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PAGE 1 DynamicsofanAnalogueofthe QuadraticFamilyon SU May16,2010 PAGE 2 DYNAMICSOFANANALOGUEOFTHEQUADRATICFAMILYON SU BY JOHNANTHONYEMANUELLO AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts UnderthesponsorshipofPatrickMcDonaldandNecmettinYildirim Sarasota,Florida May,2010 PAGE 3 Acknowledgments First,IwanttothankProfessorsPatrickMcDonaldandNecmettinYildirimfor theirsponsorshipofthisthesis,theirteaching,andtheirinsightintomathematics.I trulyappreciateallthelonghourstheyputintohelpingmepreparethisthesis. Also,ImustthankProfessorsEiriniPoimenidouandDavidMullinsforservingon myBaccalaureateCommitteeandprovidinginvaluablereccomendationsforimprovingmythesis. Iwouldalsoliketothankmymotherforraisingmeandsupportingmeforthese pasttwenty-twoyears. Lastly,Iwouldliketoexpressspecialappreciationtomygirlfriend,Kriszti.Her encouragementinspiredmetopursuemathematicsandherloveandsupporthave helpedtomakethisthesispossible. PAGE 4 DYNAMICSOFANANALOGUEOFTHEQUADRATICFAMILYON SU JohnAnthonyEmanuello NewCollegeofFlorida,2010 ABSTRACT Forthisthesisananalogueofthewellknown QuadraticFamily wasconstructed for S 3 ,theunitspherein R 4 ,usingthealgebraofunitquaternions.Becausethe unitquaternionscanbeidentiedwiththeLiegroup SU ,thefamilyprovidesa collectionofdynamicalsystemson SU .Thesedynamicsfor SU wereanalyzed withthepurposeofndinganaloguesofthewell-knownphenomenonassociatedtothe QuadraticFamily .Thiswasaccomplishedbycomputingtheorbitsofmanydierent seedvaluesfordierentparametervaluesandthencreatingagraphicalrepresentation ofthedata.Theresultsindicatethatthedynamicson SU arerichandthatthere aresomesimilaritiesbetweenitandthequadraticfamily. PatrickMcDonaldNecmettinYildirim DivisionofNaturalSciencesDivisionofNaturalSciences PAGE 5 Contents 1Introduction1 1.1StatementofProblem..........................2 1.2OutliningtheThesis...........................3 2Preliminaries4 2.1AnOverviewofDiscreteDynamicalSystems..............4 2.1.1DiscreteDynamicalSystems...................4 2.1.2MoreonFixedandPeriodicPoints...............8 2.1.3Bifurcations............................10 2.1.4Chaos...............................12 2.2TheQuadraticFamily..........................13 2.2.1When c> 1 = 4andWhen c =1 = 4................13 2.2.2When )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 PAGE 6 3TheAnalogueandaNumericalApproach30 3.1ConstructingtheAnalogue........................30 3.2NumericalApproach...........................31 4Results33 4.1NumericalResults.............................33 4.1.1InitialObservations........................33 4.2AnalyticResults.............................36 5Conclusions49 5.1AccomplishmentsandMissteps.....................50 5.2DirectionsforFurtherStudy.......................51 Appendices51 AMATLABCode52 v PAGE 7 Chapter1 Introduction Themaintopicofthisthesisis dynamicalsystems ,afascinatingsubeldofmathematicswhichisprimarilyconcernedwiththebehaviorofafunctionorfamilyof functionsonagivenspaceovertime.Dynamicalsystemscanbeboth discrete ,meaningthetimeisthoughtofasdistinctorunconnectedpoints,or continuous ,meaning timeisuninterrupted.Thisthesisisconcernedonlywiththeformer. Itcouldbearguedthatsincemost,ifnotall,dynamicalsystemsthatoccurin naturearereallycontinuoussystems,thestudyofdiscretesystemsisreallyanexercise inpedantry.Suchaclaimcouldnotbefartherfromthetruth.Infact,theresultsof discretesystemshavebeenappliedtocontinuoussystems.Eveniftheassertionwere true,thereisstillanaestheticmotivationtostudydiscretesystems.Whateverthe casemaybe,astheeldofdynamicalsystemshasgrown,therehasbeenroomfor researchersofbothdiscreteandcontinuoussystemstoworktogether. Thehistoryofdynamicalsystemsreachesasfarbacktoseventeenthandeighteenth centuriesfollowingNewton'sinventionoftheCalculus.However,mostoftheprofound developmentsofthissubeldhaveoccurredasrecentlyasthelastcentury.While manygreatmathematicianslikeJamesYorkeandStephenSmalehavedevotedmuch oftheircareerstouncoveringthemysteriesofdynamicalsystems,muchhasyetto 1 PAGE 8 beexplored.Thegoalofthisthesisistohelpshedsomelightonthedynamicsofa newmapwhichisananalogueofamorefamiliarone. 1.1StatementofProblem Thedynamicsof thequadraticfamily i.e.thefamilyoffunctionsoftheform f c : R R ,where f c x = x 2 + c wherec 2 R arewellknown.Forsomevaluesof c thedynamicsareverysimple,butthedynamicsarefarmorecomplicatedandeven chaotic forothervaluesof c ThedynamicsofanalogousmapsonotherLieGroupsarenotwellknown.Thus thequestionisasked:Whatdynamicalpropertiesofthismaparepreservedwhen denedonaspaceotherthan R ?Answeringsuchaquestionisbynomeansaneasy task.Intuition,computationaltools,andtimewillallberequired.Butmathematics isaversatileeld,sothetaskisnotaninsurmountableone. InthecurrentstudyananalogueofthequadraticfamilyontheLieGroup SU willbedened.Todoso,x C 2 R 4 with k C k6 =1,where kk denotestheEuclidean normin R 4 .Letthemap F C : SU SU ,bedenedby: F C A = A 2 + C k A 2 + C k .1 Thebasicideawillbetodetermineweatherornot.1sharesanyofitscore dynamics,suchastheexistenceofxedandperiodicpoints,withthatofthequadratic family.Toreachsuchaconclusion,rsttheorbitsforvaryingvaluesof C willbe computedformanypointsin SU vianumericalmethods.Thenumericaldatawill beanalyzed,andsomeresultswillbeprovenviaanalyticmethods. 2 PAGE 9 1.2OutliningtheThesis Beforethe real workcanbegin,thereadermustbeacquaintedwiththebasic termsandconceptsusedindynamicalsystems.Nosuchknowledgeisassumedand therstpartofChapterTwowillserveasanoverviewofdiscretedynamicalsystems andpresentthefundamentaldynamicpropertiesofthequadraticfamily.TheSecond ChapterwillalsointroducethereadertotheLieGroup SU anditsimportant algebraicandgeometricproperties. IntheThirdChapter,.1willbereintroducedandthenumericalmethodsused todescribethedynamicsoftheanaloguewillbedescribed.ChapterFourwilldescribe manyofthenumericalresultsandanalyticresults.InChapterFivetheresultswill besummarizedanddirectionsforfutureresearchwillbediscussed.MATLABthe softwareusedtogatherthenumericaldatacodewillbeprovidedintheappendix, althoughknowledgeoftheprogramlanguagewillbenecessarytoreallyappreciateit. 3 PAGE 10 Chapter2 Preliminaries Theprimarypurposeofthischapteristointroducethemajorterms,conceptsand theoremswhichwillberequiredtounderstandtheproblempresentedintheprevious chapter.Thereadershouldconsiderthissectionofthethesisabriefreviewofthe literature.Shouldfurtherinformationbedesired,thereferencesectioncontainssome neliteratureforthecurious. 2.1AnOverviewofDiscreteDynamicalSystems Inordertomaketheconceptsandtermsofdiscretedynamicalsystemsaccessible toalargeraudience,thescopeofthispreliminarychapterwillbelimitedtofunctions f : R R or g : C C .Thesefunctionsshouldbefamiliartothereaderandwill demonstratehowthedynamicsofsimple,familiarfunctionscanbequitecomplicated. 2.1.1DiscreteDynamicalSystems Therearemanygooddenitionsofdynamicalsystems.Theonechosenforthis workissimilartotheonefoundin[6]andisabitmoregeneralthantheonefound in[2]and[1].Butrst,someothernotionsmustrstbeclaried. 4 PAGE 11 Denition2.1. Let U beasetand d : U U R beadistancefunctionon U Thenthepair U;d iscalledametricspace. Thespace R n andthestandardEuclideannormisametricspacethatshouldbe familiartothereader. Denition2.2. Let X;d beametricspace, A X ,and T R .Let T t 0 ;t 1 = [ t 0 ;t 1 T T where t 1 ;t 0 2 T and t 1 >t 0 .Thenforanyxed a 2 A and t 0 2 T ,a function p t 0 ;t 1 ;a : T t 0 ;t 1 A X iscalledamotionif p t 0 ;t 0 ;a = a Here T issomeunderstoodnotionoftimeand t 0 ;a aretheinitialdata.This denitionreallysaysthatamotionisafunctionthatistheidentityon A attime t 0 Denition2.3. Let beasetofmotions,thenthetuple T;X;A; iscalleda dynamicalsystem.Further,if T = N S f 0 g then T;X;A; iscalledadiscrete dynamicalsystem. Thescopeofthischapterislimitedtodiscretedynamicalsystemsoftheform N S f 0 g ;X;A; ,where X iseither R or SU theset= F ;k;x = f k x forsomefunction f c : X X whichdependsonaparameter c and f k isthe k th iterateof f Anexampleshouldclearanymisgivingsthereadermayhavewiththepreceding denition: Example2.1. Considerthiscommonnancialsituation:Suppose $ X aredeposited intoasavingsaccountwhichearns10%annualinterest.Supposethemoneyisleftin theaccountfor n years,duringwhichtime,nomoneyiswithdrawn. Let B n denotethebalanceafter n years.Thenitiseasytosee B 0 = X and B 1 = : 1 B 0 =1 : 1 X .Tocalculate B 2 tenpercentinterestmustbeaddedto B 1 i.e B 2 = : 1 B 1 = : 1 2 X .Continuingthisiterativeprocess,itisclearthat B n = : 1 n X 5 PAGE 12 Intheexampletheparameteristheinterestrate.Whatevertheratemaybe,the overallbehaviorofthesystemissimple:as n !1 B n !1 .Ingeneral,however, thedynamicscanbefarmorecomplicatedandhighlyunpredictable.Infacteven simplereal-valuedfunctionslike f x = x 2 + c haveverycomplicateddynamics. Asthelastexampleclearlydemonstrates,studyingdynamicsinvolves iteration