Fabio Dercole and Sergio Rinaldi
Citation: Chaos 24, 023134 (2014); doi: 10.1063/1.4882685 View online: http://dx.doi.org/10.1063/1.4882685
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CHAOS24,023134(2014)
Lovestoriescanbeunpredictable:JulesetJiminthevortexoflife
FabioDercolea)andSergioRinaldib)DepartmentofElectronics,Information,andBioengineering,PolitecnicodiMilano,MilanoI3,Italy
(Received30September2013;accepted28May2014;publishedonline17June2014)
Lovestoriesaredynamicprocessesthatbegin,develop,andoftenstayforarelativelylongtimeinastationaryorfluctuatingregime,beforepossiblyfading.Althoughtheyare,undoubtedly,themostimportantdynamicprocessinourlife,theyhaveonlyrecentlybeencastintheformalframeofdynamicalsystemstheory.Inparticular,whyitissodifficulttopredicttheevolutionofsentimentalrelationshipscontinuestobelargelyunexplained.Acommonreasonforthisisthatlovestoriesreflecttheturbulenceofthesurroundingsocialenvironment.Butwecanalsoimaginethattheinterplayofthecharactersinvolvedcontributestomakethestoryunpredictable—thatis,chaotic.Inotherwords,weconjecturethatsentimentalchaoscanhavearelevantendogenousorigin.Tosupportthisintriguingconjecture,wemimicarealandwell-documentedlovestorywithamathematicalmodelinwhichtheenvironmentiskeptconstant,andshowthatthemodelischaotic.ThecaseweanalyzeisthetriangledescribedinJulesetJim,anautobiographicnovelbyHenri-PierreRochethatbecamefamousworldwideafterthesuccessofthehomonymousfilm
C2014AIPPublishingLLC.[http://dx.doi.org/10.1063/1.4882685]directedbyFranc¸oisTruffaut.V
Althoughhuntingforchaosisnotaspopularasitusedto
be,wededicatethispapertothepresentationofanewstrangeattractor.Itconcernsthemostimportantdynamicprocessinourlife—theevolutionofloveininterpersonalrelationships.1–4Moreprecisely,wesup-porttheconjecturethatromanticrelationshipscanbeunpredictable—technicallychaotic—onthesolebasisoftheinterplayofthecharactersinvolved.Thiscannotbedonewithoutamathematicalmodel,becauselovestoriesare,ingeneral,influencedbytheturbulenceofthesur-roundingsocialenvironmentandfartooshorttoallowthereconstructionofastrangeattractor.5Thefirstallu-siontotheconjecturewasmadebyStrogatz,2whomen-tionedthe“many-bodyproblem”whenpresentinghisadmittedlyrudimentarymodelofRomeoandJuliet.AmoretechnicalhintcanbefoundinapaperbySprott,6whereana€ıveextensionofStrogatz’smodeltothecaseofahypotheticaltriangleisdiscussed.Togiveacrediblesupporttotheconjecture,wefocusonarealandwell-documentedtriangularlovestory,weidentifyfromitthemainpsycho-physicaltraitsofthethreeindividuals,andweencapsulatetheminamathematicalmodelwithcon-stantenvironment.Wethenshowthatforreasonableval-uesoftheindividualtraitsthemodelcomparesfavorablywiththelovestoryandischaotic.
permanent(stationaryorfluctuating)regime,whileothers4,11focusonthephaseofmaritaldissolution.Moreover,mathe-maticalmodelshavealsobeendevelopedforafewspecific(thoughrelativelysimple)lovestories,describedintheliter-atureorinfilms.12–15Bymentioningtheanalogywiththe“many-bodyproblem”ofcelestialmechanics,Strogatz2somehowconjecturedthat
sentimentalrelationshipscanbeunpredictable—thatis,chaotic—onthesolebasisoftheinterplayofthecharactersinvolved.
Inotherwords,theconjectureisthatsentimentalchaoscanhavearelevantendogenouscomponent,andnotsimplyreflecttheturbulenceofthesurroundingsocialenvironment.Aproofofthisconjecturecanonlybebasedonamathemati-calmodel.Infact,theinteractionswithotherindividuals,aswellashealth,cultural,andeconomiccircumstances,makedifficulttoidentifytheoriginofthesentimentalturbulence.Andthoughnonlineartimeseriesanalysiscaninprinciplehelpinsolvingtheproblem,lovestoriesaretooshorttoallowthereconstructionofastrangeattractor.5Instead,bymeansofamathematicalmodelwithconstantparameters,onecaneasilycutalltheinteractionswithotherindividualsandkeeptheenvironmentconstant.
Ana€ıvesupporttotheaboveconjecturecanbefoundinanextensionofStrogatz’smodeltoahypotheticaltriangle.6However,acrediblesupportcanonlybegivenbymodelingarealandwell-documentedlovestory.Thisistheaimofthisstudy.
Thestartingpoint—theselectionofthelovestory—israthercritical.Indeed,thestorymust
–beknownworldwide,ifwewantourmessagetoreachthelargepublic;
–containsymptomsofturbulenceandunpredictability,topossiblysupportourconjecture;
–containafew,atleastqualitative,informationtoallowthevalidationofthemodel.
C2014AIPPublishingLLCV
I.INTRODUCTION
AfterStrogatz’s[1988]pioneeringpaper,2lovestories
havebeenmodeledwithincreasingsuccessintermsofdif-ferentialordifferenceequations.Manyattempts3,7–10describeanonymousstoriesfromthestateofindifference,inwhichwearewhenwefirstmeet,totheestablishmentofa
a)Authortowhomcorrespondenceshouldbeaddressed.Electronicmail:fabio.dercole@polimi.it.b)Alsoat:EvolutionandEcologyProgram,IIASA,LaxenburgA-2361,Austria.
