rance.

189







190 RENE THOM

contributions notfinished. n the ontrary,he practical" esults fC.T. are,upto now, not verystriking; valuated by the strict-positivist-criterionf the

discoveryf "newphenomena", hey educe to a few nottoosurprising) acts ngeometric pticselaboratedby M. Berry tBristol n hisworkon caustics.

In the minds of most people, C.T. reduces to what I call "elementarycatastropheheory"E.C.T.) involvinghe too) celebrated evencatastrophesnR4. For thesake ofcompleteness, et meperhapsrecall thebasic schemaof an"elementary atastrophe". uppose we have a systemS), the tates fwhich reparametrized y a point6 in a (smooth)manifoldM. Supposethatwemayact onthis ystem y varying pointu in a space U of control arameters. upposethisdeterminations defined ya map (system fequations)

(E) F(e, u) = 0,or

Fi (4j, Uk) = 0?

i= 1,29,...,n j=1,2,--,n, k=1, m..M

Inpracticallyllcases considered ncontrol heory,ne admits hat his ystem fequations may be solved-through the implicitfunction heorem,with the

assumption that the Jacobian D(F1 ... FJI)/D(61 ,... * ) is not zero; hence thepossibilityf olving2)with espect o 1,***, =p (u), throughnvpoint fthegraph fthemapF intheproduct paceM x U R'. This amounts osaying(cf. n_ m) that thisgraph s transversal o the fibers f the projectingmap(D:Mx U-> U (Condition ). Control heoryhen ries o define unctionsi(t) nsucha wayas to have thecorrespondingunctions6(t) = 'pou t)) satisfyomeoptimalityondition.

Let us suppose first hat n = 1; then (E) reduces to only one equationF(x, ui)= 0. IfaF/ax$ 0 we are inthe standard ituationT).

Elementaryatastropheheory eals,on thecontrary, ith ituationswhere

thetransversalityonditionr) fails;that uchsituations annotbe avoided-ingeneral-is shownbythefollowing ictureFig. 1)whereforn= m= 1thegraphof E) is a compact urve.Any malldeformationfF willhavepointswith erticaltangents a, b,projectingo a, /8).Ofcourse,from "control"pointofview,wewill do all thatwe can in orderto avoid (inourspace U) the"bad" points , /,where hevariation foursystems nolonger moothlyontrollable. atastrophetheory-incontrast-tries ostudyuch ituations. ereenters he dea that hereare twotypesof lack of transversality:he "finite odimension ype",and the"infiniteodimension"type.The finite ype s exemplified y the case whereaF/ax= 0, but thereexistsa higher rder derivative kF/laxk which s notzero

(with 'F/3x(O)= 0, for < k). Thenusing heWeierstrass-Malgrangerepara-tiontheorem, maybe locallyreplaced bya distinguishedolynomialnx:

A F(x, u) = x +al(u)nk+ + ak (u), (

and this ocal situation s now well understood, s we have for t a polynomial







STRUCTURAL STABILITY, CATASTROPHE THEORY 191

T F(x, u) =0

b

a

a (3 ~~~~~~~~~~U

FIG. 1

model. Moreover, fthe set of coordinates u is such that the mappingu * vu -* ai (u) into the space v of coefficientsf the genericequation of degree kxk/k +V2Xk-2+ -- + v,nis surjectivethe k-1) coefficient1being nnihilatedthrough ranslation n x), then the singularitywill be "unavoidable", i.e.,structurallytableunderanysmallCk deformationf theequationF(6, u) = 0.This s a consequenceof theglobalcontinuityf thedistinguishedolynomial sdefinedbythe local germof functions , a factexpressed n the Weierstrass-Malgrange heorem.