theactofrepeatingaprocess,whichinthiscaseistherepeatedapplicationofa function.Onewayinwhichmathematiciansstudydynamicalsystemsistostudy whereaninitialpointgoesafteracertainnumberofiterations,givenaparameter value. Denition2.4. Let f beafunctioncontinuousatapoint x 0 .Theorbitof x 0 under f isthesequence f x i g 1 i =0 ofpoints x 0 x 1 = f x 0 x 2 = f 2 x 1 = f f x 0 ,...,where f j x = f f ::: f x | {z } j )]TJ/F41 7.9701 Tf 6.586 0 Td [(times isthej-thiterationof x under f .Additionally, x 0 iscalled theseedoftheorbit. Remark2.1. Itisworthemphasizingthat f j x 6 = f x j ingeneral.Thereader mustbecarefulnottoconfusetheseverydierentcalculations. Notimmediatelyapparentisthefactthattherearemanydierentkindsoforbits. Somemaybehaveinpredictableways,whileothersdonot.Someorbitsdrifttoward innity,whileotherstendtowardapoint. Denition2.5. Apoint x iscalledaxedpointof f if f x = x .Additionally, x is saidtobeeventuallyxedafterjiterationsif f j x = f i x forall i>j where j> 1 Clearlytheorbitofaxedpointisthesimplestkindoforbitif f x = x ,then theorbitis x 0 = x;x 1 = f x = x;x 2 = f x 1 = x;:::;x n = f x n )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 = x;::: However,xedandeventuallyxedorbitsarealsothemostimportantkindoforbits anddemonstrateplaceswherethesystemismoststable.Itwillsoonbecomeclear thatxedandeventuallyxedorbitsare,ingeneral,quiterare. 6 PAGE 13 Thereisacriterionfordeterminingweatherornotafunctionhasxedpoints,but itcannotbeapplieduniversally.Furthermore,itisnotalwayseasytogureouthow manyxedpointsthereareorwheretondthem.Thus,itusuallymoreconvenient tocalculatexedpointsdirectlyseeExample2. Theorem2.1. Let [ a;b ] R andlet f :[ a;b ] [ a;b ] becontinuous.Then 9 y 2 R suchthat f y = y i.e. f hasaxedpointin [ a;b ] Proof. Considerthefunction g x = f x )]TJ/F19 11.9552 Tf 12.829 0 Td [(x ,whichisclearlycontinuous.Since f a 2 [ a;b ],itmustbetruethat f a a g a = f a )]TJ/F19 11.9552 Tf 10.726 0 Td [(a 0.Similarly f b b andthus g b = f b )]TJ/F19 11.9552 Tf 10.9 0 Td [(b 0.SobytheIntermediateValueTheorem 9 y 2 [ a;b ]such that g y = f y )]TJ/F19 11.9552 Tf 11.955 0 Td [(y =0 f y = y ,asrequired. Corollary2.1. Let [ a;b ] R andlet f :[ a;b ] R becontinuous.Supposethat [ a;b ] f [ a;b ] Then 9 y 2 [ a;b ] suchthat f y = y i.e. f hasaxedpointin [ a;b ] Thereisaproofofthecorollarygivenin[4],butitisverysimilartoproofofthe previoustheorem. Fixedandeventuallyxedorbitsaretypesoforbitswhichareconsideredstable. Thereareorbitswhichhavesomelevelofstability,butarenotxed. Denition2.6. Wesayapoint x isn-periodicif f n x = x .Thesmallestsuchn iscalledtheprimeperiodof x .Wesay x iseventuallyperiodicifthereisan m such that f i x isperiodicfor i m Periodicandeventuallyperiodicorbitsarealsoquitesimpleanddonotdemonstratethepotentialcomplexitythatmanyorbitscanhave.Thiswillbecomeclear whenthedynamicsofthequadraticfamilyarefullyinvestigated.Fornow,thereader shoulddirecthisorherattentiontothefollowingexample. Example2.2. Let f x = x 2 )]TJ/F18 11.9552 Tf 11.929 0 Td [(1 and x 0 = p 2 .Thentheorbitof x 0 is p 2 ; 1 ; 0 ; )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 ; 0 ; )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 ; 0 ; )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 ;::: .Thustheorbitof x 0 iseventuallyprimeperiodtwo. 7 PAGE 14 Itisalsopossibletondxedpointsof f .If x isaxedpointthen x 2 )]TJ/F18 11.9552 Tf 11.605 0 Td [(1= x x 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(x )]TJ/F18 11.9552 Tf 11.955 0 Td [(1=0 x = 1 p 5 2 Remark2.2. Fromhere,theawkwardtermprimeperiod n willbedropped.When thephraseperiod n "isused,itwillmeanprimeperiod n ",unlessotherwisestated. 2.1.2MoreonFixedandPeriodicPoints Therearedierentclassesofxedandperiodicpoints: Denition2.7. Let x 0 beaxedpointof f andsuppose 9 > 0 suchthatforall x satisfying j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j < ,theorbitof x convergesto f x 0 = x 0 .Then x 0 iscalledan attractingxedpoint. Denition2.8. Let x 0 beaxedpointof f andsuppose 9 > 0 suchthatforall x satisfying j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j < ,thereexists k> 0 suchthat j f n x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j < forall n PAGE 15 andas n growslarge, a n getssmallsince j a j < 1 .Thustheorbitof y under f tends towardzero.Byasimilarargument,theorbitof y under g tendsawayfromzero. Thepreviousexamplesuggeststhatperhapsslopeataxedpointdetermines whetherornotitattractsorrepelsnearbyorbits.Infactconsiderthefollowing proposition. Proposition2.1. Let f be C 1 and x 0 beaxedpointof f .If j f 0 x 0 j < 1 then x 0 isanattractingxedpoint.If j f 0 x 0 j > 1 then x 0 isarepellingxedpoint.If j f 0 x 0 j =1 thennoconclusioncanbemade. Proof. Thisisbasedonaprooffoundin[2]. Suppose j f 0 x 0 j < 1.Since f is C 1 ,theremustbeaneighborhoodof x 0 suchthat j f 0 x j < 1tooforall x inthatneighborhood.Moreprecisely,thereisa > 0such thatforall x satisfying j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j < j f 0 x j << 1. Thusanyaverageslopeonthisintervalmustalsobelessthan ,i.e. f x )]TJ/F41 7.9701 Tf 6.587 0 Td [(f x 0 x )]TJ/F41 7.9701 Tf 6.587 0 Td [(x 0 < thisisreallyjusttheMeanValueTheorem.Thismeans j f x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j = j f x )]TJ/F19 11.9552 Tf 11.955 0 Td [(f x 0 j j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j < j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j < .Bythesameargument, j f 2 x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j = j f 2 x )]TJ/F19 11.9552 Tf 11.955 0 Td [(f 2 x 0 j j f x )]TJ/F19 11.9552 Tf 11.955 0 Td [(f x 0 j .Butthismeansthat j f 2 x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j 2 j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j whichimplies j f n x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j n j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j byinduction.Therefore, f f n x g 1 n =0 convergesto x 0 and x 0 isanattractingxedpoint. Nowsuppose j f 0 x 0 j > 1.Byasimilarargument, 9 > 0suchthatforall x satisfying j x )]TJ/F19 11.9552 Tf 11.956 0 Td [(x 0 j < j f 0 x j >> 1.ThusbyMVT f x )]TJ/F41 7.9701 Tf 6.587 0 Td [(f x 0 x )]TJ/F41 7.9701 Tf 6.586 0 Td [(x 0 ,whichimplies that j f x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j = j f x )]TJ/F19 11.9552 Tf 11.955 0 Td [(f x 0 j j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j > j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j .Applyingthisinductively asucientnumberoftimes,itisclearthat f k x )]TJ/F19 11.9552 Tf 11.956 0 Td [(x 0 > k j x )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 0 j > forsome k .Thus f k x = 2 [ x 0 )]TJ/F19 11.9552 Tf 12.35 0 Td [(;x 0 + ].Thusthereisanintervalsuchthat f k x escapes after k iterations,so x 0 mustberepelling. Toprovethelastpartoftheproposition,consider g x = x 3 + x ,whichhasa xedpointat x =0.Infact g 0 =1butlet y =1 =m forsomeinteger m .Then j g y j > j y j > 0so0isrepelling. 9 PAGE 16 Nowconsider h x = x )]TJ/F19 11.9552 Tf 11.598 0 Td [(x 2 ,whichalsohasaxedpointat x =0.But h 0 =1 and0isneutralitattractsorbitsonthepositivesideof0andrepelsorbitsonthe negativesideof0 Thus,noconclusioncanbemadeif j f 0 x 0 j =1. Therearesimilarclassicationsforperiodicpoints. Denition2.10. Let x 0 beaperiodicpointofperiod n for f .Supposethat x 0 isan attractingrepellingorneutralxedpointfor f n ,then x 0 isanattractingrepelling orneutralperiodicpoint. Thereisalsoananalogoustheoremwhichgivesacriterionfordeterminingthe stabilityofaperiodicpoint. Proposition2.2. Let x 0 beaperiodicpointofperiod n for f andlet f n becontinuouslydierentiable.If j f n x 0 j < 1 then x isanattractingperiodicpoint.If j f n x 0 j > 1 then x isarepellingperiodicpoint.If j f n x 0 j =1 thennoconclusion canbemade. Theproofisomittedasitisthesameastheproofoftheanalogoustheorem involvingxedpoints. Remark2.3. Itisworthnotingthattheprevioustheoremsareusedinsometexts todeneattractingandrepellingxedperiodicpoints.Suchisthecaseinmore elementaryworkslike[1]. 2.1.3Bifurcations Bifurcationliterallymeansasplittingintotwo.Inthecontextofdynamicalsystems,thismeansasmallchangeintheparametervalueofadynamicalsystemcauses asuddenchangeintheoverallbehavior.Inthecaseofone-parameterdynamicalsystems,abifurcationoccurswhenthereisachangeinthenumberortypeofperiodic andxedpoints. 