1054-1500/2014/24(2)/023134/9/$30.00
24,023134-1
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Undertheaboveconstraints,ourchoicehasbeenthetriangularlovestorydescribedbyRocheinhis1953autobio-16graphicnovelJulesetJim.ThestorybeginsinParisafewyearsbeforetheFirstWorldWar,whereitends20yearslater.ItinvolvesKathe,herhusbandJules,andhisbestfriendJim(HelenGrund,FranzHessel,andHenri-PierreRocheinthereallife).
Roche’snovelisknownbecauseitisconsideredasoneofthemaincontributionsconveyingtheanti-bourgeoisideologyof“freelove”thatcanbecondensedinsayingthat“oneshouldnotconstrainthepeopleoneloves,butleavethemfreetoengageinotherrelationships.”ThiscentralideaofRoche’sphi-losophybecameverypopularintheseventiesandwaslaterextensivelydebatedinliteraryessays.ButthelovestoryofJulesetJimbecamefamousworldwideafterthesuccessofthe1961homonymousfilm—acelebratedmasterpieceoftheFrenchNouvelleVaguedirectedbyFranc¸oisTruffaut.
Asforthesymptomsofturbulenceandunpredictability,thereadingofthenovelgivestheimpressionthatKatheisquiteunstableanddifficulttopredict—shechangespartnerseventimesinthe20yearsofconcerns,alternatingbetweenJulesandJim.Actually,theuncertaintyoverthefuturecreatesinthetriangle(andinthereader)anincreasingten-sionthatceasesonlywhenKatheandJimcommitsuicide:
JuleswouldneverhaveagainthefearthathadbeenwithhimsincethedayhemetKate,firstthatshewoulddeceivehim—andthen,quitesimply,thatshewoulddie,forshehadnowdonethattoo(p.236intheEnglishtranslationofRoche’snovel16)ThedramaticendimaginedbyRocheishenceinter-pretableasapoeticwayofinterruptingthetortureduetoarecurrentshock—thechangeofpartner—thatbecomespar-ticularlyunsustainablebecauseunpredictable.Alsothefewavailabledataconfirmthatthepartnerchangesareirregular.
Thepaperisorganizedasfollows.InSec.II,weintroducethereadertothebasicnotionsweusetomodelsentimentalrelationships.3TheninSec.III,webuildourmodelofJulesetJim.First,wenotethatthelovestorybetweenKatheandanyoneofhertwoloversisscarcelyinfluencedbythepresenceoftheother.Thisisadirectconsequenceoftheprincipleoffreelovethatinspiresthelifeofthethreecharacters.Iftheywouldrigorouslyfollowthisprinciple,thenthetriangularrelationshipwouldbeequivalenttotwofullyseparatedpairwiserelation-ships.Inturn,thetrianglecouldbemodeledwithtwoindepend-entsubmodels,Kathe-JulesandKathe-Jim,respectively,describedinSecs.IIIAandIIIB.However,sincethethreecharactersslightlydeviatefromthepureideologyoffreelove,thetriangleisdescribedmorerealisticallyinSec.IIICbyweaklycouplingthetwosubmodels.Specifically,weintroducetwosmallparameterstotakeintoaccountthatKathedoesnotlivetwoindependentlovestoriesandthatJulesandJimareslightlycomplaisant,thefirst,andjealous,thesecond.AllourmodelingchoicesaresupportedbyliterarypassagesextractedfromRoche’snovel.
InSec.IVA,wevalidateourmodelagainsttheinforma-tionavailableinthenovel,whichweidentifyinsevenspe-cificfeatures,includingthenumberandthechronologyof
thepartnerchanges.Sincetherearenoelementsinthenovelthatcouldsuggestreasonablevaluesforthecouplingparam-eters,weperformsystematicsimulationstocheckifthereisaregionintheplaneofthetwoparametersforwhichallfea-turesaresatisfactorilyreproducedbythemodel.Theresultofthisvalidationshowsthatthisregionischaracterizedbysmallandpositivevaluesofthecouplingparameters.
Finally,tosupportourconjecture,weshowinSec.IVBthatforthevalidatedvaluesofthecouplingparameters,themodeltrajectorydescribingthestoryofJulesetJimasymp-toticallyreachesachaoticattractor.Moreover,inlessthan20years,thetrajectoryreachestheattractorandspendsclosetoitatimeinwhichpredictionsbecomeimpracticableaccordingtothecomputed(largest)Lyapunovexponent.17AbroaderdiscussionofourresultsandafewgeneralconclusionsthatcanbedrawnfromthisstudyclosethepaperinSec.V.Sincetheresultsobtainedwithamodelbasedonsubjectiveinterpretationsarenotascredibleasthosebasedonprecisephysicallaws,thereaderisinvitedtochecktherobust-nessofourconclusionsbyinteractivelysimulatingourmodelusinganonlinesimulator(seesupplementarymaterial22),whereallmodelparameterscanbesignificantlychanged.
II.MATERIALANDMETHODS
Levinger1hasbeenthefirsttousegraphstorepresentthetimeevolutionofthefeelingsofonepersonforanother.FollowingLevinger’sabstraction,andalsotominimizethenumberofequations,weassumethattheinterestofoneper-sonforanothercanbecapturedbyasinglevariable,calledfeeling.Lowandhighpositivefeelingscorrespondtofriend-shipandlove,whilenegativefeelingsindicateantagonismandhate;zerocorrespondstoindifference.Forexample,inthepairwisestorydepictedinthetoppanelsofFig.1,shedevelopsfromtheverybeginningapositivefeelingforhim,whileheisinitiallyantagonistic.Incontrast,intheotherstory(bottompanels),sheandhearealwayspositivelyinvolved,butsufferfromremarkableupsanddowns.Ofcoursethegraphsstartfromthefeelingsthattheyhaveonefortheotheratthebeginningofthestory.Thus,thestartingpointistheoriginoftheplaneofthefeelingsifthetwoindi-vidualsareinitiallyindifferenttoeachother.