The infiniteodimension ase is exemplified yF(x, u) = u - exp (-1/x2), acurve witha flat contact (infinite rder). No embeddingof F into a finiteparameterfamily (x, v) can make it stable. Such singularitiesre avoidable(structurallynstable)bya smalldeformationfF. It is theresult f thetheory fmaps J.Mather)thatfor given etof ntegers n,m), there xists nlya finitenumber f "unavoidable" singularities,lthough hedefinition f ocal equival-encehas to be purely opological nd no longer local C' isomorphismat least,ingeneral).

Hence thisgeneral dea that here re "unavoidable"catastropheituations,and that t is of fundamentalmportance o know all ofthem;whereas control

theory p to now tries oavoid them notalways, s infact ontrol heory nowsthe need ofshiftingtrategies iscontinuously).t seems clearthat fwe wanttounderstand iological phenomena,we have to understand hese catastrophiceffects:or ife tself hows a greatmasteryndealingwith hesephenomena, sshownby physiological ventssuch as heartbeat, nerveinflux, nd by mor-phological vents, uchas gastrulation,nembryology.







192 RENE THOM

NowinE.C.T. we supposethat he ocal state6 inF(6, u) is tself ivenbyanoptimality rinciple ftypeG(f, u): that s, (;) reduces to: e minimizesG(e, u)

for givenu E U, hence to systemX) oftype

adi

whichdefines he criticalpointsof G(e, u) whene varies.Caution shouldbeapplied to distinguish etweensituationswhich re describedby an optimalityprinciple nd thosegovernedby an "extremality" rinciple.Although ormallythey redirected y thesamemathematics,hey xhibit fairly ifferentharac-ter.One could saythat ituations overnedbyan optimality rinciple xhibit

fundamentallyrreversibleharacter,whereas thosegovernedbyan extremalityprincipledepend on a reversibleynamics.Typical of this ast situation s theHamilton-Jacobi heorywhichdescribesthepropagation f a wave front,ndwhich eads to an extremalityrinciple n the pace (p) ofnormalized ovectors.(The G functiontself sdefinednterms fthe nitial ata.) The globaltheory fsingularitiesffunctions, hich rises out ofE.C.T., is nowfully rown. t hasshed ight nmany oints fwaveoptics Saddle-method: he dea ofunfoldingsveryusefulhere),andthetheory as nowreached a stage due to theworkofV.Arnold ndtheMoskowschool)where ts apabilities xceed theneed for t. Thesingularitiesf codim< 16 are now known,and it has developed into a verybeautifultheory,which touches many branches of mathematics:algebraic,geometry,latonic olidclassification,impleLie grouptheory,tc.).

Formostpeople,C.T. is dentified ith .C.T. The reasonfor his sobvious:thetableofthesevenelementaryatastrophes ppears as an immediateway ofclassifying aturalsituations, nd beyond thatclassification,ittle s known.Nevertheless, heobjection hatvery ewnatural ynamics regradient ynamicsremainsvalid. To thatobjection it mayof coursebe answered thatnear anyattractor ofanydynamical ystemhere s a local Lyapunovfunction. ence insomesenseanyasymptoticegime s defined ya Lyapunovfunctionround tsattractor.n fact,fone allowssome Ck-noise, henonly heLyapunovfunctionmaybe said tohavea meaning, s theflow emains nteringnto he evelvarietyof uch Lyapunovfunction. ut then hequestion riseswhetheruchLyapunovfunctions o exhibitthe bifurcation henomenaexhibitedby the E.C.T. Theanswer s obviouslyno, as thesimplest f all possiblebifurcations,heso-calledHopfbifurcation,hich ransformssource nto repeller, reates-in general-an attractingycle around the repeller. Then the corresponding yapunovfunctionxhibits on a straightine section)a behavior of typeX4/4 +X2 /2 tox4/4 -X2/2, bifurcationmongall even functions.Hence, in such a case, thecyclic, ongradient-likeharacter fthebifurcations expressedbythefact hat