10 PAGE 17 Whiletherearemanykindsofbifurcationsfoundnaturallyinfamiliardynamical systems,thisthesiswillfocusontwo,thesaddle-nodeandtheperioddoubling,as theyapplytothequadraticfamily,whosedynamicswillbestudiedingreaterdepth inthenextsection.Assuch,denitionsasfoundin[1]willbepresentedinthis sectionandexampleswillfollowinthenextsection. Denition2.11. Let f x; beafunctionwithdomain D where isaparameter. Ifthereisan I D ,axedparametervalue 0 ,and > 0 suchthat: 1.Forall satisfying 0 << 0 + f x; hasnoxedpointsin I 2.If = 0 f x; hasonexedpoint x 0 in I ,and j f 0 x 0 ; j =1 3.Forall satisfying 0 )]TJ/F19 11.9552 Tf 12.309 0 Td [(<< 0 f x; hastwoxedpointsin I ;oneis attractingandtheotherisrepelling. Then f hasasaddle-nodebifurcationattheparametervalue 0 [1]. Itshouldbenotedthattheabovedenitioncanbealteredtodescribeasaddlenodebifurcationintheotherdirection.Thisisaccomplishedbyswitchingthedirectionsofinequalities,andmovingtheepsilontotheothersideoftheinequalitywith theoppositesign. Thesaddle-nodebifurcationissimple.AnexamplecanbefoundinProposition 2.5.Theperiod-doublingbifurcationisfarmorecomplicated. Denition2.12. Let f x; beafunctionwithdomain D where isaparameter. Then f hasaperiod-doublingbifurcationattheparametervalue 0 if 9 I D and > 0 suchthat[1]: 1.Foreach satisfying j )]TJ/F19 11.9552 Tf 11.955 0 Td [( 0 j < ,thereisauniquexedpoint x 0 2 I for f x; 2.For 0 )]TJ/F19 11.9552 Tf 11.954 0 Td [(< 0 f x; hasnoperiodtwocyclesin I and x 0 isattracting orrepellingrespectively. 11 PAGE 18 3.Forall satisfying 0 << 0 + f x; hasauniquetwo-cycle y 0 ;y 1 ;y 0 ;y 1 ;::: whichisattractingorrepellingrespectively.Also, x 0 isarepellingxedpointorattractingrespectively. 4.As 0 y i x 0 0 Justasinthecaseofthesaddle-nodebifurcation,thedirectionoftheperioddoublingbifurcationmaybereversed.Anexampleofaperiod-doublingbifurcation isgiveninProposition2.9 2.1.4Chaos Thenotionofchaoscameabouttodescribedynamicalsystemswhosebehavior cannotbepredictedwhentherearesmallchangesininitialconditions.Itshouldbe notedthatthereisnogenerallyaccepteddenitionforachaoticdynamicalsystem. Infact,therearemanydenitionswhichhavebeenacceptedbyvarioussegmentsof thescienticcommunity,andrejectedbyothers.Forthepurposeofthisthesis,the denitionin[1]willbeused. Beforethedenitioncanbegiven,thereadermustrstbecomefamiliarwith terms: Denition2.13. Let A;d beametricspaceand B A .Then B issaidtobe densein A if 8 > 0 and 8 a 2 A 9 b 2 B suchthat d a;b < Denition2.14. Let D = N S f 0 g ;X;A;F ,where F ;k;x = f k x forsome function f : A X beadynamicalsystem.If 8 x;y in A and > 0 9 z 2 A with k x )]TJ/F19 11.9552 Tf 11.956 0 Td [(z k < andan N 2 N S f 0 g suchthat f k y )]TJ/F19 11.9552 Tf 11.955 0 Td [(f k z < whenever k>N then D iscalledatransitivedynamicalsystem. Nowthedenitionofchaosasdenedin[1]canbepresentedwithoutfurther discussion: 12 PAGE 19 Denition2.15. Adynamicalsystem D issaidtobechaoticifthefollowinghold: 1.Theperiodicpointsof D aredense. 2. D istransitive. 3.Thereissome > 0 suchthat 8 x andany > 0 9 y and k suchthat if j x )]TJ/F19 11.9552 Tf 11.956 0 Td [(y j < ,then f k x )]TJ/F19 11.9552 Tf 11.955 0 Td [(f k y > or D dependssensitivelyoninitial conditions 2.2TheQuadraticFamily Nowthatmanyofthebasicconceptsofdiscretedynamicalsystemshavebeen established,theywillbeusedtoexamineaseeminglysimpledynamicalsystem,the QuadraticFamily .Recallthatthisdynamicalsystemisthefamilyofreal-valued functionsoftheform f x;c = x 2 + c ,where c isarealparameter.Manyofthe observationsmadeherearealsoobservedin[1].Itwillsoonbecomeclearthatas thevalueof c changes,thebehavioroftheorbitswillchange,drasticallysoinmany instances. Considertherathersimpleinstanceofwhen c =0. Example2.4. Let f x = x 2 .Clearly, x =0 and x =1 aretheonlyxedpoints. Since f 0 =0 < 1 0 mustbeattracting.Thismayalsobeobservedbytaking x 0 =1 =m forsomeinteger m f n x 0 =1 =m n andas n !1 1 =m n 0 .Similarly, x =1 isrepelling. 2.2.1When c> 1 = 4 andWhen c =1 = 4 Thedynamicsofthequadraticfamilyaresimplestwhen c> 1 = 4.Theyaresummarizedinthefollowingtwopropositions. 13 PAGE 20 Proposition2.3. Let f x = x 2 + c ,where c> 1 = 4 .Thenthereisno x 2 R such that f x = x i.e f hasnoxedpoints Proof. Assumeforacontradictionthat 9 x 2 R suchthat f x = x .Then x 2 )]TJ/F19 11.9552 Tf 10.415 0 Td [(x + c = 0 x = 1 p 1 )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 c 2 .But4 c> 1andthus p 1 )]TJ/F18 11.9552 Tf 11.955 0 Td [(4 c isnotreal x= 2 R ,whichcontradicts thehypothesis.Therefore,thereisno x 2 R suchthat f x = x ,asrequired. Proposition2.4. Let f x = x 2 + c ,where c> 1 = 4 .Then 8 x 2 R f n x !1 as n !1 Proof. First,itmustbeshownthat f x >x .Consider,again, g x = f x )]TJ/F19 11.9552 Tf 11.995 0 Td [(x .By thelastproposition g x 6 =0 8 x .Since g iscontinuous,thenpreciselyoneofthe followingistrue: 1. g x > 0 8 x 2. g x < 0 8 x But g 0 x =2 x )]TJ/F18 11.9552 Tf 13.019 0 Td [(1,sobytherstderivativetest, f attainsitsminimumat x 0 =1 = 2.But g = 2=1 = 4 )]TJ/F18 11.9552 Tf 12.301 0 Td [(1 = 2+ c andsince c> 1 = 4, g = 2 > 0whichmeans that f x >x 8 x 2 R Nowitissucienttoshowthat f f n x g 1 n =0 isunboundedforanychoiceof x Supposeforacontradictionthat 9 x 0 suchthat j f n x 0 j PAGE 21 When c =1 = 4thereisachangeinbehaviorformanyorbits[1]: Proposition2.5. Thevalue c =1 = 4 isasaddle-nodebifurcationfor f c x = x 2 + c Proof. Itmustbeshownthat 1.Thereexistsan suchthatforany c satisfying1 = 4 PAGE 22 consequencesofthisfactwillbeobservedingreaterdetaillater.Fornowobservethe followingfactsabout x + and x )]TJ/F18 11.9552 Tf 7.085 -4.338 Td [(. Proposition2.7. Let f x = x 2 + c where )]TJ/F18 11.9552 Tf 9.299 0 Td [(3 = 4 c< 1 = 4 andlet x + and x )]TJ/F51 11.9552 Tf 11.844 -4.338 Td [(be denedasabove.Thenthefollowingaretrue: 1. x + isrepelling. 2. x )]TJ/F51 11.9552 Tf 11.269 -4.338 Td [(isattractingfor )]TJ/F18 11.9552 Tf 9.298 0 Td [(3 = 4 PAGE 23 solutionsforxedpoints.Solet y + = 1+ p )]TJ/F39 7.9701 Tf 6.586 0 Td [(3 )]TJ/F39 7.9701 Tf 6.587 0 Td [(4 c 2 and y )]TJ/F18 11.9552 Tf 10.641 -4.339 Td [(= 1 )]TJ 6.587 6.183 Td [(p )]TJ/F39 7.9701 Tf 6.586 0 Td [(3 )]TJ/F39 7.9701 Tf 6.587 0 Td [(4 c 2 bethepotential periodtwopointsfor f Nowsuppose )]TJ/F18 11.9552 Tf 9.299 0 Td [(3 = 4 PAGE 24 interval.ByProposition2.7, x )]TJ/F18 11.9552 Tf 11.149 -4.338 Td [(isattractingfor c suchthat )]TJ/F18 11.9552 Tf 9.298 0 Td [(3 = 4 c< 1 = 4.Thus set x 0 c = x )]TJ/F18 11.9552 Tf 11.571 -4.338 Td [(andandthesecondpartofaresatised.ByProposition.8, thesecondpartofissatised. Let y )]TJ/F18 11.9552 Tf 11.688 -4.339 Td [(bethesameperiodtwopointfoundinthelastproof.Let y 0 c = y )]TJ/F18 11.9552 Tf 11.687 -4.339 Td [(and y 1 c = f y )]TJ/F18 11.9552 Tf 7.084 -4.338 Td [(.ThenbyProposition2.8,therstpartsofandaresatised. Lastlyitmustbeshownthatitmustbeshownthat y 0 c y 1 c x 0 )]TJ/F39 7.9701 Tf 6.586 0 Td [(3 = 4 = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 = 2as c !)]TJ/F18 11.9552 Tf 26.518 0 Td [(3 = 4.Butitisclearthatsince y 0 c = 1 )]TJ 6.586 6.182 Td [(p )]TJ/F39 7.9701 Tf 6.587 0 Td [(3 )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 c 2 ,lim c !)]TJ/F39 7.9701 Tf 15.055 0 Td [(3 = 4 1 )]TJ 6.586 6.183 Td [(p )]TJ/F39 7.9701 Tf 6.587 0 Td [(3 )]TJ/F39 7.9701 Tf 6.587 0 Td [(4 c 2 = )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 = 2. Thuslim c !)]TJ/F39 7.9701 Tf 15.055 0 Td [(3 = 4 y 1 c = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 = 2 2 )]TJ/F18 11.9552 Tf 12.088 0 Td [(3 = 4= )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 = 2.Thusissatisedand c = )]TJ/F18 11.9552 Tf 9.298 0 Td [(3 = 4isa period-doublingbifurcation. Thisindicatesthatthedynamicshavebecomemoreintricate.However,theseare tameintricaciesandthedynamicswillonlygetmorecomplicated. 2.2.3When c )]TJ/F18 14.3462 Tf 26.301 0 Td [(2 Therearegreatchangesinthebehaviorofthequadraticfamilywhen c = )]TJ/F18 11.9552 Tf 9.298 0 Td [(2.In thiscase, f x = x 2 )]TJ/F18 11.9552 Tf 11.632 0 Td [(2hasarepellingxedpoint x + =2.Attentionshouldbepaid totheinterval[ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2],forsomethinginterestingisgoingonhere. Proposition2.10. Let f x = x 2 )]TJ/F18 11.9552 Tf 11.511 0 Td [(2 .Thentheset )]TJ/F41 7.9701 Tf 7.314 -1.794 Td [(n = f x 2 [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2] j f n x = x g is non-empty. Proof. Considerthecasewhere n =1: f x isclearlycontinuouson[ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]and f [ )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 ; 2]=[ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2] 9 y 2 [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]suchthat f y = y ByTheorem1.Thus )]TJ/F39 7.9701 Tf 7.314 -1.793 Td [(1 6 = ; When n =2, f 2 [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]= f f [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]= f [ )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 ; 2]=[ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2] f 2 x hasaxed pointin[ )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 ; 2])]TJ/F39 7.9701 Tf 20.321 -1.793 Td [(2 6 = ; Proceedingwithaninductivehypothesis,assumethat f k [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]=[ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]i.e. )]TJ/F41 7.9701 Tf 7.314 -1.793 Td [(k isnon-empty. 