Feelingsvaryovertimebecauseoftheinterplayofcon-sumptionandregenerationmechanisms,hereconsideredastime-invariantprocesses.Thebasicconsumptionmechanismisoblivion.Itexplainswhyapersonloosesmemoryofthepartnerafterbeingabandoned.Theregenerationprocessestypicallyconsideredinminimalmodels3,7–10,12–15arethereactiontoloveandthereactiontoappeal—themixofbeauty,talent,wealth,andothertraitsthatareindependentoffeelings.
Consideracoupleanddenotebyx(t)andy(t)thefeelingsthatsheandhehaveonefortheotheratdayt.Amodelissim-plyabalanceofthefeelingsbetweenanydaytandthefollow-ingday(tþ1).Inwords,herfeelingtomorrowisequaltothatoftodayminusthelossofinterestbetweentodayandtomorrowduetooblivion,plustherechargeofinterest,againbetweentodayandtomorrow,duetoherreactionstohisloveandappeal.
ThelossofinterestduetooblivioncanbedescribedwithafunctionF(x)increasingwithx,toexpressthefact,
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023134-3F.DercoleandS.RinaldiChaos24,023134(2014)
FIG.1.Graphicalrepresentationoftwohypotheticallovestories.(Left)Feelings’timeseries.(Right)Trajectoriesintheplaneofthefeelings.
commoninnaturalsystems,thattherateatwhichagivenpropertyislostispositivelycorrelatedwiththeabundanceoftheproperty.Typically,thelossisassumedtobepropor-tionaltox,sothefunctionF(x)islinearandgivenbytheproductofaproportionalitycoefficientsfandx.Theparame-terf,calledforgettingcoefficient,representstheportionofinterestlostinonedaythroughoblivion.
Asfortherechargeofthefeeling,wedenotebyRL(y)herreactiontothepartner’slove(whereRstandsforreactionandLforlove)andbyRA(ay)herreactiontothepartner’sappeal,hereindicatedwithayandassumedtobeinvariant.AlsothereactionRA(ay)isassumedtobelinear,i.e.,RA(ay)¼rAay.
Tomodelthereactiontolove,wedistinguishbetweensecureindividuals—whoincreasetheirreactionforanyincreaseoftheloveofthepartner—andinsecureones—whoavoidhighinvolvementsbydecreasingtheirreaction(andpos-siblyreactnegatively)whentheloveofthepartnerisaboveacriticalthreshold.Secureindividualsarethereforecharacter-izedbyfunctionsRL(y)increasingwiththeloveyofthepart-ner(seeRef.3forasurvey).Amongthesefunctions,therearelinearfunctions,whichhowevercorrespondtoratherextremeindividualswithunboundedcapacityofrecharge.Incontrast,insecureindividualsarecharacterizedbyfunctionsRL(y)whicharedecreasingatsufficientlyhighvaluesofy(Fig.2,top).
Anotherimportantcharacteristicofanindividualisthepropensitytoreacttotheappealofthepartnerinabiasedway,dependingonher/hisownstateofinvolvement.Forexample,parentsoftenseetheirownchildrenmorebeautifulthantheyreallyare.Butthesamephenomenon,calledsyner-gism,hasalsobeenobservedinastudyofperceptionofphysicalattractiveness.18Inthiscase,thereactiontothepartner’sappealcanbewrittenintheform(1þS(x))RA(ay),wherethefunctionSisincreasingforpositivex(saturatingforlargex)andiszerofornegativex(Fig.2,bottom).Theoppositebehaviorisalsopossible,likeinplatonicindividu-alsdescribedbyareactiontoappealoftheform(1–P(x))RA(ay),wherePisshapedlikeSandmeasuresthelossofsexualinterestforincreasingvaluesoftheinvolvementx.
Individualswhoareneithersynergicnorplatonicarenotbiasedbytheirownfeelings.
III.THEMODELOFJULESETJIM
Inthissection,weproposeamathematicalmodelforthelovestoryofJulesetJimusingadidacticstylethatshouldmakethepaperaccessiblealsotonon-technicallyorientedreaders.Inparticular,wepresentthemodelasarulethatupdatesthefeelingsofKathe,Jules,andJimrecursivelyfromonedaytothenext.Anequivalentcontinuous-timefor-mulationofthemodelisalsopossible(intermsofordinarydifferentialequations).
Thelovestoryisreducedtoapuretriangleinaconstantenvironment.Specifically,weneglecttheinteractionsthat
FIG.2.(Top)Reactiontolovetypicalofaninsecureindividual.(Bottom)Typicalsynergismfunction.SeeTableIfortheanalyticalexpressions.
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023134-4F.DercoleandS.RinaldiChaos24,023134(2014)
Kathe,Jules,andJimhavewithotherminorcharactersdescribedbyRoche,andkeepallmodelparametersconstant.Sixvariables—thefeelingsofeachpersonfortheothers—areinprinciplerequiredinaminimalmodel.However,JulesandJimhaveadeepandpermanentfriendship:
IntwentyyearsJimandhehadneverquarrelled.Suchdisagreementsastheydidhavetheynotedindulgently(p.237)
Wethereforeconsideronlythefeelingsx1andx2ofKatheforJulesandJimandthefeelingsy1andy2ofthetwofriendsforher.
Thethreefollowalmosttoperfectiontheideologyoffreelove:
Inhermind,eachloverwasaseparateworld,andwhathappenedinoneworldwasnoconcernoftheothers(p.108)
ItseemsthusreasonabletosplitKatheintotwoinde-pendentwomen,oneinlovewithJulesandonewithJim,andtodescribethetrianglebymeansoftwoindependentsubmodelsofpairwiserelationship:theKathe-Julessubmo-del(Sec.IIIA)andtheKathe-Jimsubmodel(Sec.IIIB).