the corresponding yapunov function ndergoesa catastrophewhichis notelementary, ut is definedonly among functionswhich are equivariantwithrespect osomeaction fa symmetryroup.This imple xample hows hat hereis nochance thatE.C.T. alonemayprovide sufficientetofpossiblebifurcationsin nature an objectionalreadymade byJ. Guckenheimer,who showedthatbifurcationsmong gradient-like ields s not the same as the bifurcation f








gradientfields). t is quite obvious that othertypesof bifurcations, uch assymmetryquivariant, omposed map bifurcations,tc.,have to be taken nto

consideration.Moreover we still knowvery ittle bout theglobal problemofcatastropheheory, hich stheproblem fdynamicynthesis,.e.,howto relateintoa single ystem field f ocal dynamics.

Perhapsthe mostcomplete tudieswhichdeal with hisproblem re thoseconcernedwith themodelizing f embryological rocesses.Here perhapsthemost novel idea is that when we model such processes,there s no point inlooking-as forordinary ynamical ystems-for flowvaryingn a fixed hasespace, but, nstead, ne should ook for fixed ynamics efined na succession fphasespaces: a degreeof iberty eing tsome times riggered,nd atsome othertimes, ampedout andnonexistent. ut ofcourse hese deas have notyet eached

the tageofa possible xperimentalonfirmation,ndhence are still naninfancystage.

From myviewpoint, .T. isfundamentallyualitative, nd has as itsfunda-mental im the xplanation f nempiricalmorphology.tsepistomological tatusis the one of aninterpretative-hermeneuticheory. ence it s notobviousthat twillnecessarily evelopinto newpragmatic evelopments.

The importancef C.T. in appliedmathematics.he idea ofstructuralstability, lthoughproposed originally y technicians n applied mathematics,Andronov and Pontryagin, oes not seem to have elicited a tremendousenthusiasmmong specialists n computingechniques. believethatthere s afundamentalppositionbetween hestructuraltabilitypproach ndthe"com-puting iewpoint".When a situation s structurallytable andvery tronglyo),thismeansthat his ituationmaybe reachedregardless f the nitial ata,or witha verypoor approximationn these data. Hence computings in thatcase notnecessary, nd the situationmightbe easier to describethrough eometricaldescription,r ingeneralwith rdinaryanguage.

In contrast, henwedealwith nunstable, hreshold-likeituationwherewehave the choice between several outcomes (success or failure), hen t is offundamentalmportanceodetermine hepreciseposition fthethreshold hichseparatesthe nitialdata which ead tosuccessfrom hosewhich ead tofailure.Hence, in suchcircumstances, highly ccuratedescription f theunderlyingdynamics s necessary.Such a precise descriptionn general involvesusingphysical aws,which re theonlypossibletoolstoget uchprecisedynamics. hisshowswhythestructuraltability equirementctuallymeansvery ittle or hetechnicianwhohas togivea reliable answer o a technical roblem.

There is also the philosophicalproblempointedout by D. Berlinski n"Synthese":Why s it thatphysical aws do notthemselves bey thestructuralstability equirement? lassicalmechanics, or nstance, asto do withHamilto-

niansystems,whichare veryfar frombeingstructurallytable. This is a hugeproblem,which would ike todiscussrelativelyriefly,s a complete reatmentwouldrequire engthy evelopments.

I wouldclaim-as a principle-that nyphenomenonsassociatedwith omekind of irreversibility:ora phenomenonhas to appear, ence it has to emitsomethingwhichcan be seen (or detectedthroughome apparatusamplifying