18 PAGE 25 But f k +1 [ )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 ; 2]= f f k [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]= f [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]=[ )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 ; 2].Thus)]TJ/F41 7.9701 Tf 54.528 -1.793 Td [(k +1 6 = ; Byinduction,)]TJ/F41 7.9701 Tf 80.962 -1.793 Td [(n 6 = ;8 n Corollary2.2. Let f x = x 2 )]TJ/F18 11.9552 Tf 12.301 0 Td [(2 .Thentheset )]TJ/F41 7.9701 Tf 7.315 -1.794 Td [(n = f x 2 [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2] j f n x = x g has 2 n elements. Averyclevergeometricproofforthiscorollarycanbefoundin[1]. Theprecedingpropositionandcorollaryshowthatbetween c = )]TJ/F18 11.9552 Tf 9.299 0 Td [(5 = 4and c = )]TJ/F18 11.9552 Tf 9.298 0 Td [(2, f x gainsaninnitenumberofperiodicpoints,aprofound,perhapsunexpected change. Inthecaseof c< )]TJ/F18 11.9552 Tf 9.298 0 Td [(2,noticethat x + = 1+ p 1 )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 c 2 > )]TJ/F39 7.9701 Tf 6.586 0 Td [(1+ p )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 c 2 = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 = 2+ p )]TJ/F19 11.9552 Tf 9.299 0 Td [(c> )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 = 2 )]TJ 12.371 9.89 Td [(p 2 > )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 >c .Letting I =[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(x + ;x + ],noticethat f = c isnotin I .In fact,thereexists > 0suchthatforall x 2 )]TJ/F19 11.9552 Tf 9.299 0 Td [(; f x isnotin I .Thustheorbit ofany x 2 )]TJ/F19 11.9552 Tf 9.299 0 Td [(; tendstoward 1 Let= f x 2 I j f n x 2 I 8 n 2 N g .Clearly )]TJ/F19 11.9552 Tf 9.298 0 Td [(; ,soitisnotatrivial set.Infactitcanbeshownthatisactuallythe CantorMiddle-ThirdsSet [1]. Moreimportantly,isasetonwhichthequadraticfamilyischaotic. Theorem2.2. Let f c x = x 2 + c and asdenedabove.Suppose c< )]TJ/F18 11.9552 Tf 10.494 6.694 Td [( 5+ p 2 4 Then f c ischaoticon Theproofcanbefoundin[1].However,thisresultissomewhatunsatisfying sincemaybesmall".But,ifthecasewhere c = )]TJ/F18 11.9552 Tf 9.299 0 Td [(2isanalyzed,itisclearthatit ispossiblefor f c tobechaoticonalargerset. Proposition2.11. Let f )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 x = x 2 )]TJ/F18 11.9552 Tf 11.955 0 Td [(2 ischaoticon [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2] Proof. Itcaneasilybeveriedthat f )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 istransitive,anddependssensitivelyoninitial conditions.Corollary2.2suggeststhattheperiodicpointsmaybedensein[ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]. Toprovethis,let x 1 ;x 2 2 [ )]TJ/F18 11.9552 Tf 9.298 0 Td [(2 ; 2]andwithoutlossofgenerality,assumethat x 1 PAGE 26 f n )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 [ x 1 ;x 2 ] [ x 1 ;x 2 ].Therefore 9 x 2 [ x 1 ;x 2 ]suchthat f n )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 x = x ,whichmeans x isaperiodicpointfor f )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 .Nowchoosing x 1 ;x 2 suchthat j x 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(x 2 j < forany > 0, itisclearthat x isarbitrarilycloseto x 1 and x 2 .Thustheperiodicpointsaredense. Ifthereisonlyoneideatounderstand,itisthatthedynamicsforthequadratic familyvaryquiteabitdependingonthevalueof c .Nowthatthedynamicsof thequadraticfamilyhavebeendescribed,attentionmustnowbepaidtotheother preliminaries. 2.3TheComplexQuadraticFamily Recallthattheprimarygoalofthisthesisistodiscoverwhetherornotthedynamicalpropertiesof TheQuadraticFamily haveanalogueswhenthedynamicalsystem isdenedonaspaceotherthan R .Fortunately,thedynamicsoffunctionsofthe form g c z = z 2 + c; .1 where z c 2 C havealsobeenstudied.Thus,thedynamicsofthisfamilywillprovide informationcriticaltoachievingthisgoal.Thoughthedynamicsarerichformany valuesof c ,onlythevalue c =0willbeanalyzedsinceitgivesagoodexampleofthe kindofdynamicalbehaviormostpertinenttothisthesis. 2.3.1When c =0 Thedynamics g 0 z = z 2 ,alsoknownasthesquaringfunction,aresimple[1].In fact,theorbitofanypoint C isveryeasytowrite.Let z 0 2 C betheseed.Since z 0 = re i ,where r = j z 0 j and =arg z 0 ,thenitisclear g n 0 z 0 = r 2 n e i 2 n Thus,theultimatebehaviorofanyorbitdependsonthevalueof r .If r> 1, 20 PAGE 27 Figure2.1:TheFilledJuliaSetfor g 0 is S 1 then r 2 n !1 as n !1 ,whichmeans j g n 0 j!1 as n !1 .However,if r< 1, g n 0 z 0 0as n !1 .If r =1,then g n 0 z 0 2 S 1 forall n .Thus g 0 z isboundedi j z j 1.Theregioncontainingsuchpointshasaspecialname. Denition2.16. The lledJuliaset ofadynamicalsystem,denoted J isthesetof seedswhoseorbitsarebounded.Theboundaryof J iscalledthe Juliaset [1]. Thus,theJuliasetfor g 0 is S 1 seeFigure2 : 1. Additionally,theresomethingspecialaboutthepoint0. Proposition2.12. Let g 0 z = z 2 .Then 0 isanattractingxedpoint. Proof. Clearly g 0 =0.Nowlet =1and z = re i 2 C .Thenif j z )]TJ/F18 11.9552 Tf 11.955 0 Td [(0 j = j z j = r< ,itmustbethecasethat g n 0 z 0as n !1 Nowthecasewhere g 0 z = z 2 isrestrictedto S 1 mustbeinvestigated.Inthis case,thereareperiodicpointsandtheyareeasytond. Proposition2.13. Let g 0 z = z 2 .Thenthepoint z =cos + i sin 2 S 1 ,where = 2 k 2 n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 forsomeinteger k ,isaperiodicpointofprimeperiod n for g 0 21 PAGE 28 Proof. Thisproofcanbefoundin[1]. Firstitmustbeshownthat z isaxedpointfor g n 0 z .But g n 0 z =cos n + i sin n .Since2 n =2 n 2 k 2 n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 = 2 k 2 n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 mod2 ,thencos n =cos and cos n =cos .Thus g n 0 z = z Additionally,forall1 ` PAGE 29 Proof. Therstpartoftheproofoftransitivityshowsthattheiteratedimageofany arceventuallycoversthecircle[1].Thusif e i 1 e i 2 areclose,itispossiblefor g 0 to map e i 1 e i 2 topointsclosetopointsonadiameterofthecircle. Corollary2.3. Let g 0 bedenedasabove.Then g 0 ischaoticon S 1 Proof. ByPropositions2.14,2.15,and2.16, g 0 ischaoticon S 1 Theprecedingcompletelydescribesthedynamicsof g 0 = z 2 .Itisclearthat thedynamicsareverysimpleinsideandoutsidetheunitcircleandthattheonly interestingdynamicsoccuron S 1 2.4Propertiesof SU 2.4.1AlgebraicPropertiesof SU Thereadershouldbefamiliarwiththenotionofagroup[3]. Denition2.17. Agroupisapair G; where G isasetand abinaryoperation satisfyingthefollowing: 1.Whenever g 1 ;g 2 2 G g 1 g 2 2 G closure 2.Forall g 1 ;g 2 ;g 3 2 G g 1 g 2 g 3 = g 1 g 2 g 3 associativity 3. 9 e 2 G suchthat g e = e g = g 8 g 2 G existenceofanidentity 4. 8 g 2 G 9 h 2 G ,suchthat g h = h g = e existenceofinverses Ofnotableimportanceisthenotionofclosureofthesetundertheoperation. Lateritwillbecomeclearthatclosureisneededinordertoconstructatrueanalogue ofthequadraticmap. Example2.5. Let GL ; C bethesetof 2 2 invertiblematricesisagroupwith respecttomatrixmultiplication. 23 PAGE 30 Perhapsforeigntothereader,isthenotionofaunitarymatrix[5]. Denition2.18. Let A be n n matrixwithcomplexentries. A issaidtobeunitary if A A = AA = I where A istheconjugatetransposeof A TheSpecialUnitaryGroup SU isthegroupwithrespecttomatrixmultiplicationof2 2unitarymatriceswithcomplexentrieswhichhavedeterminant1.In fact, SU isasubgroupof GL ; C Asitturnsout,matricesin SU canbeexpressedinaverysimplemanner. Theorem2.3. Let A 2 SU .Then A = 2 6 4 a )]TJETq1 0 0 1 366.073 522.767 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 366.073 512.791 Td [(b b a 3 7 5 where a a + b b =1 foraunique pair a;b 2 C 2 Proof. Let A = 2 6 4 ac bd 3 7 5 ,where a;b;c;d 2 C But A isunitary AA = I ,soitmustbethecasethat A = A )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 .Since det A =1, A )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 = 2 6 4 d )]TJ/F19 11.9552 Tf 9.299 0 Td [(c )]TJ/F19 11.9552 Tf 9.299 0 Td [(ba 3 7 5 .Thus, a = d and )]TJETq1 0 0 1 394.065 361.968 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 394.065 351.992 Td [(b = c and A = 2 6 4 a )]TJETq1 0 0 1 170.879 320.921 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 170.879 310.945 Td [(b b a 3 7 5 Additionally1=det A = a a + b b Toseeuniqueness,suppose A = 2 6 4 a )]TJETq1 0 0 1 330.9 255.034 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 330.9 245.058 Td [(b b a 3 7 5 and A = 2 6 4 c )]TJETq1 0 0 1 438.336 255.034 cm[]0 d 0 J 0.478 w 0 0 m 6.083 0 l SQBT/F19 11.9552 Tf 438.336 245.058 Td [(d d c 3 7 5 suchthat a 6 = c and b 6 = d .Butthismeansthat 2 6 4 a )]TJETq1 0 0 1 313.274 203.826 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 313.274 193.85 Td [(b b a 3 7 5 6 = 2 6 4 c )]TJETq1 0 0 1 385.728 203.826 cm[]0 d 0 J 0.478 w 0 0 m 6.083 0 l SQBT/F19 11.9552 Tf 385.728 193.85 Td [(d d c 3 7 5 .Thus A 6 = A ,whichisa contradictionanduniquenessfollows. Thelastpropositionimpliesthatthereisaone-to-onecorrespondencebetween SU andasubsetof C 2 .Itisawellknowfactthat C 2 isequivalentgeometrically 24 PAGE 31 speakingto R 4 .Thus,thereisalsoaone-to-onecorrespondencesbetween