However,Rochedescribesspecificbehaviorsonthepartofthetwofriendsthatviolatetherigidprincipleoffreelove.JulesiscomplaisantwithJim—heispleasedwhenKatheiswithJimbecauseherealizesthismakesherhappier.Thischaracteristic,peculiartoJules,isconsistentwithhispla-tonicnature(seeSec.IIIA)andiswelldescribedbyRoche:
‘…I’mterrifiedoflosingher,Ican’tbeartolether
gooutofmylife.Jim—loveher,marryher,andletmegoonseeingher.WhatImeanis,ifyouloveher,stopthinkingthatI’malwaysinyourway’(p.27)Althoughjealousyisatoddswiththeideologyoffreelove,JimisslightlyjealousofJules:
Shebestowedhergraciousnessoneachinturn…andJimwasjealous(p.97)
Thetriangleishencedescribedbycouplingthetwosub-modelsthroughsuitableparametersthatmeasurethesmalldevi-ationsofthethreecharactersfromtheprincipleoffreelove.
A.ThesubmodelKathe-Jules
Katheisapassionatewoman,andthoughcharmedbyJules,sheisatthesametimeannoyedbyhisplatonicnature:
Shehadbeendrawnbyhismind,hisgiftoffantasy.Butsheneeded,inadditiontoJules,amaleofherownsort(p.90)
Forthisreason,herreactionRLtoJules’loveisoftheinsecuretype(Fig.2,top).
Moreover,Katheisdefinitelyanenthusiasticperson,soherreactiontoJules’appealisamplifiedbythefactor(1þS),whereSisKathe’ssynergism(Fig.2,bottom).
Inconclusion,assumingthatKathe’sforgettingandreactiontoappealarelinear,herequationis
x1ðtþ1Þ¼x1ðtÞÀfx1ðtÞþRLðy1ðtÞÞþð1þSðx1ðtÞÞÞrAa1:
(2)
ThemodelofthecoupleKathe-Julesisthereforecom-posedofEqs.(1)and(2).ThemodelcanbeusedrepeatedlytocomputethetimeevolutionofthefeelingsofKatheandJules.Forthis,wemustfirstassignreasonablevaluestoallparame-ters,takingintoaccountallpossibleindicationspresentinthenovel.Forexample,wetakeKathe’sappealagreaterthanJules’onea1,becausesheis,byfar,morefascinatingthanhim.Similarly,weassumesheforgetsfasterthanhim,f>f1,beingthemoreunstableinthecouple.Ofcoursethespecificvalueswehaveselectedremainratherarbitraryandbasedonoursub-jectiveinterpretations.AllthedetailsaboutthefunctionsRL,S,andPandtheparametervaluescanbefoundinTablesIandII.
Now,assumingthatthedaytheymeetforthefirsttime,sayt¼0,KatheandJulesarecompletelyindifferentonetoeachother,wecanfixx1(0)¼y1(0)¼0andusethetwoequa-tionstocomputethevaluesofthetwofeelingsthenextday,thusobtainingx1(1)¼rAa1andy1(1)¼rA1a.Itisinterestingtonotethatonlyappealmattersatthebeginningofalovestory,sincefeelingsarestilllatent.Togoontothenextday,itissufficienttoincreasetimeofoneunitandusethesameequa-tionswrittenfort¼1tocomputethefeelingsatdayt¼2.Notethatalsotheforgettingfunctionsandthereactionstolovearenowinvolved.Repeatingthesameoperationsfort¼2,3,…,wecancomputethefeelingsofKatheandJulesatday3,4,…,andcontinuelikethisformonthsoryears.Theresultscanbeeasilyportrayedtoshowtheevolutionofthelovestoryinatimeintervalofinterest.InthiswayweobtainthegraphsinFig.3(top),wherethepointsindicatedwith1,2,and3repre-sentthefeelingsofKatheandJulesattheendofthefirst,sec-ond,andthirdyearoftheirrelationship.
KatheandJulesarealwayspositivelyinvolved,buttheirlovestorydoesnotreachaplateau.Indeed,astimegoeson,theirfeelingstendtooscillatewithaperiodofabout4years,moreprecisely3yearsand10months.Atthebeginningoftheirrelationship,KatheandJulesareincreasinglyinvolved,untilKathehasthefirstinversioninhertrend.Accordingtothemodel,theseinversionsarerecurrent.
B.ThesubmodelKathe-Jim
ThemainpeculiarityofJulesistobeplatonic:Really,Julesishappy,inhisownway,andjustwantsthingstogoon.He’sseeingyouoften,inidylliccircumstances,andhe’slivingonhope(p.24)
HethereforereduceshisreactiontoKathe’sappealwhenheismoreinlovewithher,i.e.,hisreactiontoappealisdampedbythefactor(1–P),wherePisJules’platonicity(shapedasinFig.2,bottom).
Inaccordancewithhisplatonicnature,Julesisasecurelover,andassuminglinearforgettingandreactionfunctions,theequationregulatinghisfeelingforKatheisthefollowing:y1ðtþ1Þ¼y1ðtÞÀf1y1ðtÞþr1x1ðtÞþð1ÀPðy1ðtÞÞÞrA1a:
(1)
ThemaincharacteristicofJimistobeinsecure,asall“DonJuan”aretoavoiddeepinvolvements:
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023134-5F.DercoleandS.RinaldiChaos24,023134(2014)
FIG.3.ThehypotheticallovestoriespredictedbytheKathe-Jules(top)andbytheKathe-Jim(bottom)submodels.