194 RENE THOM

human ision). hen he pparent ime eversibilityfphysicalawsonly howsthat hese awsdo notdescribehenomena ythemselves,utmore ccurately

change fframes etweenbservers.heydescribe,o to speak,how he amelocal irreversiblehenomenon aybe perceived y differentbservers. orinstance ewtonian ynamics,hich ives iseto Kepler's aws and planetarymotions,oesnot n tselfescribe phenomenology.orthemotion fcelestialbodies sa phenomenonnlynasmuchs these odies re ighted y he un-atypicallyrreversiblehenomenonue to the rreversibleransferrom ravita-tional nergyoradiativenergy. oreover,ven planetwhich urnsround nobscure un does not ead to a phenomenon:or,nKeplerian erms, here salways canonical ransformationfphase pacewhichmaps planet ttime tothe same planet t time '> t. Thiscanonical ransformation,t is true, s not

compatible ith he rojectionqi, i) * (qi)on the sual onfigurationpace.Butsuch ncompatibilityay ead only ophenomenologyf ne s able to fix he qi)coordinate,hat s, to localize theplanet.Such a localization peration anidempotentrojection) lways equires oupling ith source f energytheemission fwhichs rreversible).

Puttinghings ore bruptly,would are o ay hat he ime eversibilityfphysicalaws sprobablyomore hanhe xpressionf sociologicalonstraint,namelyommunicationetweeneveral bservers.orthis onstraintsnothingmore thanthe inguisticonstraintetweenmembersf the same linguisticcommunity:hen eople peak he ame anguage,heyhare he ame emanticuniverse:ecause, othe ame entence,hey ave oput he amemeaningor tleast, pproximatelyhe ame). n fact, nyobserver asto communicateithhimself-withis wnpast.Henceheneeds o have hepossibilityfcomparinghisway f ookingt theuniverset time 1with he ookhehad attime o t1.This equires commontandardfdescription,permanent ay fparametriz-ing he tates ftheworld. encereversibilityfthedynamics.

The ame act anbeexpressedn nother ay: upposewehave dynamicalsystemS) describedy flow on a smoothmanifold . What oes tmean ospeakof a stationary-orn asymptotic-regimefthisdynamic?his basic

problemasno atisfactorynswer. ut etusdefinehis egime y he ttractorofthedynamicowards hich hegiven rajectorystending ith tendingoinfinity.hen the asymptoticynamics n theattractor has to be time-reversible,or he perationf akinghe imitor e +ooadmits nly tationarydynamicsssolutions.ence fwewant odescribetable objects r ituations"inour rreversibleniverse,nlyhosewhicharrytime-reversibleynamicsrepossible andidates.t salsopossiblehathis ynamicssnot mooth,hatt wesits eversibilityo compensationetweenwo ntagonisticrreversibleynamics.Thissuggestshat he motivationorHamiltonianynamics ayperhaps efoundnquantummechanics;amelybservehat heharmonicscillatorlow,

associated ith heHamiltonian = p2 q2 s alsothegradientfthefunctionV= pqwith especto thehyperbolic etricS2 dq2_dp2.This anbe viewedasthe esultf conflictetweenwo ntagonisticynamicsupportedn the ,qaxesrespectivelyor s n zero-sumame etween wo layers).f uch viewscorrect,nly ery ew f he ossible amiltonianystemsould e"quantized",namelyhose orwhich is the quare f hemodulus f holomorphicunction








of he omplexariablesj= qi+ ip,.Only hose ould e considereds"natural".In somesense, he ackofstructuraltabilityxhibitedythephysicalaws s

nothingut he xpressionf onstraints,amelyhose ssociated ithhe lobalconservationfspace-time,espite he easeless hangesnspacedue to ocalinteractions.

The quantitativespects fcatastropheheory. ecauseof therelationfE.C.T. toHamilton-Jacobiheory,t sobvious hat omephysicalhenomenaare amenableto the catastrophicchema. quoted earlier he caustics fgeometricptics. omepartial ifferentialquations eaddirectlyothe ameE.C.T. scheme; iemann'squation ,=f(u), has solutionsiven y heE.C.T.schema s shown yPeterLax muchbefore atastropheheoryxisted. utgeneralizationsfsuchresultsequire aution.As soon as theunderlyingub-strate as ocal ymmetryroperties,hen he heoryastobe changedccord-inglyn order o takeaccount f theconstraintsue to this ymmetry.hephenomenafphase andphase ransitions)re not menable irectlyo catas-tropheheory.ne could ay hat .C.T. describeshe ehavior f sort f ther,a materiarimawithoutny pecificroperty.orthat eason, nanymediumE.C.T. still eeps ome alidityata sort fglobal, ualitativeiewpoint).ut hepreciseuantitativeawswillngeneral ot xist. ypicalxamplesf hat re hecriticalhenomena, here .C.T. isnothingut he lassical andau ormeanfield)heory,heoryhicheads o ncorrectharacteristicxponents.ere gain,