SU andasubsetof R 4 Thereismoreevidencetosuggestsuchacorrespondence.Toseethis,consider theQuaterniongroupoforder8, Q 8 = f 1 ; i; j; k g .Itisagroupwithrespect tomultiplicationaccordingtothetablebelow: 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1 Itiseasytoseethatthisgroupisgeneratedbytheelements i j ,and k Thisideacanbeextendedtocreateanalgebraicstructurewhichisbasedonlinear combinationsofelementsof Q 8 .Dene Q 8 [ R ]tobetheset f x + y i + z j + w j j x;y;z;w 2 R g .Giventhat i 2 = j 2 = k 2 = )]TJ/F18 11.9552 Tf 9.298 0 Td [(1itseems reasonabletosee Q 8 [ R ]asanextensionof C .Thisis,infact,trueandwillgivethe meanstomakethegeometricconnectionsbetween SU and S 3 assuggestedabove. First,notethat C isinaone-to-onecorrespondencewith R 2 .Thisiseasytosee ifanarbitraryelement z 2 C ,where z = a + bi .Therealpartof z correspondsto the x -coordinate,whiletheimaginarypartof z correspondstothe y -coordinate,i.e. a + bi = a;b .Continuingalongtheselines,itiseasytoseethat = a + bi + cj + dk = a;b;c;d .Thus, Q 8 [ R ] = R 4 Nowconsiderasubsetof Q 8 [ R ],theunitquaternions: Q 8 [ R ]= f x + y i + z j + w j j x;y;z;w 2 R ;x 2 + y 2 + z 2 + w 2 =1 g .Thenitis clearthat Q 8 [ R ] = f x;y;z;w 2 R 4 j x 2 + y 2 + z 2 + w 2 =1 g ,i.e. Q 8 [ R ] = S 3 Nowitmustbeshownthat Q 8 [ R ] = SU .Toseethis,considerthissetof matrices: 25 PAGE 32 = 8 > < > : I = 2 6 4 10 01 3 7 5 ; I = 2 6 4 i 0 0 i 3 7 5 ; J = 2 6 4 0 )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 10 3 7 5 ; K = 2 6 4 0 i i 0 3 7 5 9 > = > ; Itiseasilyveriedthat IJ = )]TJ/F33 11.9552 Tf 9.298 0 Td [(JI = K ,andthussatisesthequaternionstructure describedabove.Thenconsiderthefollowingfunction : Q 8 where = f I; I ; J ; K g ,denedby 1= I i = I j = J k = K Whatisimportantisthat respectsthestructureof Q 8 and inthefollowing sense: Denition2.19. Let G H begroupsand : G H abijection.Itissaidthat isanisomorphismifforall x;y 2 G xy = x y Asaconsequenceofthegroupisomorphism between Q 8 and ,thereisa groupisomorphismbetween Q 8 [ R ]and [ R ]= f xI + y I + z J + w K j x 2 + y 2 + z 2 + w 2 =1 g .Thisgivestheappropriate correspondencebetween S 3 and SU Thereismoretothisrelationship.Infact,thereisanunderlyingrelationship whichmakesthetopologyof SU equivalenttothatof S 3 2.4.2TopologicalPropertiesof SU Asatopologicalspace, SU isequivalent"totheunitspherein R 4 S 3 .This notionmustbemademoreprecise.Thissectionwillaccomplishthisgoal. Itwasshownthatanymatrixin SU canbewrittenasvectorin S 3 i.e.if A = 2 6 4 a )]TJETq1 0 0 1 170.879 193.438 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 170.879 183.462 Td [(b b a 3 7 5 where a = x 1 + ix 2 and b = x 3 + ix 4 then x 1 ;x 2 ;x 3 ;x 4 isthevector representation. Establishingthiscorrespondenceservestoallowaneasytransitionbetweenthe dierentrepresentationsof SU .Inthissense,thematrixrepresentationisusefulfor 26 PAGE 33 computationsasmatrixmultiplicationandadditionarewellunderstoodconcepts, whilethevectorrepresentationisusefultounderstandthegeometricimplicationsof theseoperations. Firstsomedenitionsareneeded. Denition2.20. Let G; beagroupwhichisalsoatopologicalspace.Suppose thatthefunctions G G G : x;y 7! x y and G G : x 7! x )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 arecontinuous. Then G iscalledatopologicalgroup. Recallthat SU isasubgroupof GL ; C .Itiswell-knowthat GL ; C isan opensettopologicallyequivalentto C 4 .Thus,thisinducesatopologyon SU Example2.6. Thegroup SU isatopologicalgroup.Thisisbecausematrixmultiplicationisreallyjustavectorofpolynomialsintheentries,andpolynomialsare alwayscontinuous.Takinginversesisalsocontinuoussinceitisaprocesswhichinvolvesdeterminantsandrearrangingentries,whichisatworstarationalfunction. Sincerationalfunctionsarecontinuouseverywheretheyaredened,itfollowsthat suchfunctionsarecontinuouson SU Denition2.21. Let X and Y betopologicalspacesand f : X Y afunction satisfying: 1. f isabijection. 2. f iscontinuous. 3. f )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 iscontinuous. Then f iscalledahomeomorphismand X and Y arecalledhomeomorphic. Example2.7. Let g : SU S 3 denedby 2 6 4 x 1 + ix 2 )]TJ/F19 11.9552 Tf 9.299 0 Td [(x 3 + ix 4 x 3 + ix 4 x 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(ix 2 3 7 5 7!h x 1 ;x 2 ;x 3 ;x 4 i 27 PAGE 34 Then g clearlysatisestheconditionsofahomeomorphism. Giventheestablishedcorrespondencebetween SU and S 3 andusingtheestablishednotionsofsequencesandconvergenceon R 4 ,itisnaturaltotalkabout sequencesandconvergenceon SU .Infact, SU isclosedinthefollowingsense: Proposition2.17. Let f X n g beanysequencein SU suchthat X n X forsome X 2 M 2 C .Then X 2 SU Proof. Thisistheprooffoundin[5]. Firstobservethatthelimit X mustbeunitary.Since X n isin SU 8 n then X n X n = I 8 n .Thus lim n !1 X n X n = I .But X n X and X n X as n !1 XX = I ,whichmeans X isunitary. But X n = 2 6 4 a n )]TJETq1 0 0 1 223.898 440.369 cm[]0 d 0 J 0.478 w 0 0 m 10.613 0 l SQBT/F19 11.9552 Tf 223.898 430.393 Td [(b n b n a n 3 7 5 and X = 2 6 4 a )]TJETq1 0 0 1 338.885 440.369 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 338.885 430.393 Td [(b b a 3 7 5 .Thus, a n a a n a b n b and b n b .But1= a n a n + b n b n 8 n .Therefore,1= lim n !1 a n a n + b n b n = a a + b b whichmeans X 2 SU ,asrequired. Nowitisclearthehomeomorphismspreservetopologicalproperties.Thus,the familiartopologicalnotionsof R 4 and S 3 forthatmattermaybeappliedto SU 2.4.3OtherPropertiesof SU Thissubsectionisdevotedtoexploringsomeothercharacteristicsof SU ,which, althoughnotentirelyrelevant,areveryinteresting. Observethattheeigenvaluesofanygiven S 2 SU areconjugates[5]. Proposition2.18. Let S 2 SU ,then S hasnon-zeroeigenvalues 1 and 2 such that 2 = 1 .Additionally, 1 2 =1 28 PAGE 35 Proof. Recall S = 2 6 4 a )]TJETq1 0 0 1 249.808 712.042 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 249.808 702.066 Td [(b b a 3 7 5 .Tocalculatetheeigenvalues,constructthecharacteristicpolynomialandnditsroots. p S t =det tI )]TJ/F19 11.9552 Tf 13.158 0 Td [(S = t )]TJ/F19 11.9552 Tf 11.955 0 Td [(a )]TJETq1 0 0 1 517.451 660.834 cm[]0 d 0 J 0.478 w 0 0 m 4.977 0 l SQBT/F19 11.9552 Tf 517.451 650.858 Td [(b bt )]TJETq1 0 0 1 521.637 633.771 cm[]0 d 0 J 0.478 w 0 0 m 6.145 0 l SQBT/F19 11.9552 Tf 521.637 626.95 Td [(a Thismeans p S t = t 2 )]TJ/F18 11.9552 Tf 12.513 0 Td [( a + a t + a a + b b = t 2 )]TJ/F18 11.9552 Tf 12.513 0 Td [( a + a t +1,whichhasroots 1 ; 2 = a + a p a + a 2 )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 2 ,whichareconjugates.Toseethesecondassertionconsider a + a + p a + a 2 )]TJ/F39 7.9701 Tf 6.587 0 Td [(4 2 a + a )]TJ/F24 11.9552 Tf 6.586 8.027 Td [(p a + a 2 )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 2 whichisclearlyequalto a + a 2 )]TJ/F39 7.9701 Tf 6.586 0 Td [( a + a 2 )]TJ/F39 7.9701 Tf 6.586 0 Td [(4 4 =1,as required. Anotherinterestingpropertyof SU isthatofunitarydiagonalizability. Theorem2.4. Let S 2 SU witheigenvalues .Then 9 U 2 SU suchthat U SU = 2 6 4 0 0 3 7 5 Proof. Let ~u = 2 6 4 x 1 x 2 3 7 5 beaneigenvectorof S for ,where x 1 ;x 2 2 C .Theniteasily veriedthat ~v = 2 6 4 )]TJETq1 0 0 1 245.595 294.36 cm[]0 d 0 J 0.478 w 0 0 m 11.384 0 l SQBT/F19 11.9552 Tf 245.595 287.539 Td [(x 2 x 1 3 7 5 isaneigenvectorof S for Set U =[ ~u~v ],then U 2 SU .Asimplecalculationshows U SU = 2 6 4 0 0 3 7 5 ,asrequired. 29 PAGE 36 Chapter3 TheAnalogueandaNumerical Approach 3.1ConstructingtheAnalogue Recall,themaingoalofthisthesisistoconstructananalogueof f x = x 2 + c for SU andtoanalyzeitsdynamics.Toreachthisgoal,itisnecessarydistillthe importantcomponentsofthequadraticfunctionandndanalogouscomponentsto useon SU Itiseasytoseethatthequadraticfunctioninvolvessquaringrealnumbers.Since SU isgroupwithrespecttomatrixmultiplicationitseemslogicalfortheanalogue tosquareanelementof SU inthenaturalway.Byclosureofthegroup,squaresof elementswillalsobeinthegroup.Thequadraticfunctionalsofeaturesanaddition byaconstant.Ananalogousoperationforthisnewcontextwouldbetoaddby aconstant2 2unitarymatrixwithcomplexentries.Sofartheanaloguelooks somethinglike G X = X 2 + C where X 2 SU and C 2 R 4 .Howeverthereis sometroublesinceforall C 6 =0, X 2 + C= 2 SU .Thegoodnewsisthatthereisa solutionwhichutilizesthegeometricpropertiesof SU 30 PAGE 37 Ifthevectorrepresentationsof SU and U ; C areused,then G X = X 2 + C isalsoavectorin R 4 whichcanbenormalizedif G X 6 =0,i.e.thezeromatrix. Then G X k G X k isaunitvector,whichmeansitisanelementof S 3 ,whichmeansitisan elementof SU .Thus, F C X = X 2 + C k X 2 + C k ispreciselytheanaloguethatisneeded. However,if G X =0,noamountofmanipulationwillmakeitaunitvector.Thus, anysuchmatrix C whichgives G X =0forany X 2 SU mustnotbeused.But if C= 2 SU ,then C 6 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(X 2 forany X 2 SU andthus G X =0. Remark3.1. Sinceanydiagonalmatrixin