‘Oh,when,’shesaidtohimoneday,—‘whenareyougoingtostopgivingmebitsofyourselfandgivemeeverything?’(p.207)
Thus,hisreactionRL2toKathe’sloveisnonlinearandshapedasinFig.2(top).Assumingthathisforgettingandreactiontoappealarelinear,Jim’sequationisthen
y2ðtþ1Þ¼y2ðtÞÀf2y2ðtÞþRL2ðx2ðtÞÞþrA2a:
(3)
KatheissecureinherrelationshipwithJim(becauseheisnotplatonic)andsynergic.ThisisthereforeKathe’sequation,x2ðtþ1Þ¼x2ðtÞÀfx2ðtÞþrLy1ðtÞþð1þSðx2ðtÞÞÞrAa2;
(4)
whereSisagainKathe’ssynergism.
Inconclusion,themodelofthecoupleKathe-JimiscomposedofEqs.(3)and(4).Againparametersmustbefixedatreasonablevalues,e.g.,Jim’sappeala2smallerthanKathe’sone,thoughlargerthanJules’appeal—Jimbeingacharming“DonJuan;”andJim’sforgettingbeingfasterthanJules’one,f2>f1,inagreementwiththe“DonJuan”natureofJim(seeTableII).Onceallparametersarefixed,themodelcanberepeatedlyusedtocomputethetimeevolutionofthefeelingsofKatheandJim.TheresultisinFig.3(bot-tom).Inthiscasetoo,theinvolvementsofKatheandJimincreaseduringthefirstphaseoftheirrelationshipandthentendinafewyearstowardaswingingregimewithaperiodof3yearsand4months.Thistime,thefirsttoinvertthepositivetrendisJim,whobeinginsecurerefusestoodeepinvolvements.
C.Themodelofthetriangle
Toimplementthefirstchange,weassumethatKathe’sforgettingcapabilitiesdependuponherstateofinvolve-ment.Moreprecisely,weassumethatatanygiventimesheforgetslessquicklytheloversheismoreinvolvedwith.Thisisrealizedbymultiplying,inthetwoequationsforKathe(seebelow),herforgettingcoefficientfbyafac-torwhichisgreaterthan1inoneequationandsmallerthan1intheother.Inordertodeviateonlyslightlyfromthefree-loveprinciple,emustbeasmallpositiveparameter.
JulesdoesnotsufferwhenKatheismoreinlovewithJim.Actually,heispleasedbecauseheseesKathemorehappy.Asalreadysaid,thispeculiarcharacteristicisconsist-entwiththeplatonicnatureofJulesandiswelldescribedbyRoche.InordertotakeJules’complaisanceintoaccount,hisreactiontoKathe’sloveisamplifiedbyafactorgreaterthan1whensheismoreinlovewithJim,namely,whenx2isgreaterthanx1(seeJules’equation).
InordertotakeJim’sjealousyintoaccount,hisreactiontoKathe’sloveisdumpedbyafactorsmallerthan1whensheismoreinlovewithJules,namely,whenx1isgreaterthanx2(seeJim’sequation).Forsimplicity,Jules’complai-sanceandJim’sjealousyarequantifiedbythesamepositiveparameterd,whichmustalsobesmallifweliketoavoidlargedeviationsfromthefree-loveprinciple.
Inconclusion,themodelofthetriangleiscomposedofthefollowingfourdifferenceequations:x1ðtþ1Þ¼x1ðtÞÀfexpðeðx2ðtÞÀx1ðtÞÞÞx1ðtÞ
þRLðy1ðtÞÞþð1þSðx1ðtÞÞÞrAa1;ðKatheforJulesÞ
x2ðtþ1Þ¼x2ðtÞÀfexpðeðx1ðtÞÀx2ðtÞÞÞx2ðtÞ
þrLy1ðtÞþð1þSðx2ðtÞÞÞrAa2;ðKatheforJimÞ
y1ðtþ1Þ¼y1ðtÞÀf1y1ðtÞþr1x1ðtÞexpðdðx2ðtÞÀx1ðtÞÞÞþð1ÀPðy1ðtÞÞÞrA1a;
y2ðtþ1Þ¼y2ðtÞÀf2y2ðtÞ
þRL2ðx2ðtÞÞexpðdðx2ðtÞÀx1ðtÞÞÞþrA2a;
ðJulesÞðJimÞ
Themodelofthetriangleisobtainedbyweaklycou-plingthetwosubmodelsKathe-JulesandKathe-Jim.Forthisweintroducethefollowingextra-characteristicsinthebehaviorsofthethreeindividuals:
–Kathedoesnotliveinfullyseparatedworlds;–JulesiscomplaisantwithJim;–JimisjealousofJules.