this ailurefE.C.T. is due to the act hat he hermodynamicunctionsave osatisfy-preciselyecauseof their tatisticalefinition-subtleequirementswhen heyreconsidereds functionsefined nspatial oordinates.hey realways erynear harmonicunctions,s theyhaveto damptheir ariationsthroughiffusion.

In fact, hephysicalpplicationsfC.T. arenotverymportant,ecausepreciselyn a physicalubstrate,heevolution fwhich s very ccuratelydescribedyphysicalaws, hepresence fcatastrophes aybe immediatelydeduced romhe quations. enceC.T. itselfsdispensable.

It is mportanto knownow o what xtent he ocalpolynomial odels f

E.C.T. maybe fitnto mpiricalata. t isvery emptingodo so on a precisequantitativeasis.Of course uch fitmay edone; thasalready eendone nbiochemistryprotein enaturation,ozak), and also forsomesociologicalphenomenamodelingfprisoniots) yE. C. Zeeman ndhis oworkers.mustconfess hat uch ttemptsfapplyinguantitatively.C.T. (particularlyn avery soft" ubstrateuch s sociology)o not eem omevery eliable. havesome a priori"bjectionsowardsivingnexplicituantitativexpressionortheunfoldingarametersf catastrophe.yobjection tems romhe act hatthe homogeneityequirements in generalnot satisfied orthe modeledphenomenon;ence twill epreposterousoexpect or heu functionsxplicit

homogenousxpressions,eeded f hephenomenons nvariantith espect ospacedilatations.nd fwe donothave uch homogeneityroperty,hen heufunctionsave d hocexpressions hichwilldependnanarbitraryay ntheunits sedtomeasure hemagnitudesnteringntheformula.

Nowit willprobably e excessive o denythepossibilityf applyingquantitativeit f catastrophechemeoempirical ata.But such procedure







196 RENE THOM

maynotbe more nor ess)trusted han nyother pproximationrocedureappliednnumericalnalysissuch s the nterpolationf continuousunction

by polynomialy he east quaremethod)nd ts alidityhould econfirmedneach pecificase;and bviously,ecannotxpectuch pproximatingevices othrowny ight n themechanismsnderlyinghe tudied henomena.he adhoc-and probablyllusory-characterfsuch echnicss particularlybviouswhen heunderlyingoordinatesescribe sychologicaleelingsuch s aggres-sivenessrfrustration,orwhich oobvious ay fmeasurementanbefound.

Nevertheless,e cannotdismiss hepossibilityhata fine uantitativestudy-ona relativelyhard" ubstrate,lose tophysics-may elptodecidebetween everal atastrophe odels.Very requently,heresnouniquenessfthegeneratingatastrophe,lthoughptonownoexamples knownf uch

study. precise uantitativetudy fsuch casemayhelpto make choicebetweenmodels,nddecide or furtherualitative odeling.hismay etruealsofornterpretingtatisticalata, s explainedelow.