SU X = 2 6 4 a 0 0 a 3 7 5 canbeidentied withanelementof S 1 ,thentheanalogueisequivalenttothecomplexquadraticfamily g c = z 2 + c restrictedtoareal-valuedparameteronalldiagonalmatricesin SU Thusif C =0 ,thentheanaloguemustbechaoticon S 1 3.2NumericalApproach Nowthattheanalogueisconstructed,thenextstepistotestthebehaviorof many,manydierentorbitsovermanydierentparametervalues.Thisisaproblem bestsolvedbycomputers.Therearemanydierentmathematicalsoftwarepackages, manyofwhichcouldbeusedforthispurpose.However,sinceMATLABwasdesigned formatrixcalculationsandisaverysimpleprogramminglanguage,itisidealforthis situation. Thecodeusedtosolvethisproblemcanbefoundintheappendix,andconsists ofthreeparts.Therstpartchoosesseedsfrom SU ,iteratesthemthousandsof timeseachandfordierentvectors C tosimplifythecalculations,allthe C -vectors willbenon-unitvectorsalongthe x -axis.Thus C = h c; 0 ; 0 ; 0 i where c 6 =1.The dataiscollectedinarrayswhichcontainthevaluesofthecomponentsoftheseeds andtheiriterates,aswellasthevaluesof c 31 PAGE 38 Moreprecisely,aunitvectorin R 4 ,say h x;y;z;w i ischosen.Forthepurposeof producingamanageablegraphicalrepresentationofthedata, R 4 willbethoughtof as C 2 C 2 .Thus h x;y;z;w i = h e i ;e i' i p 2 .Inconstructingthisseedvector and will bechosenfrom[0 ; 2 ,insuchawaythatthesquare[0 ; 2 [0 ; 2 isthroughly covered.Thenthisseedisiteratedon F twothousandtimeseachforvarying c values between )]TJ/F18 11.9552 Tf 9.299 0 Td [(2and2withstepsize : 04thevalues c = 1areomitted. Thesecondpartofthecodegoesthroughthedataandforeachseedand c -value triestondperiodicbehaviorwithintheorbit.Toaccomplishthis,ittakesthelast valuecalculatedintheorbitandcheckstoseeifanyothervaluesareequivalenttoit moduloasmallerrorof10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(6 .Ifsuchbehaviorisfoundtheprocessstopsandthe period"isdenotedbythenumberofstepsneededtoarriveatsuchapoint.Ifno suchbehaviorisfound,period"issetequalto0. Thelastpartgeneratesgraphicsforanalyzingthedata.Ittakesinthreepieces ofdata: 1.theseedvector. 2.thevalueof c 3.theperiod"calculatedinthesecondpartofthecode. ThenusingapreexistingmoduleintheMATLABsoftwarecalled imagesc ,whichisa scaledataanddisplayimageobject,asortofheatmapiscreatedwherebyasquare ofuniformsizeisplacedaroundeachofthepoint ;' chosenforeachoftheseeds andiscoloreddependingontheperiod". Thisgraphicalrepresentationofthedatawillreveal potential placeswherethe dynamicsareinterestingforparticularvaluesof c .Theword potential isuseddeliberatelyasthismethod,likeallnumericalmethods,aresubjecttonumericalerror.So theseresultswillnotbeexact,butthemethodbeingusedhereisthemostecient meanstotestthehypothesis. 32 PAGE 39 Chapter4 Results Therearetwokindsofresultswhichwillbepresentedinthefollowingsections: numericandanalytic.Thenumericresultsdescribedcomedirectlyfromananalysis ofthedatacollectedviatheMATLABcalculations.Thereadermustbecautioned nottotaketheseresultsasfact,sinceoverthecourseofthousandsandthousands ofcalculations,roundingerrorcanskewthedata.Assuch,theanalyticresultsare presentedtoverifysomeofthenumericalresults. 4.1NumericalResults 4.1.1InitialObservations Athoroughreviewofthenumericaldatayieldssomeinterestingobservations. Observation1. Thereseemstobeacriticalintervalofactivity.Morepreciselymost oftheperiodicbehavioroccursfor c 2 )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 : 9 ; 1 andoutsidethatintervaltheperiodic behaviorisrareorevennon-existent. Observation2. For j c j largemostoralloftheorbitstestedareeventuallynumericallyxed. 33 PAGE 40 a b c Figure4.1:LargeValuesof j c j Remark4.1. Someexplanationofthepicturesisrequired.Thehorizontalaxisgives thevalueof andtheverticalaxisgivesthevalueof ,where h e i ;e i i' i p 2 2 SU isthe seed.Thecolorscaleonthesidegivesthecolorcorrespondingtotheperiodcalculated intheMATLABprogram. Cautionmustbeplacedwithcallingsuchorbitsxedherexedismeantinthe literalsenseasdenedinapreviouschapterand numericallyxed ismeantinthe 34 PAGE 41 sensethatthecomputingprogramhasdenedit.Withinthecomputingprogramis somecodewhichaccountsfornumericalerrorinthecalculations.Thus,ifanorbit isgettingclosetoapointandyetneverattainsit,theprogramwillcalliteventually xed.Therearepoints,however,whichareindeedxedananalyticprooffollowsin thenextsection. Observation3. Thereareeventuallynumericallyperiodicpointsofmanydierent periods. Again,duecaremustbeexercisedbeforelabelingthesepointsperiodicasrounding maymakeorbitsappearperiodic.Thereareorbitswhicharetrulyperiodican analyticprooffollowsinthenextsection. Observation4. Thereisalargeregionofeventuallynumericallyperiod 3 points whichgrowsas c increasesfrom c = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 : 9 until c = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 : 7 andcontractsas c increases to c = )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 : 3 atwhichpointtheregioncompletelyvanishes. ThisphenomenonisdisplayedinFigures4.3and4.4.Inordertomaketheregion visible,thegraphicsdisplaythenaturallogoftheperiodonthecolorspectrumNote: log 1 : 099.Beadvisedthatthecolorcorrespondingtologisamediumshade ofblue. Observation5. Forsomevaluesof c thedynamicsseemtobechaotic.Thegures seemtosupportthisobservation. Toprovethat F ischaoticonallof SU isnotcurrentlypossibleasleast accordingtothedenitionofchaosgivenin[1],sincenotenoughisknowaboutthe periodicpointstodetermineweatherornottheyaredensein SU .Itis,however, possibletoprove F ischaoticon S 1 SU SeeProposition4.5. 35 PAGE 42 a b c d Figure4.2:ManyDierentPeriods 4.2AnalyticResults Havingreviewedthenumericalresults,itcannotbeconcludedimmediatelythat alloftheseresultsarelegitimate.Infact,numericalerrorcancertainlyskewthe results.Thus,thesepotentialresultsshouldbeprovenanalytically. Therstanalyticresulttoproveisthatforlargevaluesof c allorbitstendtoward C 36 PAGE 43 a b c Figure4.3:ThePeriodThreeRegion Proposition4.1. Let F A = A 2 + C k A 2 + C k ,where C = c I and let F i A = F F F :: F A :: | {z } i )]TJ/F41 7.9701 Tf 6.586 0 Td [(times bethei-thiterate.Thenfor k C k sucientlylarge F i X C k C k = h 1 ; 0 ; 0 ; 0 i as i !18 X 2 SU Proof. Let X 0 beanelementof SU andlet X 1 = F X 0 .Itissucienttoshow that X 1 iscloserto C than X 0 .Proceedwithdirectcomputations: 37 PAGE 44 a b c d Figure4.4:TheVanishingPeriodThreeRegion 38 PAGE 45 Since X 0 2 SU X = 2 6 4 a 0 )]TJETq1 0 0 1 293.881 712.042 cm[]0 d 0 J 0.478 w 0 0 m 9.709 0 l SQBT/F19 11.9552 Tf 293.881 702.066 Td [(b 0 b 0 a 0 3 7 5 where a 0 = x 0 1 + ix 0 2 = re i b 0 = x 0 3 + ix 0 4 = se i' =arg a 0 ,and =arg b 0 r = j a 0 j ,and s = j b 0 j Itiseasytoseethat X 0 2 = 2 6 4 r 2 e i 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F19 11.9552 Tf 9.299 0 Td [(rs e )]TJ/F41 7.9701 Tf 6.587 0 Td [(i + + e i )]TJ/F41 7.9701 Tf 6.587 0 Td [(' rs e i + + e i )]TJ/F41 7.9701 Tf 6.586 0 Td [( r 2 e )]TJ/F41 7.9701 Tf 6.587 0 Td [(i 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(s 2 3 7 5 .1 Thenset Y = 2 6 4 r 2 e i 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(s 2 + c )]TJ/F19 11.9552 Tf 9.298 0 Td [(rs e )]TJ/F41 7.9701 Tf 6.587 0 Td [(i + + e i )]TJ/F41 7.9701 Tf 6.586 0 Td [(' rs e i + + e i )]TJ/F41 7.9701 Tf 6.586 0 Td [( r 2 e )]TJ/F41 7.9701 Tf 6.586 0 Td [(i 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(s 2 + c 3 7 5 : .2 Thuswehavethat X 1 = F X 0 = Y k Y k .Usingthevectorrepresentation,itisclear that X 1 = h r 2 cos )]TJ/F41 7.9701 Tf 6.587 0 Td [(s 2 + c;r 2 sin ;rs cos + +cos )]TJ/F41 7.9701 Tf 6.586 0 Td [( ;rs sin + +sin )]TJ/F41 7.9701 Tf 6.587 0 Td [( i p )]TJ/F39 7.9701 Tf 6.587 0 Td [(2 r 2 s 2 cos +2 r 2 c cos + s 4 )]TJ/F39 7.9701 Tf 10.488 0 Td [(2 s 2 c + c 2 + r 4 +4 rs cos 2 Toshowthat X 1 iscloserto C than X 0 ,comparetheangles.Let 1 and 1 denote thetwoanglesbetween C and X 1 .Soset ~a 1 = h r 2 0 cos 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(s 2 0 + c;r 2 0 sin 0 i p )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 r 2 0 s 2 0 cos 0 +2 r 2 0 c cos 0 + s 4 0 )]TJ/F18 11.9552 Tf 11.956 0 Td [(2 s 2 0 c + c 2 + r 4 0 +4 r 0 s 0 cos 2 0 .3 Thencos 1 = ~a 1 ~e 1 k ~a 1 k ,where ~e 1 = h 1 ; 0 i .Aftersomesimplealgebra, cos 1 = r 2 0 cos 0 )]TJ/F41 7.9701 Tf 6.587 0 Td [(s 2 0 + c s [ )]TJ/F40 5.9776 Tf 5.756 0 Td [(2 r 2 0 s 0 +2 r 2 0 c cos 0 + s 2 0 )]TJ/F42 5.9776 Tf 5.756 0 Td [(c 2 + r 4 0 +4 r 0 s 0 cos 2 0 ] [ )]TJ/F40 5.9776 Tf 5.756 0 Td [(2 r 2 0 s 2 0 +2 r 2 0 c cos 0 + s 2 0 )]TJ/F42 5.9776 Tf 5.756 0 Td [(c 2 ] 1+ r 2 0 c )]TJ/F40 5.9776 Tf 5.756 0 Td [(4 r 2 0 s 2 0 +4 r 0 s 0 cos 2 0 39 PAGE 46 Itiseasytoseethatthedenominatorwillhaveagreaterpowerof c thanthatof thenumerator.Thenfor c sucientlylarge,cos 1 cos 0 Byasimilarcalculation,cos 1 cos 0 .Therefore X 1 iscloserto C than X 0 for c sucientlylarge. Thenextthingtoproveistheexistenceofxedpoints.Butbeforearigorous