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TABLEI.Nonlinearfunctions(specifiedfornonnegativefeelings).Character
Symbol
Expression
8>1Àððy1ÀsIÞ=yIÞ2> 8 2>> 2>> ify1!sIify1 KatheRL(y1)Kathe’sreactiontoJules’love S(x) JulesP(y1) JimRL2(x2) TABLEII.Modelparameters.CharacterKathe ContextForgetting ReactiontoloveRL(y1) SymbolfrLrIyLsIyIrAssSxSaf1rL1rA1psPyPa1f2rI2xLsI2xIrA2a2Value2/3651/36580/365102.510.51/365291201/3651/3650.5/36510142/36520/36510911/3655 Description Kathe’sforgettingcoefficient Kathe’sreactioncoefficienttoJim’sloveKathe’s-to-JulesmaximuminsecurenessSensitivityofKathe’sreactiontoJules’loveKathe’s-to-JulesinsecurenessthresholdSensitivityofKathe’s-to-JulesinsecurenessKathe’sreactioncoefficienttoappealKathe’smaximumsynergismKathe’ssynergismthresholdSensitivityofKathe’ssynergismKathe’sappeal Jules’forgettingcoefficient Jules’reactioncoefficienttoloveJules’reactioncoefficienttoappealJules’maximumplatonicityJules’platonicitythresholdSensitivityofJules’platonicityJules’appeal Jim’sforgettingcoefficientJim’smaximuminsecureness SensitivityofJim’sreactiontoloveJim’sinsecurenessthresholdSensitivityofJim’sinsecurenessJim’sreactioncoefficienttoappealJim’sappeal ReactiontoappealS(x) Jules AppealForgetting ReactiontoloveReactiontoappealP(y1) Jim AppealForgettingRL2(x2) ReactiontoappealAppeal anddiffersfromtheensembleofthetwoindependentsubmo-delsforthepresenceofthetwosmallcouplingparameterseandd(seeTablesIandIIfortheanalyticalexpressionsandthereferencevaluesoftheotherparameters). IV.RESULTS A.Validationofthemodel (i) (ii)(iii)(iv) Inthetwentyyearsofconcern,Kathechangespartnerseventimes,alternatingbetweenJulesandJim; thechronologyofthepartnerchangesiswelldocu-mentedbyRoche; duringthefirstyearsKatheismoreattractedbyJules(shemarrieshim); attheverybeginningofthestory,KatheismoreattractedbyJim,whomissesastrategicdate:,thingsmightIfKateandJimhadmetatthecafe haveturnedoutverydifferently(p.80) WenowvalidateourmodelofJulesetJimagainstthefollowingquantitative/qualitativefeatureswehaveidentifiedinthenovel: This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 110.65.147.240 On: Tue, 24 Nov 2015 00:31:15 023134-7F.DercoleandS.RinaldiChaos24,023134(2014) (v) Jim’supsanddownsaremorerelevantthanthoseofJules: “Jimwaseasyforhertotake,buthardtokeep.Jim’slovedropstozerowhenKate’sdoes,andshootsuptoahundredwithhers.Ineverreachedtheirzeroortheirhundred”(p.231) (vi) ThedropsininterestofKatheforJulesanticipatethoseofJulesforKathe: ThedangerwasthatKatewouldleave.Shehaddoneitoncealready…andithadlookedasifshedidn’tmeantoreturn…Shewasfullofstressagain,Julescouldfeelthatshewasworkingupforsomething(p.89) (vii) ThedropsininterestofJimforKatheanticipatethoseofKatheforJim: HehimselfwasincapableoflivingformonthsatatimeinclosecontactwithKate,italwaysbroughthimintoastateofexhaustionandinvoluntaryrecoilwhichwasthecauseoftheirdisasters(p.189) Wekeepallparameters(excepteandd)atthevaluesofTableIIandwefirstlookforpairs(e,d)forwhichfeature(i)isreproducedbythemodel.Forthiswefixadensegridinthe(e,d)planeandwesystematicallysimulateourmodelforeachpointofthegrid,alwaysstartingfromthestateofindiffer-ence—sinceJulesandJimaretogetherwhentheyareintroducedtoKathe—andstoppingthesimulationafter20years.Thepairs(e,d)intheovershadedregioninFig.4arethoseforwhichthemodelpredictssevenchangesofpartner—sevenchangesofsignofKathe’sunbalancex1–x2aftershemarriesJules. Andfortheparticularvaluesofeanddcorrespondingtothewhitedotinthefigure,thepredictedchronologyofthepartnerchangesisinbestagreementwith(ii).Kathe’sunbal-anceisgraphedinFig.5(bottom-left)andthecorrelationbetweentheseveninstantssuggestedbythemodelandthoseindicatedbyRocheis0.97!(Fig.5,bottom-right). WethencomparethemodelpredictionsofFig.5withfeatures(iii)–(vii).Feature(iii)iswellpredictedbecausex1>x2inthefirstyearsofthestory(seeKathe’sunbalance).Feature(iv)isalsopredicted,evenifnotvisibleatthescaleofthefigure.Infact,Jules’appealislowerthanthatofJim(a1 FIG.4.ThelargestLyapunovexponentofthemodeltrajectorystartingfromx1(0)¼x2(0)¼y1(0)¼y2(0)¼0(computationbasedonthediscreteQRstandardalgorithm19andcheckedtobeconsistentwithothernonnegativeinitialconditions).Theexponentispositive(red)forchaoticattractors;zero(yellow)forquasi-periodicattractorsandbifurcatingcycles;negative(green)forstablecycles.For(e,d)intheovershadedregionthemodelpre-dictsthatKathechangespartnerseventimesin20years(seeFig.5,bottom). intheovershadedregionofFig.4.Then,wehavesystemati-callysimulatedourmodelbyperturbing(upto10%)thepa-rametersoftheKathe-JulesandKathe-Jimsubmodels,andwehavecheckedthatitwasalwayspossibletofitfeatures(i)–(vii)withsmallvaluesofthecouplingparameterseandd.Thereadercanverifytherobustnessofourresultsbyusingtheonlinesimulator(seesupplementarymaterial22),whereallmodelparameterscanbesignificantlychanged. B.Supportoftheconjecture