It ismy onvictionhat hedomain menable oquantitativenalysis asbeen-inrecent ears-grosslyxaggeratedthenterestsf he omputerndus-tryreperhaps ot ntirelyoreignothis tate f ffairs).venfor problemfphysicalature,hephysicalawsgiveyou onstraintshich ave obesatisfied,butdonot ufficengeneralo determinehe volutionfthe ystem.or fwesubmitny hysical ediumnadomain toboundaryonditions,hentwill efalsengeneralhat tsfurthervolution ill verywhereatisfy globalpartialdifferentialquationE).What appens-in eneral-is hat here ill eanopendense ubdomainDO)where he quation E) is satisfied,heremainingom-plementaryet D - DO)willbe a setof"catastrophe"oints,therwisetated,"defects"f he tructure;his etwill ontainregular" oints ormingnopensetD1, where ome ystemfP.D.E.'s (E1) associatedwithE) willhavetobesatisfied;nd hen nD-1 -D1,defectsf he efects,ome therystemE2)willhave obesatisfied,tc.Uptonow, oalgorithmxists odeterminehisequence(thistratification)fdefects,or he ssociatedequence fP.D.E.'s.This swhywe can ensiblyelieve hat completeualitativelassificationf defects" ill

have obe found efore satisfactoryuantitativepproachmay erealized.Ofcourse,t salways ossiblen concreteasestorely na statisticalnalysis,utthenwe cannever e absolutelyure hat global ailurecatastrophe )fourgivenystem illnothappen.)

Thequalitativenalysis. atastropheheorys not mathematicalheory;tis alsonot "scientificheory",s wehave een hat y tselft annot rovidenycluetoexternaleality. evertheless,tclaims o have omethingosay aboutphenomena.hestatementshat .T. allows ne toproduce reof he ollowingnature:If, nthe ntervalf imeto, 1), he ystemxhibitedomemorphology

(Mot), henone has toexpect hat na furthernterval t1, 2) t willexhibit omemorphologyM2)."9Such statementannever econsidereds anabsolutelyertainrediction,

such sthe nesderived romhysicalaws.Thefuture orphologyMi2)derivesfromMl) by nhypothesisbout he implicityf he nderlyingynamics.f hepredictionsrealized, henheresnothingobesurprisedbout. f he rediction








failsthatmay appen)nd morphology12differentrom 2does ppear,hisis nteresting,ecauset hows hat ur riginalssumptionsere oo imple,nd

somenew lementf omplicationas obe ntroducednto he icture.aradox-ically, ne could aythatC.T. ismore nterestinghen t fails hanwhen t ssuccessful.his ype f"qualitative"easoning,hich llows netoreconstructpiecewise heunderlyingynamicf a systems basicallyn "interpretative"theory an hermeneuticheory, edantically).o myview, his s the mostinterestinglement roughtnbyC.T. Whetheruch n approach-whichyitselfoesnoteadtoprediction-mayetermedcientificrmerelyphilosophi-cal", s a matterfdefinition.t scertainlycientificy he echniquessed-ifnotby tspractical esults.

Althoughoclear-cutxample f this rocedures known ptonow, ne

may ope hat .T.may every sefuln tatistics.or t sfair o ay hattatisticspresentlysvery ar romcompleteheory. hat inallys statisticsbout?fwesubmitsystemoa series f xperiments,eendfinallyith cloud fpointsntheEuclideanpaceU ofobservables.tatisticssnothingut he nterpretationof louds fpoints. ow hemost bviouspproacho nterpretuch cloudnUwillbe to constructspaceM ofhidden arameters,ndthenntheproductM x U, to form deterministicergodic) ynamicalystem such that heprojectionn U of an invariantquiprobable istributionn rcM x U maygeneratehegiven istributionfpointsnU.Strangelynough,tdoesnot eemthat tatisticsid conceivets ask hatway.Usuallytatisticsscillates etween