proofispresented,thereadermaywishtoseeaconcreteinstanceofaxedpoint. Example4.1. Let C = 2 6 4 )]TJ/F18 11.9552 Tf 9.299 0 Td [(20 0 )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 3 7 5 .Then F X hasxedpointat h)]TJ/F18 11.9552 Tf 13.948 0 Td [(1 ; 0 ; 0 ; 0 i Toseethisfactproceedwiththecalculations.Let X = 2 6 4 )]TJ/F18 11.9552 Tf 9.298 0 Td [(10 0 )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 3 7 5 .Then X 2 = 2 6 4 10 01 3 7 5 ,whichmeansthat F X = X = 2 6 4 )]TJ/F18 11.9552 Tf 9.298 0 Td [(10 0 )]TJ/F18 11.9552 Tf 9.298 0 Td [(1 3 7 5 andthusitisxed. Infactforanyvectoroftheform ~a = h x;y; 0 ; 0 i in SU thereis c whichxes ~a Proposition4.2. Let F A = A 2 + C k A 2 + C k .Thenthereexistsadiagonalmatrix ~ X in SU suchthatforsome C F ~ X = ~ X Proof. Thisproof,likethelast,usesthegeometryof SU : Let X beadiagonalmatrixin SU .Then X hasavectorform h x ;y ; 0 ; 0 i2 S 1 .Itiseasytoseethattheactioninducedbytheanaloguedoublesthesizeofthe angle, ,between h x ;y i andthe x -axis.Sothenitmustbeshownthat 9 c such thattheadditionof h c; 0 i nulliesthechangeinthesizeof Let h x;y i beunitvectorwhoseanglewiththe x -axisis =2 .Thus F X = h x + c;y i .Usingsimilarrighttrianglelaws,itisclearthatthefollowingmusthold: y y = x + c x .4 40 PAGE 47 Thenitisclearthat y x = y x + c .5 tan 2 = y x + c .6 Usingtheadditiveangleformulasforthetangentfunction i.e.tan =tan )]TJ/F41 7.9701 Tf 6.699 -4.977 Td [( 2 + 2 = 2 tan 2 1 )]TJ/F18 11.9552 Tf 6.586 -0.996 Td [( tan 2 2 .But tan = y x .7 y x = 2 )]TJ/F41 7.9701 Tf 11.95 -4.427 Td [(y x + c 1 )]TJ/F27 11.9552 Tf 11.955 9.684 Td [()]TJ/F41 7.9701 Tf 11.95 -4.428 Td [(y x + c 2 .8 Aftersomealgebra,theequationbecomes: y 2 1 x + c 2 )]TJ/F18 11.9552 Tf 11.955 0 Td [(2 x 1 x + c )]TJ/F18 11.9552 Tf 11.955 0 Td [(1=0.9 whichisquadraticin )]TJ/F39 7.9701 Tf 12.068 -4.977 Td [(1 x + c .Thususingthequadraticformula, 1 x + c = 2 x p 4 x 2 +4 y 2 2 y 2 .10 c = y 2 x p y 2 + x 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(x .11 whichisalwaysreal.Soif h x ;y i arechosensothat c 6 = 1,then F xes h x ;y i asrequired. Thereisaninterestingcorollary: Corollary4.1. Let ~a = h 0 ; 0 ;z;w i beanelementof SU .Then ~a iseventually xedby F foranyvalueof c 6 = 1 Proof. Thematrixrepresentationof ~a is A = 2 6 4 0 )]TJ/F19 11.9552 Tf 9.298 0 Td [(z + iw z + iw 0 3 7 5 41 PAGE 48 Then A 2 = 2 6 4 )]TJ/F18 11.9552 Tf 9.298 0 Td [( z + iw z )]TJ/F19 11.9552 Tf 11.955 0 Td [(iw 0 0 )]TJ/F18 11.9552 Tf 9.299 0 Td [( z + iw z )]TJ/F19 11.9552 Tf 11.955 0 Td [(iw 3 7 5 = 2 6 4 )]TJ/F18 11.9552 Tf 9.299 0 Td [(10 0 )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 3 7 5 But h)]TJ/F39 7.9701 Tf 9.88 0 Td [(1+ c; 0 ; 0 ; 0 i kh)]TJ/F39 7.9701 Tf 14.114 0 Td [(1+ c; 0 ; 0 ; 0 ik = h 1 ; 0 ; 0 ; 0 i ,dependingonthevalueof c ,bothofwhichare eventuallyxedforany c Thus, ~a iseventuallyxed,asrequired. Nowamoregeneraltheoremmustbeproven,butrstalemma. Lemma1. TheContractionMappingPrinciple. Let M;d beacomplete metricspace.Let T : M M beacontractionmappingon M ,meaningthereexists r< 1 suchthat d T x ;T y PAGE 49 Noticethatsince T isacontractionmapping,itsatisesaLipschitzconditionand isthuscontinuous.Thenitmustbethecasethat T lim n !1 T n x =lim n !1 T n +1 x Thus T x =lim n !1 T n +1 x = x ,i.e. x isaxedpoint. Toseethat x isunique,supposethereisanotherxedpoint, y .Thensince T isacontractionmapping, d x ;y = d T x ;T y r d x ;y ,whichisa contradictionsince r< 1. ThisLemmaindicatesthatiftheanalogueisacontractionmappingonanysubspaceof SU ,thenitmusthaveattractingxedpoints.First,amoregeneral versionofattractingxedpointswillbedened. Denition4.1. Let M;d beametricspace.Let G : M M withaxedpoint x 0 2 M .Suppose 9 > 0 anda k 2 N suchthatforall x satisfying d x;x 0 < d G n x ;x 0 < forall n>k .Then x 0 iscalledanattractingxedpoint Nowtheexistenceofattractingxedpointsmaybeproven: Proposition4.3. Let F A C = A 2 + C k A 2 + C k C 2 R 4 .Thenthereexists ~ X in SU such thatforsome C 2 R 4 F C ~ X = ~ X i.e.thisdynamicalsystemhasxedpoints. Additionally, ~ X isanattractingxedpoint. Proof. Let B X beaballofradius about X andlet M = B X SU ,for some X 2 SU and > 0.Suppose F isrestricted M .Thusitwillbesucientto showthat 9 C suchthat F isacontractionmappingon M Let Y;Z 2 M Y 6 = Z a = k Y 2 + C k and b = k Z 2 + C k .Consider k F Y )]TJ/F19 11.9552 Tf 11.955 0 Td [(F Z k = Y 2 + C a )]TJ/F41 7.9701 Tf 13.15 4.708 Td [(Z 2 + C b Aftersomealgebra,itisclearthat k F Y )]TJ/F19 11.9552 Tf 11.955 0 Td [(F Z k = bY 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(aZ 2 + C b )]TJ/F19 11.9552 Tf 11.955 0 Td [(a ab .14 43 PAGE 50 But j b )]TJ/F19 11.9552 Tf 11.955 0 Td [(a j = jk Z 2 + C k)-222(k Y 2 + C kjjk Z 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(Y 2 kj bythetriangleinequality. Additionally, jk Z 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(Y 2 kj = k Z 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(Y 2 k = k Z )]TJ/F19 11.9552 Tf 11.956 0 Td [(Y kk Z + Y k Also, bY 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(aZ 2 = Y )]TJ/F19 11.9552 Tf 11.955 0 Td [(Z bY + aZ + b )]TJ/F41 7.9701 Tf 6.586 0 Td [(a YZ Y )]TJ/F41 7.9701 Tf 6.586 0 Td [(Z Thus, k F Y )]TJ/F19 11.9552 Tf 11.955 0 Td [(F Z k k Y )]TJ/F41 7.9701 Tf 6.587 0 Td [(Z k ab h bY + aZ + b )]TJ/F41 7.9701 Tf 6.586 0 Td [(a YZ Y )]TJ/F41 7.9701 Tf 6.587 0 Td [(Z +2 k C k i Soset r = 1 ab h bY + aZ + b )]TJ/F41 7.9701 Tf 6.587 0 Td [(a YZ Y )]TJ/F41 7.9701 Tf 6.586 0 Td [(Z +2 k C k i .Nowitmustonlybeshownthat r PAGE 51 producingamessy"systemofequationsthatisnoteasilysolvedwithalgebraic methods.Theanalytictoolusedforthexedpointcase, TheContractionMapping Theorem canbeusedtoshowthat F n hasxedpoints.Butanyxedpointfor F isalsoaxedpointfor F n .Thuscontractionmappingmaynotproduceperiodic points. However,itispossibleeveneasytondperiodicpointswhen C =0.Recall thatsince F 0 isequivalenttothedoublingmapon S 1 ,thefollowingtwopropositions aretrue: Proposition4.4. Let F C A = A 2 + C k A 2 + C k andlet C =0 .Thenthepoint X = h cos ; sin ; 0 ; 0 i2 S 1 SU ,where = 2 k 2 n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 forsomeinteger k ,isa periodicpointofprimeperiod n for F 0 Proposition4.5. Let F 0 bedenedasabove.Then F 0 ischaoticon S 1 Thenextstepistoprovetheexistenceofperiodicpointsinabitmoregenerality. Infact,thereareperiodicpointsfor F C ,where C = c I ,in S 1 SU when c 6 =0. Toprovethis,alemmaisneeded: Lemma2. Let h : 2 ; 3 4 [0 ; ] beacontinuousandontofunction,where < 4 .Then 9 x 2 2 ; 3 4 suchthat h x = x Proof. ByCorollary2.1theresultfollows. Proposition4.6. Let < 1 3 andlet C = ; 0 anddene F : S 1 S 1 by F A = A 2 + C k A 2 + C k then 9 X 2 S 1 suchthat 2 < arg X < 3 4 and X isaperiod2 point. Proof. Firstconsiderthecasewhen C = )]TJ/F19 11.9552 Tf 9.299 0 Td [(; 0 Let g : 2 ; 3 4 S 1 bedenedby g = h sin ; cos i .Clearly g iscontinuous. Nowlet F bedenedasabove.Then F g : 2 ; 3 4 S 1 isalsocontinuous. Additionally, F 2 g 2 = ; 0and F 2 g 3 4 =cos )]TJ/F19 11.9552 Tf 11.955 0 Td [( ; sin )]TJ/F19 11.9552 Tf 11.955 0 Td [( ,where << 1 3 < 4 45 PAGE 52 Considertheprincipalvalueargumentfunctionarg: S 1 [0 ; 2 ,whichiscontinuousandonto.Then h =arg F 2 g : 2 ; 3 4 A [0 ; 2 iscontinuousand onto.But h 2 =arg ; 0=0andsimilarly h 3 4 = )]TJ/F19 11.9552 Tf 12.17 0 Td [( ,where < 4 .Thus A =[0 ; )]TJ/F19 11.9552 Tf 11.956 0 Td [( ]andbythepreviouslemma, 9 2 2 ; 3 4 suchthat h = .Thus, X = h cos ; sin i isaxedpointfor F 2 .Nowitmustbeshownthat X isnot axedpointfor F Butarg F g 2 ; 3 4 = )]TJ/F19 11.9552 Tf 11.955 0 Td [(; 3 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( .Since )]TJ/F19 11.9552 Tf 11.955 0 Td [(; 3 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( [0 ; )]TJ/F19 11.9552 Tf 11.955 0 Td [( ]= ,it mustbethecasethat X = 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(; 3 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( andthus X cannotbeaxedpointfor F Therefore, X isaperiodtwopointfor F ,asrequired. Asimilarargumentworkswhen C =+ ; 0 Infact,thereisananalogouslemmaandpropositionforperiod n points: Lemma3. Let h n : h 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 ; n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 2 n i [0 ; ] n> 2 beacontinuousandonto function,where < 2 n .Then 9 x 2 h 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 ; n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 2 n i suchthat h x = x Proof. Again,Corollary2.1givestheresult. Proposition4.7. Let < 1 3 n andlet C = ; 0 anddene F : S 1 S 1 by F A = A 2 + C k A 2 + C k then 9 X 2 S 1 suchthat 2 n < arg X < 3 2 n +1 and X isaperiod n point. Proof. Firstconsiderthecasewhen C = )]TJ/F19 11.9552 Tf 9.299 0 Td [(; 0 Let g : 2 n ; 3 2 n +1 S 1 bedenedby g = h sin ; cos i .Clearly g iscontinuous. Nowlet F bedenedasabove.Then F g : 2 n ; 3 2 n +1 S 1 