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Dynamics of an Analogue of the Quadratic Family on Su (2)

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004246/00001

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Title: Dynamics of an Analogue of the Quadratic Family on Su (2)
Physical Description: Book
Language: English
Creator: Emanuello, John Anthony
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2010
Publication Date: 2010

Subjects

Subjects / Keywords: Quadratic Family
Dynamical Systems
Chaos
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: For this thesis an analogue of the well known Quadratic Family was constructed for S3, the unit sphere in R4, using the algebra of unit quaternions. Because the unit quaternions can be identified with the Lie group SU(2), the family provides a collection of dynamical systems on SU(2). These dynamics for SU(2) were analyzed with the purpose of finding analogues of the well-known phenomenon associated to the Quadratic Family. This was accomplished by computing the orbits of many different seed values for different parameter values and then creating a graphical representation of the data. The results indicate that the dynamics on SU(2) are rich and that there are some similarities between it and the quadratic family.
Statement of Responsibility: by John Anthony Emanuello
Thesis: Thesis (B.A.) -- New College of Florida, 2010
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: McDonald, Patrick; Yildirim, Necmettin

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Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2010 E5
System ID: NCFE004246:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004246/00001

Material Information

Title: Dynamics of an Analogue of the Quadratic Family on Su (2)
Physical Description: Book
Language: English
Creator: Emanuello, John Anthony
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2010
Publication Date: 2010

Subjects

Subjects / Keywords: Quadratic Family
Dynamical Systems
Chaos
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: For this thesis an analogue of the well known Quadratic Family was constructed for S3, the unit sphere in R4, using the algebra of unit quaternions. Because the unit quaternions can be identified with the Lie group SU(2), the family provides a collection of dynamical systems on SU(2). These dynamics for SU(2) were analyzed with the purpose of finding analogues of the well-known phenomenon associated to the Quadratic Family. This was accomplished by computing the orbits of many different seed values for different parameter values and then creating a graphical representation of the data. The results indicate that the dynamics on SU(2) are rich and that there are some similarities between it and the quadratic family.
Statement of Responsibility: by John Anthony Emanuello
Thesis: Thesis (B.A.) -- New College of Florida, 2010
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: McDonald, Patrick; Yildirim, Necmettin

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2010 E5
System ID: NCFE004246:00001


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DynamicsofanAnalogueofthe QuadraticFamilyon SU May16,2010

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DYNAMICSOFANANALOGUEOFTHEQUADRATICFAMILYON SU BY JOHNANTHONYEMANUELLO AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts UnderthesponsorshipofPatrickMcDonaldandNecmettinYildirim Sarasota,Florida May,2010

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Acknowledgments First,IwanttothankProfessorsPatrickMcDonaldandNecmettinYildirimfor theirsponsorshipofthisthesis,theirteaching,andtheirinsightintomathematics.I trulyappreciateallthelonghourstheyputintohelpingmepreparethisthesis. Also,ImustthankProfessorsEiriniPoimenidouandDavidMullinsforservingon myBaccalaureateCommitteeandprovidinginvaluablereccomendationsforimprovingmythesis. Iwouldalsoliketothankmymotherforraisingmeandsupportingmeforthese pasttwenty-twoyears. Lastly,Iwouldliketoexpressspecialappreciationtomygirlfriend,Kriszti.Her encouragementinspiredmetopursuemathematicsandherloveandsupporthave helpedtomakethisthesispossible.

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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

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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
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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

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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

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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

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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

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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

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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

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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

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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

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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 .Then x 0 iscalledarepellingxedpoint. Toputitsimply,attractingxedpointsaresonamedbecauseorbitswithseeds nearbyareattracted"towardsthemandinthecaseofrepellingxedpointsnearby orbitsarerepelled"away.However,therearexedpointswhichfailtobeeither attractingorrepelling. Denition2.9. Let x 0 beaxedpointof f .Itissaidthat x 0 isneutralif x 0 is neitherattractingnorrepelling. Inthecaseofneutralxedpoints,somenearbyorbitsmaybeattracted,while othersarerepelled.AnexampleofsuchaxedpointisgiveninProposition2.1. Example2.3. Let f x = ax and g x = bx ,where j a j < 1 and j b j > 1 .Clearlyboth havexedpointsat x 0 =0 .Itisnothardtoseethat 0 isattractingfor f butrepelling for g .Toseethisconsidertheorbitof y = 1 m forsome m 2 Z .Then f n y = a n m 8