twoviewpoints:ither hegiven loud shouldbe concentratedlonga sub-manifold f U, thusdescribing system f quantitativeaws governingheobservables,rthecloud hould e concentratedn a central oint, ndnoiseonly xplains hedeviation rom his enter. ut whathappens f thecloudexhibitsomeclear-cutmorphology,uch s boundaryines, orners,rtriplepoints? tatisticss completelyopeless n front f these ituations,nd thespecialistas todevise ad hoc"explanationsor hese ingularitiesf hapeofthedistribution.atastropheheory hich nows ow heprojectionfcriticalvalues ooksgenerically, ay nterprethese accidents s singularitiesf aprojection romMx U- U, and maydevise he simplestossiblemodel M)accountingor hese ingularities.or nstance,f n a planeR2a cloud fpointsexhibitsharp orderine:withncreasef he ensityowardshe orderFig. )insteadf diffuseatternn both ides Fig.3), then twillbe morenaturalodefinehe loud f ase 1 as coming romhe rojectionf "fold",whereasuchaninterpretationould e irrelevantncase 2. Moreover,he nterpretationfdata n high imensionalpaceU may e made ifficulty he act hat ointsregiven y heiroordinatessa table frealnumbers; ence hemorphologyf hecloudmaywellbevery ifficultorecognize.

Therole f he tatistician-inrontf hese ata-is fundamentallyhe ame

as the ole f hedivinerntheprimitiveocieties. e hastofindhe easons orsuch nd uch urprisinghenomena.encehis interpretative",hermeneutic"function. ow such taskmay lways e consideredlso ina gametheoreticframework.nany ame,t s offundamentalmportanceor playeroguess hestrategyf hispartner.he hermeneuticnterpretationaybe regardeds anattemptoplay gainst malevolentevilwhowants o hide rom ouhis rump







198 RENE THOM

FIG. 2A.

M

FIG 2B. Fold ssociated ithig.

FIG. 3

cards, is trategy,ndhis ules fbehavior. ow he sual robabilisticpproachwith is mphasisnstandard istributionsGauss,Poissonaws), ssumes hatthe evil nly as very udimentaryrain,uch s the runkard'sehavior hichgeneratesrownian otion. completetatisticalheory illnecessarilyallforintermediateituationsetween stricteterminism,nda completelyncoher-ent ehavior.atastropheheory,ithts undamentallyorphologicalpproach

may every sefulntryingocopewithuch ituations,here urdevilhas amore oherent ehavior, globalfinalityor nstance, ventually ixedwithnoise.

Butperhapsneof hemostnterestingspectsfC.T. s ts ndifferenceithrespecto thepropertiesftheunderlyingubstrate.tmaygive uite lot ofinsightithoutnvolvingheneed f specificnformationnthe ubstratetself.








In that espect, atastropheheorymay eemvery rrogantospecialists, hohave aken reat ains oacquire rich nddetailednformationn this ubstrate

itself. his s perhaps reasonwhyC.T. did make so little rogress mongbiologists,hereneverthelesstsprospectsookamong hemostpromising.want ogivehere notherxample, orrowedromeology.

Asanexamplef hisypefmorphologicalnalysis,etmedescribe ere he"tectonicsfplates" n geology rom hecatastropheheory ointofview.Suppose hat n the urfaces2 of the arth,here xists velocitylowwhichsometimesmaybe multivalued;ach determinationeing ssociatedwithminimumf potentialunction (x; u) as in theE.C.T. scheme. etus admitfirsthe alidityf Maxwell's ule: n achpoint , he ocalminimumhence helocalvelocity)s the ector which ives he bsoluteminimumfV(x,u). Then,

"generically",he et f atastropheoints here wo elocitylowsollidewill eagraph avingsonly ingularitiesree ndpointsFig. ), boundaryinesFig. ),andtriple ointsFig.6).Moreoverherewill ebasicallywo ypesfboundarylines: fterubtractionf he verage elocity,eget itheronverginginesFig.7), or divergingines Fig. 8). The first ne givesrise to orogenesis,s thecompressed aterialends o be liftedpwardsnthe hird imensionheight).

