混沌理论chaostheory中英文

ChaosTheory:ABriefWhatexactlyischaos?Thename"chaostheory"comesfromthefactthatthesystemsthatthetheorydescribesareapparentlydisordered,butchaostheoryisreallyaboutfindingtheunderlyingorderinapparentlyrandomdata.Whenwaschaosfirstdiscovered?Thefirsttrueexperimenter haoswasameteorologist,namedEdwardLorenz.In1960,hewasworkingontheproblemofweatherprediction.Hehadacomputersetup,withasetoftwelveequationstomodeltheweather.Itdidn'tpredicttheweatheritself.Howeverthiscomputerprogramdidtheoreticallypredictwhattheweathermightbe.Figure1:Lorenz'periment:thedifferencweenthestartingvaluesofthesecurvesisonly.000127.(IanStewart,DoesGodyDice?TheMathematicsofChaos,pg.141)Onedayin1961,hewantedtoseeaparticularsequenceagain.Tosavetime,hestartedinthemiddleofthesequence,insteadofthebeginning.Heenteredthenumberoffhisprintoutandlefttoletitrun.Whenhecamebackanhourlater,thesequencehadevolveddifferently.Insteadofthesamepatternasbefore,itdivergedfromthepattern,endingupwildlydifferentfromtheoriginal.(Seefigure1.)Eventuallyhefiguredoutwhathappened.Thecomputerstoredthenumberstosixdecimalcesinitsmemory.Tosavepaper,heonlyhaditprintoutthreedecimalces.Intheoriginalsequence,thenumberwas.506127,andhehadonlytypedthefirstthreedigits,.506.Byallconventionalideasofthetime,itshouldhaveworked.Heshouldhavegottenasequenceveryclosetotheoriginalsequence.Ascientistconsidershimselfluckyifhecangetmeasurementswithaccuracytothreedecimalces.Surelythe andfifth,impossibletomeasureusingreasonablemethods,can'thaveahugeeffectontheoutcomeoftheexperiment.Lorenzprovedthisideawrong.Thiseffectcametobeknownasthebutterflyeffect.Theamountofdifferenceinthestartingpointsofthetwocurvesissosmallthatitiscomparabletoabutterflyflapitswings.Theflapofasinglebutterfly'swingtodayproducesatinychangeinthestateoftheatmosphere.Overaperiodoftime,whattheatmosphereactuallydoesdivergesfromwhatitwouldhavedone.So,inamonth'stime,atornadothatwouldhavedevastatedtheIndonesiancoastdoesn'thappen.Ormaybe hatwasn'tgoingtohappen,does.(IanStewart,DoesGodyDice?TheMathematicsofChaos,pg.141)Thisphenomenon,commontochaostheory,isalsoknownassensitivedependenceoninitialconditions.Justasmallchangeintheinitialconditionscandrasticallychangethelong-termbehaviorofasystem.Suchasmallamountofdifferenceinameasurementmightbeconsideredexperimentalnoise,backgroundnoise,oraninaccuracyoftheequipment.Suchthingsareimpossibletoavoidineventhemostisolatedlab.Withastartingnumberof2,thefinalresultcanbeentirelydifferentfromthesamesystemwithastartingvalueof2.000001.Itissimplyimpossibletoachievethislevelofaccuracy-justtryandmeasuresomethingtothenearestmillionthofan Fromthisidea,Lorenzstatedthatitisimpossibletopredicttheweatheraccuray.However,thisdiscoveryledLorenzontootheraspectsofwhateventuallycametobeknownaschaosLorenzstartedtolookforasimplersystemthathadsensitivedependenceoninitialconditions.Hisfirstdiscoveryhadtwelveequations,andhewantedamu oresimpleversionthatstillhadthisattribute.Hetooktheequationsforconvection,andstrippedthemdown,makingthemunrealisticallysimple.Thesystemnolongerhadanythingtodowithconvection,butitdidhavesensitivedependenceonitsinitialconditions,andtherewereonlythreeequationsthistime.Later,itwasdiscoveredthathisequationspreciselydescribedawaterwheel.Atthetop,waterdripssteadilyintocontainershangingonthewheel'srim.Eachcontainerdripssteadilyfromasmallhole.Ifthestreamofwaterisslow,thetopcontainersneverfillfastenoughtoovercomefriction,butifthestreamisfaster,theweightstartstoturnthewheel.Therotationmightbecomecontinuous.Orifthestreamissofastthattheheavycontainersswingallthewayaroundthebottomanduptheotherside,thewheelmightthenslow,stop,andreverseitsrotation,turningfirstonewayandthentheother.