Tosupportourconjecture,weneedtoshowthat,forrea-sonableparametersettings,thetrajectoryofourmodelorigi-natingatthestateofindifferenceconvergestoachaoticattractorandthattheassociatedunpredictabilityisatworkinthefirst20years. Focusingonourvalidatedparametersetting(TableIIand(e,d)atthewhitedotinFig.4),weobtainthechaoticattractordepictedinFig.6forwhichweestimateaLyapunovexponentof0.07yr–1.Thecharacteristictimeofdivergenceofnearbytrajectoriesafterwhichpredictionsbecomeimpracticable(theinverseoftheLyapunovexpo-nent17)ishenceabout15yr.Moreover,fromFig.5,weseethattheattractorisreachedonlyafewyearsafterthebegin-ningofthelovestory,sothatwecanconcludethatunpre-dictabilitycanbefeltbeforetheendofthestory. TheLyapunovexponenthasbeencomputedforallpairs(e,d)consideredinFig.4(seethecolor-code),andtheresultisthetypicalbifurcationdiagramexpectedforweaklycoupledoscillators.Forextremelyweakcoupling,themodelattractorisatorus(seetheyellowregionclosetoe¼d¼0).Then,forlargercoupling,thetwooscillatorscansynchron-izeonacycleontorus,andthisoccursinthewell-known This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 110.65.147.240 On: Tue, 24 Nov 2015 00:31:15 023134-8F.DercoleandS.RinaldiChaos24,023134(2014) FIG.5.Thelovestorypredictedbythevalidatedmodel(eanddatthewhitedotinFig.4,otherparametersasinTableII).(Toppanels)TimeseriesofthefeelingsandtrajectoryprojectionsintheplanesoftheKathe-JulesandKathe-Jimsubmodels.(Bottompanels)Kathe’spreferenceandcomparisonwiththenovel. Arnoldtongues(theverythingreenishregions).Increasingthecoupling,theattractorundergoesacomplexstructureofbifurcations—notdiscussedindetail—thatdescribetheclas-sicaltorus-destructionroutetochaos.ThegenericityofFig.4confirmsoncemoretherobustnessofourresults. Notethatonlyaweakcouplingallowstosupporttheconjecture,sincethemodelattractorisperiodicifthecou-plingistoostrong,whereastheuncoupledensembleoftheKathe-JulesandKathe-Jimsubmodelsdescribesaperiodicorquasi-periodiclovestory.Interestingly,chaoscanbefoundfore¼0,butnotford¼0,suggestingthatthecomplaisanceofJulesandthejealousyofJimarethekeyelementstriggeringthecomplexityoftheirstory. V.DISCUSSIONANDCONCLUSIONS FIG.6.Projectionsofthechaoticattractorreachedbythevalidatedmodel(thelimitcyclesofFig.3aresuperimposedforcomparisonwiththeuncoupledensembleoftheKathe-JulesandKathe-Jimsubmodels). Asiswellknown,evenfrompersonalexperience,senti-mentalrelationshipsareinfluencedbythesocialenviron-mentinwhichindividualslive.Itisthereforenotsurprisingifthefeelingscharacterizingromanticrelationshipsinturbu-lentenvironmentscannotbepredicted.Moresubtleandinterestingistheideadiscussedinthispaper:lovestoriescanbeunpredictableeveninconstantenvironments,i.e.,onthesolebasisofthecharactersinvolved.ThisideahasbeenconjecturedinRef.2andthensupportedinRef.6throughana€ıvemathematicalmodelofahypotheticallovestory. Herewehaveprovedtheconjecturebymakingrefer-encetotherealandwelldocumentedtriangularlovestory,involvingKathe,Jules,andJim,describedbyRocheinhis 161953novelJulesetJim. Therearefiveaspectsofourstudythatareworthtobementionedbecauseofgeneralinterest.Thefirstconcernsthemethodofanalysis,whichisgeneralandconsistentwithstand-ardpsychoanalysis.First,themainpsycho-physicaltraitsoftheindividualsinvolvedareidentified,inthiscasefromacarefulreadingofthenovel.Then,thesecharacteristicsareencapsu-latedinamathematicalmodel—theformalanalogueoftheverbaldescriptionsmoretraditionallyusedinpsychology—whichisvalidatedbycomparingthemodelpredictionswiththemostrelevantfeaturesofthelovestory.Theresultisverysatis-factory:forsuitablevaluesoftheparameters,thematchingbetweenthestorypredictedbythemodelandthestory This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 110.65.147.240 On: Tue, 24 Nov 2015 00:31:15 023134-9F.DercoleandS.RinaldiChaos24,023134(2014) describedbyRochegoesbeyondwhatistypicallyexpectedinthecontextofsocialdynamics.ThisallowedustoprovetheconjecturebysimplycheckingthatthevalidatedmodelhasapositiveLyapunovexponent.Specifically,weshowthatboththetimeneededtoreachthestrangeattractorfromtheinitialstateofindifferenceandthecharacteristictimeofdivergenceofnearbytrajectoriesintheattractoraresignificantlysmallerthanthelengthofthelovestory(20years),sothatitispossibletoinferthatKathe,Jules,andJimhadhighchancestofeeltheunpredictabilityoftheirstory. Asasecondinterestingaspect,wefoundthatunpredict-abilityistriggeredbyminorandalmosthiddentraitsofthethreecharacters:Katheforgetsslightlymorequicklytheloversheislessinvolvedwith,whileJulesandJimareslightlycomplaisantandjealous.Inthegeneralcontextofsystemstheory,thisconfirmsthatsmallparameterscanplaystrategicrolesinpromotingcomplexdynamics,whileinthespecificcontextofinterpersonalrelationshipsthisjustifiestheinterestthatpsychoanalystshaveinapparentlyminordetails. Athirdaspectweliketomentionisconcernedwiththestructureofourmodel.Itiscomposedoftwooscillators(thesubmodelsKathe-JulesandKathe-Jim),whichinterferethroughweakcouplingmechanisms.Thisiscommoninseveralfieldsofscience,wheresystemscanbeviewedasinterconnectedoscillatingunits.Forexample,inecologyeachconsumerpopulationhasafavoriteresourcebutcanalsofeedonasecondaryspecies,which,inturn,canbethefavoriteresourceforanotherconsumer.Thus,complexfoodwebsarenaturallydescribedasconsumer-resourceunitsinterconnectedthroughthefeedingpreferences.Amodeloftwoconsumerscompetingfortworesourceshasthereforethesamestructurethanthemodelconsideredinthispaper.Thisisofgreatpotentialinterest,becausesomeofthegeneralresultsobtainedinmathematicalecology,20and/orresultsinthetheoryofcoupledoscillators,21couldguidethemodelingofcomplexinterpersonalrelationships. Thefourthaspecttoberemarkedisthatalovestorycanbechaoticwithoutnecessarilyinvolvingthreeindividuals,asinthecasestudiedinthispaper.Indeed,sentimentalchaoscanbepresentinthemorestandardsituationinvolvingtwoindividuals,providedatleastoneischaracterized,inadditiontotheromanticsphere,byasecondimportantemotionalcompartment.Thisistypicalofindividualsinvolvedincreativeprofessions,whereinspiration,satisfaction,andself-esteemcaninterferewiththeromanticsphere.Andsinceamodelofthissituationwouldbeatleastthree-dimensional,instabilities(chaos)caneasilyarise.Forexample,thedesta-bilizingeffectofinspirationhasbeenpointedoutintheromanticrelationshipbetweenPetrarch,thefamouspoetofthe13thcentury,andhismistress.12Finally,thelastgeneralmessageweliketoextractfromourstudyisthefactthatamathematicalstudycanbeusedtohighlightthegeniusofanartist—inthiscaseFranc¸oisTruffaut,oneoftheprominentdirectorsofthe“NouvelleVague”—whofeaturedRoche’snovelinhismostimportantfilm,JulesetJim,madein1961afterdiscussingtheideawithRoche.JeanneMoreauandOskarWerner,alreadywellknown,playedKatheandJules,whileHenriSerre,selectedbecauseofacertainresemblancetoRoche,playedJim. Truffautomitsmanyminorcharactersofthenovel,thuscon-sideringanalmoststeadysocialenvironment,butsuccess-fullyreproducesthefeelingsbetweenHelenGrundandthe twofriends.IndeedHelenGrund,theonlyoneofthethreewhocouldwatchthefilmafterHesselandRochepassedaway,wrotetoTruffaut: Butwhatdispositioninyou,whataffinitycouldhaveenlightenedyoutothepointofrecreating—inspiteoftheoddinevitabledeviationandcompromise—theessentialqualityofourintimateemotions? Truffautmagistrallyadds,hereandthere,explicitele-mentspointingtothefactthatlovestoriescanbeturbulentbecauseofattractingandrepellingforces.Sincethediscus-sionoftheseoriginalelementswouldbringustoofar,weonlymentionherethemostexplicitreferencetoattractionandrepulsion,Letourbillondelavie(thevortexoflife),thesoundtracksungbyJeanneMoreau.Thissongisundoubt-edlyabeautifulhymntochaos,characterizedbyrecurrentphasesofconvergenceanddivergence.FurtherdetailsonthegeniusofFranc¸oisTruffautinusingthemetaphorofstretch-ingandfoldingwillbepublishedelsewhere. ACKNOWLEDGMENTS TheauthorsaregratefultoJose-ManuelRey,whosecommentssignificantlyimprovedthepresentation. G.Levinger,J.Exp.Soc.Psychol.16,510(1980).S.H.Strogatz,Math.Mag.61,35(1988).3S.RinaldiandA.Gragnani,inTheComplexMattersofMind,editedbyF.Orsucci(WorldScientific,Singapore,1998),pp.87–104.4J.M.Gottman,J.D.Murray,C.Swanson,R.Tyson,andK.Swanson,TheMathematicsofMarriage:DynamicNonlinearModels(MITPress,Cambridge,MA,2002).5H.D.I.Abarbanel,AnalysisofObservedChaoticData(Springer-Verlag,NewYork,1996).6J.C.Sprott,NonlinearDynam.Psychol.LifeSci.8,303(2004).7D.H.FelmleeandD.F.Greenberg,J.Math.Sociol.23,155(1999).8X.LiaoandJ.Ran,Chaos,SolitonsFractals31,853(2007).9J.Wauer,D.Schwarzer,G.Q.Cai,andY.K.Lin,Appl.Math.Comput.188,1535(2007).10S.Rinaldi,F.DellaRossa,andF.Dercole,Int.J.BifurcationChaos20,2443(2010).11J.-M.Rey,PLoSOne5,e9881(2010).12S.Rinaldi,SIAMJ.Appl.Math.58,1205(1998).13S.Rinaldi,F.DellaRossa,andP.Landi,PhysicaA392,3231(2013).14S.Rinaldi,P.Landi,andF.DellaRossa,Int.J.BifurcationChaos23,1330038(2013).15S.Rinaldi,F.DellaRossa,andP.Landi,NonlinearDynam.Psychol.LifeSci.18,199(2014).16H.-P.Roche,JulesetJim(EditionsGallimard,Paris,1953)(inFrench;Englishtrans.byP.Evans,MarionBoyars,1998).17S.H.Strogatz,NonlinearDynamicsandChaos(Addison-Wesley,Reading,MA,1994).18J.A.Simpson,S.W.Gangestad,andM.Lerma,J.Pers.Soc.Psychol.59,1192(1990).19L.Dieci,R.D.Russell,andE.S.V.Vleck,SIAMJ.Numer.Anal.34,402(1997).20EncyclopediaofTheoreticalEcology,editedbyA.HastingsandL.Gross(UniversityofCaliforniaPress,Berkeley,2012).21A.Pikovsky,M.Rosenblum,andJ.Kurths,Synchronization:AUniversalConceptinNonlinearSciences(CambridgeUniversityPress,Cambridge,UK,2001).22Seesupplementarymaterialathttp://dx.doi.org/10.1063/1.4882685fortheonlinesimulatorandcreditsandquotes. 21 This article is copyrighted as indicated in the article. 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