isalsocontinuous. Additionally, F n g 2 n = ; 0and F 2 g 3 2 n +1 =cos )]TJ/F19 11.9552 Tf 11.955 0 Td [( ; sin )]TJ/F19 11.9552 Tf 11.955 0 Td [( ,where << 1 3 n < 4 46 PAGE 53 Recallarg: S 1 [0 ; 2 iscontinuousandonto.Then h n =arg F n g : 2 n ; 3 2 n +1 A [0 ; 2 iscontinuousandonto.But h 2 n =arg ; 0=0and similarly h 3 2 n +1 = )]TJ/F19 11.9552 Tf 12.242 0 Td [( ,where < 4 .Thus A =[0 ; )]TJ/F19 11.9552 Tf 11.956 0 Td [( ]andbytheprevious lemma, 9 2 h 2 n ; 2 n +1 i suchthat h = .Thus, X = h cos ; sin i isaxed pointfor F 2 .Nowitmustbeshownthat X isnotaxedpointfor F N forany 0 N PAGE 54 Thelastpropositionalsosaysthattheperiodicpointsof F C ,where C = ; 0, where j j < 1 3 ,isdensein S 1 .Thuswehavethat F C mustbechaoticforsuchvalues of C 48 PAGE 55 Chapter5 Conclusions Thisthesissetouttoinvestigatethedynamicsofanalogueofthequadraticfamily i.e.thefamilyoffunctionson R whichareoftheform f x;c = x 2 + c where c isa parameterontheLiegroup SU .Morepreciselythismap F C : SU SU isdenedby F C X = X 2 + C k X 2 + C k ,where C isanon-unitvectorin R 4 Oneoftheprimarygoalsofthisthesiswastondsomeparallelsinthedynamics betweenthequadraticfamilyanditsanalogueon SU .Thereissomeevidence suggestingthatsuchparallelsexistforcertaincharacteristics.Howeverthereisalso evidencesuggestingthattheAnaloguehasuniquepropertieswhichdonotresemble anypropertiesofthethequadraticfamily.Tomakeanybroadconclusionswouldbe hastyatthispoint.Researchonthisanalogueisinitsinfancy,butseveralthingscan besaidonthesubject: 1.Thereisnumericalandanalyticevidencetosuggestthatforarelativelylarge c allorbitsundertheanaloguebecomearbitrarilycloseto h 1 ; 0 ; 0 ; 0 i afterenough iterations.Forthequadraticfamilya c largerthan 1 4 willforceallorbitstoward 1 2.Theprecedingalsoindicatestheexistenceofattractingxedpointsforthe analogue,atraitwhichitshareswiththequadraticfamily. 49 PAGE 56 3.Theanalogueisprobablyalsoachaoticdynamicalsystemonallof SU for k C k small. 5.1AccomplishmentsandMissteps Thisthesisaccomplishedseveralthings: 1.IntroducedananalogueofthequadraticfamilyonaLiegroup. 2.Producedsomenumericaldatawhichsuggestthatthedynamicsoftheanalogue arerichandworthinvestigating,includingtheperiodthreeregion. 3.Providedrigorousproofofsomeofthedynamicsoftheanalogueobservedin thenumericaldata. 4.Providedsomeinsightintoparallelsbetweenthequadraticfamilyandtheanalogue. 5.Provedthattheanalogueischaoticon S 1 when c I isinaneighborhoodof zero. Itisevidentthatmuchmoremustbedoneinordertofullyunderstandthedynamicsofthisanalogue.Directionsforsuchfurtherresearchareoutlinedinthe section. Perhapstheonlymisstepwasanoverrelianceonnumericalmethods,whichatthis pointisne.Aspreviouslystated,thesearetherstresultstobegatheredforthis dynamicalsystem.Thusitismoreimportanttopointoutpotentialcharacteristicsof theanalogueinordertodemonstratetheworthinessoffurtherresearch.Asresearch progresses,theseresultscanbeveriedandmorewillbelearnedabouttheanalogue. 50 PAGE 57 5.2DirectionsforFurtherStudy Asindicatedinthenumericaldata,theanaloguehasveryrichdynamics.While thisthesishasinvestigatedsomeofthebehavior,thereisfarmorelefttobeseen. Hereisalistwiththerstbeingthemostimportantofotherpotentialdirections researcherscangowiththisanalogue: 1.Gathermoredataandformoreseedvectors.Perhapsuseafastermachine withgreatercomputingcapabilitiestoreduceroundingerror.Thiswillhelp gathermoreinformationaboutthelocationofperiodicpoints,whichwillbe instrumentalinprovingthedensityornon-densityoftheperiodicpointsin SU .Ultimatelythiswilldetermineweatherornottheanalogueischaotic on SU 2.Usemoreexactmethodsforprovingexistenceofxedandperiodicpointsin thegeneralcases. 3.Searchforotherattractingxedpointsandtrytondrepellingxedpoints. 4.Gathernumericaldataforgeneral C 2 R 4 5.Investigatetheperiodthreeregioningreaterdetailtodetermineifittrulyexists andisnotarelicofnumericalerror. 6.Augmentthetheanaloguesothatitaddsthetranslationvectorforevennumberediteratesandsubtractsitforodd-numberediteratesthiswouldgive periodbehaviorforpointslike h 1 ; 0 ; 0 ; 0 i whichisalwaysxedintheoriginal analogue.Perhapsuseonevectorforeven-numberediteratesandadierent oneforodditerates. 51 PAGE 58 AppendixA MATLABCode BelowistheMATLABcodeusedtogeneratethenumericaldataforthisthesis. Therearecommentswithinthecodewhichexplainthefunctionsofthedierentparts. 1 clearall 2 clc 3 A= zeros 2,2; 4 NumRun=3000; 5 MaxPeriod= round NumRun 10/100; 6 Mat= zeros NumRun+1,2,2; 7 C0= )]TJ/F18 11.9552 Tf 9.832 0 Td [(2; 8 Cn=2; 9 NumFramesC=40; 10 M=40; 11 StepSize=1/M; 12 % 13 %t1,t2arethevariablesforparamaterizingtwocircles, whichwillbecome 52 PAGE 59 14 %theseeds.A1,A2andB1,B2representthereal,imaginary componentsof 15 %theentriesofamatrixinSU2.Adatafileisbeing createdtostore 16 %thenumericaldatacollected. 17 % 18 t1=[0:2/M:2 pi ]; 19 t2=[0:2/M:2 pi ]; 20 A1=1/ sqrt 2 cos t1; 21 A2=1/ sqrt 2 sin t1; 22 B1=1/ sqrt 2 cos t2; 23 B2=1/ sqrt 2 sin t2; 24 a1v='v'; 25 a2v='v'; 26 b1v='1'; 27 b2v='1'; 28 FileNameData=['D' num2str M'A1='a1v'A2='a2v'B1=' b1v'B2='b2v'C0=' num2str abs C0'Cn=' num2str abs Cn 'NfC=' num2str NumFramesC'NR=' num2str NumRun'.mat' ]; 29 fig1= figure 1; 30 winsize1:2=[00]; 31 winsize= get fig1,'Position'; 32 set gca ,'xlim',[ )]TJ/F18 11.9552 Tf 10.219 0 Td [(11],'ylim',[ )]TJ/F18 11.9552 Tf 10.219 0 Td [(11],'nextplot','replace',' Visible','off' 33 Stepp=Cn )]TJ/F18 11.9552 Tf 15.477 0 Td [(C0/NumFramesC; 34 for ik=1:NumFramesC 53 PAGE 60 35 ik 36 cc2=C0+ik Stepp; 37 CValuesik=cc2; 38 [Out]=Createcc2,M,A1,A2,B1,B2,MaxPeriod,NumRun,A ,ik; 39 OutTum:,:,ik=Out; 40 end 41 save FileNameData 1 function [Out]=Createcc2,M,A1,A2,B1,B2,MaxPeriod,NumRun,A,ik 2 % 3 %A1,A2,B1,B2arearrayswhichcreatetheseedvalueswe willstartwith. 4 %Cisourtranslationvectorwhichrangesfromto. 5 %Mischoseninthepreviousportionofthecodetobethe numberofseeds. 6 % 7 C=[cc20;0cc2]; 8 Toll=1e )]TJ/F18 11.9552 Tf 9.965 0 Td [(6; 9 Out= zeros M,M; 10 for i0=1:M 11 for j0=1:M 12 % 13 %Inthisportionofthecodewearescalingtheseed vectorso 14 %thatitisunitlengthtoensurethematrixisin SU2 54 PAGE 61 15 % 16 a10=A1i0; 17 a20=A2j0; 18 19 b10=B1j0; 20 b20=B2j0; 21 ScaleFac= sqrt a10^2+a20^2+b10^2+b20^2 ; 22 if ScaleFac~=0 23 a1=a10/ScaleFac; 24 a2=a20/ScaleFac; 25 b1=b10/ScaleFac; 26 b2=b20/ScaleFac; 27 I= sqrt )]TJ/F18 11.9552 Tf 9.938 0 Td [(1; 28 a=a1+I a2; 29 b=b1+I b2; 30 A=[a )]TJ/F17 11.9552 Tf 9.302 0 Td [(conj b;b conj a]; 31 Mat1,1:2,1:2=A; 32 % 33 %HereweplugintheseedMatriciesintothe functionand 34 %theyarethenscaledagaintounitlength 35 % 36 for i=1:NumRun 37 fcn=A A+C; 38 L= sqrt sum sum fcn:,1. conj fcn:,1; 55 PAGE 62 39 Fs=fcn/L; 40 A=Fs; 41 Mati+1,1:2,1:2=Fs'; 42 end 43 % 44 %Onceeachseedisiteratedthegivennumberof times,wetake 45 %thelastpointandseeifitisclosewithin1e )]TJ/F51 11.9552 Tf 9.552 0 Td [(6toanyof 46 %itspredescessorsitwillonlygobackasfar asMaxPeriod 47 %predecessors.Theseed,period,and correspondingseedvalue 48 %aresavedinthedatafile. 49 % 50 Period=0 MaxPeriod; 51 for j=1:MaxPeriod 52 Err=Mat end ,1,1 )]TJ/F18 11.9552 Tf 8.525 0 Td [(MatNumRun )]TJ/F18 11.9552 Tf 9.31 0 Td [(j,1,1. conj Mat end ,1,1 )]TJ/F18 11.9552 Tf 8.525 0 Td [(MatNumRun )]TJ/F18 11.9552 Tf 9.31 0 Td [(j,1,1; 53 if Err < =Toll 54 Period=j; 55 break 56 end 57 end 58 % 59 %Thisgeneratestheaxesforthegraphic 60 % 56 PAGE 63 61 Outi0,j0=Period; 62 end 63 end 64 end 1 2 clearall 3 clc 4 % 5 %Thislastportionofthecodeloadsthedatafileand generatesthe 6 %pictures.Itusesimagesc,whichisapredefinedmodulein theMATLAB 7 %library.Incommentbelowisthecodewhichcreatespictures withoutalog 8 %filterandimmediatelybelowisthecodewhichcreatesthe pictureswith 9 %saidfilter.Thefilterhelpsmakeitpossibletosee pointsoflower 10 %periodicity,whichmaybehidden. 11 % 12 load D40A1=vA2=vB1=1B2=1C0=2Cn=2NfC=40NR=30000 13 for ij=1:NumFramesC 14 ij 15 %imagescxaxisij,:,yaxisij,:,OutTum:,:,ij; 16 imagesc t11: end )]TJ/F18 11.9552 Tf 9.64 0 Td [(1,t21: end )]TJ/F18 11.9552 Tf 9.64 0 Td [(1, log OutTum:,:,ij; 17 colorbar 18 Counterr=['C=' num2str CValuesij]; 57 PAGE 64 19 title Counterr 20 set gca ,'FontSize',20 21 pause 22 end 58 PAGE 65 Bibliography [1]RobertL.Devaney. AFirstCourseinChaoticDynamicalSystems:Theoryand Experiment .Addison-WeselyPublishingCompany,1992. [2]RobertL.Devaney. AnIntroductiontoChaoticDynamicalSystems .Westview Press,secondedition,2003. [3]DavidS.DummitandRichardM.Foote. AbstractAlgebra .Wiley,thirdedition, 2003. [4]RichardA.Holmgren. AFirstCourseinDiscreteDynamicalSystems .Universitext,secondedition,1996. [5]RogerA.HornandCharlesR.Johnson. MatrixAnalysis .CambridgeUniversity Press.,1985. [6]AnthonyN.MichelandKainingWang. QualitativeTheoryofDynamicalSystems: TheRoleofStabilityPreservingMappings .MarcelDekker,Inc.,1995. [7]RobertS.Strichartz. TheWayofAnalysis .JonesandBarlettPublishers,revised edition,2000. 59 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||