(JamesGleick,Chaos-MakingaNewScience,pg.29)Figure2:TheLorenzAttractor(JamesChaos-MakingaNewScience,pg.Theequationsforthissystemalsoseemedtogiverisetoentirelyrandombehavior.However,whenhegraphedit,asurprisingthinghappened.Theoutputalwaysstayedonacurve,adoublespiral.Therewereonlytwokindsoforderpreviouslyknown:asteadystate,inwhichthevariablesneverchange,andperiodicbehavior,inwhichthesystemgoesintoaloop,repeatingitselfindefini y.Lorenz'sequationsweredefini yordered-theyalwaysfollowedaspiral.Theyneversettleddowntoasinglepoint,buts etheyneverrepeatedthesamething,theyweren'tperiodiceither.HecalledtheimagehegotwhenhegraphedtheequationstheLorenzattractor.(Seefigure2)In1963,Lorenzpublishedapaperdescribingwhathehaddiscovered.He ludedtheunpredictabilityoftheweather,anddiscussedthetypesofequationsthatcausedthistypeofbehavior.Unfortuna y,theonlyjournalhewasabletopublishinwasameteorologicaljournal,becausehewasameteorologist,notamathematicianoraphysicist.Asaresult,Lorenz'sdiscoveriesweren'tacknowledgeduntilyearslater,whentheywererediscoveredbyothers.Lorenzhaddiscoveredsomethingrevolutionary;nowhehadtowaitforsome discoverhim.Anothersysteminwhichsensitivedependenceoninitialconditionsisevidentistheflipofacoin.Therearetwovariablesinaflipcoin:howsoonithitstheground,andhowfastitisflip.Theoretically,itshouldbepossibletocontrolthesevariablesentirelyandcontrolhowthecoinwillendup.Inpractice,itisimpossibletocontrolexactlyhowfastthecoinflipstocontrolitenoughtoknowthefinalresultsofthecointoss.Asimilarproblemoccursinecology,andthepredictionofbiologicalpopulations.Theequationwouldbesimpleifpopulationjustrisesindefini y,buttheeffectofpredatorsandalimitedfoodsupplymakethisequation orrect.Thesimplestequationthattakesthisintoaccountisthefollowing:nextyear'spopulation=r*thisyear'spopulation*(1-thisyear'sInthisequation,thepopulationisanumberbetween0and1,where1representsumpossiblepopulationand0represent tion.Risthegrowthrate.Thequestionwas,howdoesthisparame ffecttheequation?Theobviousansweristhatahighgrowthratemeansthatthepopulationwillsettledownatahighpopulation,whilealowgrowthratemeansthatthepopulationwillsettledowntoalownumber.Thistrendistrueforsomegrowthrates,butnotforeveryone.bifurcationdiagramforthepopulationequation.(JamesGleick,Chaos-MakingaNewScience,pg.71)Onebiologist,RobertMay,decidedtoseewhatwouldhappentotheequationasthegrowthratevaluechanges.Atlowvaluesofthegrowthrate,thepopulationwouldsettledowntoasinglenumber.Forinstance,ifthegrowthratevalueis2.7,thepopulationwillsettledownto.6292.Asthegrowthrate reased,thefinalpopulationwould reaseaswell.Then,somethingweirdhappened.Assoonasthegrowthratepassed3,thelinebrokeintwo.Insteadofsettlingdowntoasinglepopulation,itwouldjumpbetweentwodifferentpopulations.Itwouldbeonevalueforoneyear,gotoanothervaluethenextyear,thenrepeatthecycleforever.Raisingthegrowthratealittlemorecausedittojumpbetweenfourdifferentvalues.Astheparameterrosefurther,thelinebifurcated(doubled)again.Thebifurcationscamefas