FIG. 4 FIG. 5

FIG. 6

FIG. 7 FIG.8

The econd ives ise o"rifts",eepfurrowsn he arth'srust.his ery impleinterpretationlready xplains uite lotof he arthmorphology;nparticular,itsettles ythenegativehetrickyuestion-still ebated mong pecialists-whetherlates ave obeconsidereds"solid".As soon s onegets riftnding







200 RENE THOM

onto freend like he outhernxtremityf heAfricanift),heresnopointnbelievinghat latesmay eundeformable.

Ifwe ook t the ituation ore losely, ewill indutthat situationikethat nFig.9 where he shock ine s notorthogonalo thecompressionordilatation)irectionsobviouslynstable. heonlyway omake t table s toget

FIG. 9

FIG. 10

to a broken etofsegments erpendicularo thecompressionordilatation)directionoinedbyfault ines longwhich he woflowslip long achother.Such

stepwisetructuresfound

reciselylongheMid-Atlanticcean

ridge.Such lipdiscontinuitiesrefoundnfluid ynamicsMachreflexion),ndtheyoccur ach ime elocitylays fundamentalole nthe ocal tate. his xampleclearlyhowswhat anbeexpectedromhe seofC.T. and nparticular.C.T.).A globalunderstandingfthemorphology,ubject olocalrefinementsakingintoaccount heprecise articularitiesf the medium. recent esult fV.Kl6emannd G. ToulousetwoOrsay hysicists) ay oa longway owardsheclassificationfstructurallytabledefects.f a medium as localsymmetries(describedy pseudogroupG)), thenhe unction(x,u) reachestsminimuminx along n orbithomogeneouspace)H underheG isotropyroup.Hence

E.C.T. is no onger alid, ut he table efectsre submanifolds of odimKsuchthat he ink rof W,whichs a (K -i) sphere,smapped n the ocalminimumrbit by mapo- -> whichs nothomotopico zero ingeneralfibration).his rinciplerovidesvery owerful ay oclassifytable efectsnorderedmedia-liquidcrystals,uperfluidelium,tc. t shows heprofoundrelAitionetweentabilityndtransversality.








Conclusion. erhaps hemost nterestingeature f C.T. is itsability odescribenalogies.nsome ense, .T. could e called he heoryf nalogies; y

associatingith local ituationsingularityf he ocaldynamics,tprovidesnalgebraic ay fformalizinghentuitiveotionf nalogousituations.ven henotion f "program"may be to some extent ut intoa purely eometricframework.histheorys admittedlyncomplete,t dealsonlywith actions"describedrammaticallyy verbs.Analogies ealingwithnouns re farmoredifficultoanalyze: heywouldmply generalheoryfregulationor ystems:external bjects nd concepts eferringo them.Of course, heproblem fregulation,hich as hemain im fN. Wiener's ybernetics,s still ery ar utof sight.But as thecatastropheheoreticalpproachdoes notexclude hepossibilityf consideringhe temporal volution f a structureand of its

regulationtself),tprovidesbetter ay oapproachhe roblemfformalizinglife ynamics:or he urelyechnologicalpproachf yberneticsoesnot llowsuchgeneticonsiderations:machine asnoembryology.t east atastrophetheoryoesnotprecludeuch nattempt.hestudyf nalogy idnotprogresssinceAristotelianogics,stheBoolean pproach, ithts urelyxtensionalayofdefiningogic,has turnedtsbackto a true yntheticogicallowing heconsiderationfmeaning.t sthere,nthis ery hilosophicalroundhat seethemain nterestf catastropheheory.f at the end of the 17thcentury,somebody ouldhave ome, xhibitingaylor's xpansionheorem,ndsayingthat uch theorem ouldhelp ciencenfindingewphenomena,uch claim

would ave ppeared idiculous.utnevertheless,aylor'sxpansionormulasa highly seful heorem,ven f, n itself,t neverdid lead to any specificexperimentaliscovery.am nclinedo believe hat he tatus f C.T. isofthesamenatureifnot fthe ame mportance).

thom. structural stability, catastrophe theory, and applied mathematics. the john von neumann

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