ndfasteruntilsuddenly,chaosappeared.Pastacertaingrowthrate,itbecomesimpossibletopredictthebehavioroftheequation.However,uponcloserinspection,itispossibletoseewhitestrips.Lookingcloseratthesestripsrevealslittlewindowsoforder,wheretheequationgoesthroughthebifurcationsagainbeforereturningtochaos.Thisself-similarity,thefactthatthegraphhasanexactcopyofitselfhiddendeepinside,cametobeanimportantaspectofchaos.AnemployeeofIBM,BenoitMandelbrotwasamathematicianstudyingthisself-similarity.Oneoftheareashewasstudyingwascottonpricefluctuations.Nomatterhowthedataoncottonpriceswasyzed,theresultsdidnotfitthenormaldistribution.Mandelbroteventuallyobtainedalloftheavailabledataoncottonprices,datingbackto1900.WhenheyzedthedatawithIBM'scomputers,henoticedanastonishingThenumbersthatproducedaberrationsfromthepointofviewofnormaldistributionproducedsymmetryfromthepointofviewofscaling.Eachparticularpricechangewasrandomandunpredictable.Butthesequenceofchangeswastonscale:curvesfordailypricechangesandmonthlypricechangesmatchedperfectly. redibly,yzedMandelbrot'sway,thedegreeofvariationhadremainedconstantoveratumultuoussixty-yearperiodthatsawtwoWorldWarsandadepression.(JamesGleick,Chaos-MakingaNewScience,pg.86) yzednotonlycottonprices,butmanyotherphenomenaaswell.Atonepoint,hewaswonderingaboutthelengthofacoastline.Amapofacoastlinewillshowmanybays.However,measuringthelengthofacoastlineoffamapwillmissminorbaysthatweretoosmalltoshowonthemap.Likewise,walkingalongthecoastlinemissesmicroscopicbaysinbetweengrainsofsand.Nomatterhowmuchacoastlineismagnified,therewillbemorebaysvisibleifitismagnifiedmore.Chaos-Figure4:TheKochcurveGleick,Chaos-MakingaNewScience,pg.Onemathematician,HelgevonKoch,capturedthisideainamathematicalconstructioncalledtheKochcurve.TocreateaKochcurve,imagineanequilaltriangle.Tothemiddlethirdofeachside,addanotherequilaltriangle.Keeponaddingnewtrianglestothemiddlepartofeachside,andtheresultisaKochcurve.(Seefigure4)AmagnificationoftheKochcurvelook actlythesameastheoriginal.Itisanotherself-similarfigure.TheKochcurvebringsupaninterestingparadox.Eachtimenewtrianglesareaddedtothefigure,thelengthofthelinegetslonger.However,theinnerareaoftheKochcurveremainslessthantheareaofacircledrawnaroundtheoriginaltriangle.Essentially,itisalineofinfiniengthsurroundingafinitearea.Togetaroundthisdifficulty,mathematiciansinventedfractaldimensions.Fractalcomesfromthewordfractional.ThefractaldimensionoftheKochcurveissomewherearound1.26.Afractionaldimensionisimpossibletoconceive,butitdoesmakesense.TheKochcurveisrougherthanasmoothcurveorline,whichhasonedimension.S eitisrougherandmorecrinkly,itisbet ttakingupspace.However,it'snotasgoodatfillingupspaceasasquarewithtwodimensionsis,s eitdoesn'treallyhaveanyarea.SoitmakessensethatthedimensionoftheKochcurveissomewhereinbetweenthetwo.Fractalhascometomeananyimagethatdisystheattributeofself-similarity.Thebifurcationdiagramofthepopulationequationisfractal.TheLorenzAttractorisfractal.TheKochcurveisfractal.Duringthistime,scientistsfounditverydifficulttogetworkpublishedaboutchaos.S theyhadnotyetshowntherelevancetoreal-worldsituations,mostscientistsdidnotthinktheresultsofexperiments haoswereimportant.Asaresult,eventhoughchaosisamathematicalphenomenon,mostoftheresearchintochaoswasdonebypeopleinotherareas,suchasmeteorologyandecology.Thefieldofchaossproutedupasahobbyforscientistsworkingonproblemsthatmaybehadsomethingtodowithit.Later,ascientistbythenameofFeigenbaumwaslookingatthebifurcationdiagramagain.Hewaslookingathowfastthebifurcationscome.Hediscoveredthattheycome rate.Hecalculateditas4.669.Inotherwords,hediscoveredtheexactscaleatwhichitwasself-similar.Makethediagram4.669timessmaller,anditlookslikethenextregionofbifurcations.Hedecidedtolookatotherequationstoseeifitwaspossibletodetermineascalingfactorforthemaswell.Muchtohissurprise,thescalingfactorwa actlythesame.Notonlywasthiscomplicatedequationdisyingregularity,theregularitywa actlythesameasamuchsimplerequation.Hetriedmanyotherfunctions,andtheyallproducedthesamescalingfactor,4.669.Thiswasarevolutionarydiscovery.Hehadfoundthatawholeclassofmathematicalfunctionsbehavedinthesame,predictableway.Thisuniversalitywouldhelpotherscientistseasilyyzechaoticequations.Universalitygavescientiststhefirsttoolstoyzeachaoticsystem.Nowtheycoulduseasimpleequationtopredicttheoutcomeofamorecomplexequation.Manyscientistswereexploringequationsthatcreatedfractalequations.Themostfamousfractalimageisalsooneofthemostsimple.ItisknownastheMandelbrotset(picturesofthemandelbrotset).Theequationissimple:z=z2+c.ToseeifapointispartoftheMandelbrotset,justtakeacomplexnumberz.Squareit,thenaddtheoriginalnumber.Squaretheresult,thenaddtheoriginalnumber.Repeatthatadinfinitum,andifthenumberkeepsongoinguptoinfinity,itisnotpartoftheMandelbrotset.Ifitstaysdownbelowacertainlevel,itispartoftheMandelbrotset.TheMandelbrotsetistheinnermostsectionofthepicture,andeachdifferentshadeofgrayrepresentshowfaroutthatparticularpointis.OneinterestingfeatureoftheMandelbrotsetisthatthecircularhumpsmatchuptothebifurcationgraph.TheMandelbrotfractalhasthesameself-similarityseenintheotherequations.Infact,zoomingindeepenoughonaMandelbrotfractalwilleventuallyrevealanexactreplicaoftheMandelbrotset,perfectineverydetail.Fractalstructureshavebeennoticedinmanyreal-worldareas,aswellasinmathematician'sminds.Bloodvesselsbranchingoutfurtherandfurther,thebranchesofatree,theinternalstructureofthelungs,graphsofstockmarketdata,andmanyotherreal-worldsystemsallhavesomethingommon:theyareallself-similar.ScientistsatUCSantaCruzfoundchaosinadripwaterfaucet.Byrecordingadripfaucetandrecordingtheperiodsoftime,theydiscoveredthatertainflowvelocity,thedripnolongeroccurredateventimes.Whentheygraphedthedata,theyfoundthatthedripdidindeedfollowapattern.Thehumanheartalsohasachaoticpattern.Thetimweenbeatsdoesnotremainconstant;itdependsonhowmuchactivitya ng,amongotherthings.Under onditions,theheartbeatcanspeedup.Underdifferentconditions,theheartbeatserratically.Itmightevenbecalledachaoticheartbeat.The ysisofaheartbeatcanhelpmedicalresearchersfindwaystoputanabnormalheartbeatbackintoasteadystate,insteadofuncontrolledchaos.Researchersdiscoveredasimplesetofthreeequationsthatgraphedafern.Thisstartedanewidea-perhaps encodesnotexactlywheretheleavesgrow,butaformulathatcontrolstheirdistribution. ,eventhoughitholdsanamazingamountofdata,couldnotholdallofthedatanecessarytodeterminewhereeverycellofthehumanbodygoes.However,byusingfractalformulastocontrolhowthebloodvesselsbranchoutandthenervefibersgetcreated,hasmorethanenoughinformation.Ithasevenbeenspeculatedthatthebrainitselfmightbeorganizedsomehowaccordingtothelawsofchaos.Chaosevenhasapplicationsoutsideofscience.Compu rthasbecomemorerealisticthroughtheuseofchaosandfractals.Now,withasimpleformula,acomputercancreateabeautiful,andrealistictree.Insteadoffollowingaregularpattern,thebarkofatreecanbecreatedaccordingtoaformulathatalmost,butnotquite,repeatsitself.Musiccanbecreatedusingfractalsaswell.UsingtheLorenzattractor,DianaS.Dabby,agraduatestudentinelectricalengineeringattheMassachusettsInstituteofTechnology,hascreatedvariationsofmusicalthemes.("BachtoChaos:ChaoticVariationsonaClassicalTheme",ScienceNews,Dec.24,1994)ByassociatingthemusicalnotesofapieceofmusiclikeBach'sPreludewiththexcoordinatesoftheLorenzattractor,andrunningacomputerprogram,shehascreatedvariationsofthethemeofthesong.Mostmusicianswhohearthenewsoundsbelievethatthevariationsareverymusicalandcreative.Chaoshasalreadyhadalastingeffectonscience,yetthereismuchstilllefttobediscovered.Manyscientistsbelievethattwentiethcenturysciencewillbeknownforonlythreetheories:relativity,quantummechanics,andchaos.Aspectsofchaosshowupeverywherearoundtheworld,fromthecurrentsoftheoceanandtheflowofbloodthroughfractalbloodvesselstothebranchesoftreesandtheeffectsofturbulence.Chaoshasinescapablybecomepartofmodernscience.Aschaoschangedfromalittle-knowntheorytoafullscienceofitsown,ithasreceivedwidespreadpublicity.Chaostheoryhaschangedthedirectionofscience:intheeyesofthegeneralpublic,physicsisnolongersimplythestudyofsubatomicparticlesinabillion-dollarparticleaccelerator,butthestudyofchaoticsystemsandhowtheywork.ChaosFromWikipedia,theThisarticleisaboutchaostheoryinMathematics.ForotherusesofChaostheory,seeChaosTheory(disambiguation).ForotherusesofChaos,seeChaos(disambiguation).Chaostheoryisafieldofstudyinmathematics,withapplicationsinseveraludingphysics,economics,biology,andphilosophy.Chaostheorystudiesthebehaviorofdynamicalsystemsthatarehighlysensitivetoinitialconditions,aneffectwhichispopularlyreferredtoasthebutterflyeffect.Smalldifferencesininitialconditions(suchasthoseduetoroundingerrorsinnumericalcomputation)yieldwidelydivergingoutcomesforchaoticsystems,renderinglong-termpredictionimpossibleingeneral.[1]Thishappenseventhoughthesesystemsaredeterministic,meaningthattheirfuturebehaviorisfullydeterminedbytheirinitialconditions,withnorandomelementsinvolved.[2]Inotherwords,thedeterministicnatureofthesesystemsdoesnotmakethempredictable.[3[4]Thisbehaviorisknownasdeterministicchaos,orsimplychaos.o2.1SensitivitytoinitialDensityofperiodicStrange2.5MinimumcomplexityofachaoticsysDistinguishingrandomfromchaoticCulturalSeeo2.1SensitivitytoinitialDensityofperiodicStrange2.5MinimumcomplexityofachaoticsysDistinguishingrandomfromchaoticCulturalSee oo8.18.28.3Semitechnicalandpopular9ExternalChaostheoryisappliedinmanyscientificdisciplines:mathematics,programming,microbiology,biology,computerdynamics,psychology,androbotics.Chaoticbehaviorhasbeenobservedinthelaboratoryinavarietyof udingcircuits,lasers,oscillatingchemicalreactions,fluiddynamics,andmechanicalandmao-mechanicaldevices,aswellascomputermodelsofchaoticprocesses.Observationsofchaoticbehaviorinnature ludechangesinweather,[5]thedynamicsofsalitesinthesolarsystem,thetimeevolutionofthemaicfieldofcelestialbodies,populationgrowthinecology,thedynamicsoftheactionpotentialsinneurons,andmolecularvibrations.Thereissomecontroversyovertheexistenceofchaoticdynamicsintetectonicsandineconomics.[12][13][14]AsuccessfulapplicationofchaostheoryisinecologywheredynamicalsystemssuchastheRickermodelhavebeenusedtoshowhowpopulationgrowthunderdensitydependencecanleadtochaoticdynamics[citationneeded].Chaostheoryisalsocurrentlybeingappliedtomedicalstudiesofepilepsy,specificallytothepredictionofseeminglyrandomseizuresbyobservinginitialconditions.[15]Quantumchaostheorystudieshowthecorrespondenc weenquantummechanicsandclassicalmechanicsworksinthecontextofchaoticsystems.[16]Recently,anotherfield,calledrelativisticchaos,[17]hasemergedtodescrbesystemsthatfollowthelawsofgeneralrelativity.ThemotionofNstarsinresponsetotheirself-gravity(thegravitationalN-bodyproblem)isgenericallyInelectricalengineering,chaoticsystemsareused ommunications,randomnumbergenerators,andencryptionInnumericalysis,theNewton-Raphsonmethodofapproximatingtherootsofafunctioncanleadtochaotic tionsifthefunctionhasnorealroots.[19][edit]ChaoticThemapdefinedbyThemapdefinedbyx→4x(1–x)andy→x+yifx+y<1(x+y–1otherwise)dis yssensitivitytoinitialconditions.Heretwoseriesofxandyvaluesdivergemarkedlyovertimefromatinyinitialdifference.ommonusage,"chaos"means"astateofdisorder".[20]However, haostheory,thetermisdefinedmoreprecisely.Althoughthereisnouniversallyacceptedmathematicaldefinitionofchaos,acommonlyuseddefinitionsaysthat,foradynamicalsystemtobeclassifiedaschaotic,itmusthavethefollowingproperties:[21]itmustbesensitivetoinitialitmustbetopologicallymixing;itsperiodicorbitsmustbeTherequirementforsensitivedependenceoninitialconditionsimpliesthatthereisasetofinitialconditionsofpositivemeasurewhichdonotconvergetoacycleofanylength. sitivitytoinitialSensitivitytoinitialconditionsmeansthateachpointinsuchasystemisarbitrarilycloselyapproximatedbyotherpointswithsignificantlydifferentfuturetrajectories.Thus,anarbitrarilysmallperturbationofthecurrenttrajectorymayleadtosignificantlydifferentfuturebehaviour.However,ithasbeenshownthatthelasttwopropertiesinthelistaboveactuallyimplysensitivitytoinitialconditions[22][23]andifattentionisrestrictedtointervals,thesecondpropertyimpliestheothertwo[24]( ternative,andingeneralweaker,definitionofchaosusesonlythefirsttwopropertiesintheabovelist[25]).Itisinterestingthatthemostpracticallysignificantcondition,thatofsensitivitytoinitialconditions,isactuallyredund thedefinition,beingimpliedbytwo(orforintervals,one)purelytopologicalconditions,whicharethereforeofgreaterinteresttomathematicians.Sensitivitytoinitialconditionsispopularlyknownasthe"butterflyeffect",socalledbecauseofthetitleofapapergivenbyEdwardLorenzin1972totheAmericanAssociationfortheAdvancementofScienceinWashington,D.C.entitledPredictability:DoestheFlapofaButterfly’sWingsinBrazilsetoffaTorna nTexas?Theflap representsasmallchangeintheinitialconditionofthesystem,whichcausesachainofeventsleadingtolarge-scalephenomena.Hadthebutterflynotflappeditswings,thetrajectoryofthesystemmighthavebeenvastlydifferent.Aconsequenceofsensitivitytoinitialconditionsisthatifwestartwithonlyafiniteamountofinformationaboutthesystem(asisusuallythecaseinpractice),thenbeyondacertaintimethesystemwillnolongerbepredictable.Thisismostfamiliarinthecaseofweather,whichisgenerallypredictableonlyaboutaweekahead.[26]TheLyapunovexponentcharacterisestheextentofthesensitivitytoinitialconditions. twotrajectoriesinphasespacewithinitialseparationdivergeseparationvector.Thus,thereisawholespectrumofLyapunovexponents—thenumberofthemisequaltothenumberofdimensionsofthephasespace.Itiscommontojustrefertothelargestone,i.e.tothe Lyapunovexponent(MLE),becauseitdeterminestheoverallpredictabilityofthesystem.ApositiveMLEisusuallytakenasanindicationthatthesystemischaotic.Therearealsomeasure-theoreticmathematicalconditions(discussedinergodictheory)suchasmixingorbeingaK-systemwhichrelatetosensitivityofinitialconditionsandchaos.[4][edit]TopologicalTopologicalmixing(ortopologicaltransitivity)meansthatthesystemwillevolveovertimesothatanygivenregionoropensetofitsphasespacewilleventuallyoverlapwithanyothergivenregion.Thismathematicalconceptof"mixing"correspondstothestandardintuition,andthemixingofcoloreddyesorfluidsisanexampleofachaoticsystem.Topologicalmixingisoftenomittedfrompopularaccountsofchaos,whichequatechaoswithsensitivitytoinitialconditions.However,sensitivedependenceoninitialconditionsalonedoesnotgivechaos.Forexample,considerthesimpledynamicalsystemproducedbyrepeatedlydoublinganinitialvalue.Thissystemhassensitivedependenceoninitialconditionseverywhere,s eanypairofnearbypointswilleventuallybecomewidelyseparated.However,thi amplehasnotopologicalmixing,andthereforehasnochaos.Indeed,ithasextremelysimplebehaviour:allpoint cept0tendtoinfinity.[edit]DensityofperiodicDensityofperiodicorbitsmeansthateverypointinthespaceisapproachedarbitrarilycloselybyperiodicorbits.Topologicallymixingsystemsfailingthisconditionmaynotdisysensitivitytoinitialconditions,andhencemaynotbechaotic.Forexample,anirrationalrotationofthecircleistopologicallytransitive,butdoesnothavedenseperiodicorbits,andhencedoesnothavesensitivedependenceoninitialconditions.[27]Theorbits.Forexample,→→(orapproximay0.3454915→0.9045085→isan(unstable)orbitofperiod2,andsimilarorbitistforperiods4,8,16,etc.(indeed,foralltheperiodsspecifiedbySharkovskii'stheorem).[28]exhibitsaregularcycleofperiodthreewillalsodisyregularcyclesofeveryotherlengthaswellascompleychaoticorbits.[edit]StrangeUnlikefixed-pointattractorsandlimitcycles,theattractorswhicharisefromchaoticsystems,knownasattractors,havegreatdetailandcomplexity.Strangeattractorsoccurinbothcontinuousdynamicalsystems(suchastheLorenzsystem)andinsomediscretesystems(suchastheHénonmap).Otherdiscretedynamicalfixedpoints–Juliasetscanbethoughtofasstrangerepellers.BothstrangeattractorsandJuliasetstypicallyhaveafractalstructure,andafractaldimensioncanbecalculatedforthem.[edit]MinimumcomplexityofachaoticDiscretechaoticsystems,suchasthelogisticmap,canexhbitstrangeattractorswhatevertheirHowever,thePoaré-Bendixsontheoremshowsthatastrangeattractorcanonlyarisedimensionallinearsystemsareneverchaotic;foradynamicalsystemtodisychaoticbehaviourithastobeeithernonlinear,orinfinite-dimensional.ThePoaré–Bendixsontheoremstatesthatatwodimensionaldifferentialequationhasveryregularbehavior.TheLorenzattractordiscussedaboveisgeneratedbyasystemofthreedifferentialequationswithatotalofseventermsontherighthandside,fiveofwhicharelineartermsandtwoofwhicharequadratic(andthereforenonlinear).Anotherwell-knownchaoticattractorisgeneratedbytheRosslerequationswithseventermsontherighthandside,onlyoneofwhichis(quadratic)nonlinear.Sprott[30]foundathreedimensionalsystemwithjustfivetermsontherighthandside,andwithjustonequadraticnonlinearity,whichexhibitschaosrtainparametervalues.ZhangandHeidel[31][32]showedthat,atleastfordissipativeandconservativequadraticsystems,threedimensionalquadraticsystemswithonlythreeorfourtermsontherighthandsidecannotexhibitchaoticbehavior.Thereasonis,simplyput,thatsolutionstosuchsystemsareasymptotictoatwodimensionalsurfaceandthereforesolutionsarewellbehaved.Whilethe aré–Bendixsontheoremmeans ontinuousdynamicalsystemonEuclideannecannotbechaotic,two-dimensionalcontinuoussystemswithnon-Euclideangeometrycanexhibitchaoticbehaviour.[citationneeded]Perhapssurprisingly,chaosmayoccuralsoinlinearsystems,providedtheyareinfinite-dimensional.[33]Atheoryoflinearchaosisbeingdevelopedinthefunctional ysis,abranchofmathematicalysis.AnearlyproponentofchaostheorywasHenriPo aré.Inthe1880s,whilestudyingthethree-bodyproblem,hefoundthattherecanbeorbitswhicharenonperiodic,andyetnotforeverreasingnorapproachingafixedpoint.[34][35]In1898JacquesHadamardpublishedaninfluentialstudyofthechaoticmotionofaparticleglidingfrictionlesslyonasurfaceofconstantnegativecurvature.[36]Inthesystemstudied,"Hadamard'sbilliards",Hadamardwasabletoshowthatalltrajectoriesareunstableinthatallparticletrajectoriesdivergeexponentiallyfromoneanother,withapositiveLyapunovexponent.Muchoftheearliertheorywasdevelopedalmostentirelybymathematicians,underthenameofergodictheory.Laterstudies,alsoonthetopicofnonlineardifferentialequations,werecarriedoutbyG.D.Birkhoff,[37]A.N.Kolmogorov,[38][39][40]M.L.CartwrightandJ.E.Littlewood,[41]andStephenS