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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

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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

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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

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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

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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

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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 0, 9 N suchthat f k x 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(f l x 0 < whenever k;l N .Theninparticular f N x 0 )]TJ/F19 11.9552 Tf 11.956 0 Td [(f N +1 x 0 < .But f N +1 x 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(f N x 0 = f f N x 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(f N x 0 Alsoafterdoingsomealgebra,itisclearthat: f N x 0 = x 2 0 + c 2 + c 2 ::: 2 + c> 2 n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 c x 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 0 + x 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(2 0 Thenitmustalsobetruethat f f N x 0 )]TJ/F19 11.9552 Tf 11.956 0 Td [(f N x 0 > n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 c x 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 0 + x 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(2 0 2 +1 = 4 )]TJ/F18 11.9552 Tf 11.955 0 Td [(2 n )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 c x 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 0 + x 2 n )]TJ/F40 5.9776 Tf 5.756 0 Td [(1 )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 0 > 1 = 4. Whichgivesacontradiction.Therefore f f n x g 1 n =0 doesnotconverge,soitmust bethecasethat f f n x g 1 n =0 isunbounded.Thus, f n x !1 as n !1 14

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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 1and j f 0 c x )]TJ/F18 11.9552 Tf 7.085 -4.339 Td [( j < 1,whichmeans x + is attractingand x )]TJ/F18 11.9552 Tf 10.986 -4.338 Td [(isrepelling.Thusholds. Proposition2.6. Let f x = x 2 +1 = 4 .Then f f n x 0 g! 1 = 2 if j x 0 j 1 = 2 and f f n x 0 g!1 otherwise. Theproofislefttothereadersinceitiseasy. Nowitshouldbesucientlyclearthatthedynamicsareverysimplealthough notsosimpletoprovewhen c 1 = 4.Thisisnotatallindicativeofthedynamics fordierentvaluesof c 2.2.2When )]TJ/F18 14.3462 Tf 11.159 0 Td [(2
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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 1. For x )]TJ/F18 11.9552 Tf 11.016 -4.339 Td [(theargumentisalsoeasy. f 0 x )]TJ/F18 11.9552 Tf 7.085 -4.339 Td [(=1 )]TJ 11.975 9.392 Td [(p 1 )]TJ/F18 11.9552 Tf 11.955 0 Td [(4 c .If )]TJ/F18 11.9552 Tf 9.298 0 Td [(3 = 4
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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 1and j f 0 x )]TJ/F18 11.9552 Tf 7.085 -4.338 Td [( j > 1,whichmeans x + and x )]TJ/F18 11.9552 Tf 10.987 -4.338 Td [(arerepellingxedpointsi.e.istrue. Itshouldbeapparentthatsomethingprofoundhappensat c =3 = 4[1]: Proposition2.9. Let f x = x 2 + c .Then, f undergoesaperiod-doublingbifurcation at c =3 = 4 Proof. Itmustbeshownthat 9 > 0and I R suchthat: 1.Foreach c satisfying j c +3 = 4 j < ,thereisauniquexedpoint x 0 c 2 I for f x;c 2.For )]TJ/F18 11.9552 Tf 9.298 0 Td [(3 = 4 )]TJ/F19 11.9552 Tf 12.924 0 Td [( c< )]TJ/F18 11.9552 Tf 9.299 0 Td [(3 = 4, f x;c hasnoperiodtwocyclesin I and x 0 c is attractingorrepellingrespectively. 3.Forall c satisfying )]TJ/F18 11.9552 Tf 9.299 0 Td [(3 = 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(
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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

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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
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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

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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

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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 ` 0then 9 N suchthat e i 2 N )]TJ/F19 11.9552 Tf 11.956 0 Td [(e i 2 < .Therefore g 0 istransitive. Proposition2.16. Let g 0 bedenedasabove.Then g 0 dependssensitivelyoninitial conditionson S 1 22

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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

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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

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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

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= 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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a b c d Figure4.2:ManyDierentPeriods 4.2AnalyticResults Havingreviewedthenumericalresults,itcannotbeconcludedimmediatelythat alloftheseresultsarelegitimate.Infact,numericalerrorcancertainlyskewthe results.Thus,thesepotentialresultsshouldbeprovenanalytically. Therstanalyticresulttoproveisthatforlargevaluesof c allorbitstendtoward C 36

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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

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a b c d Figure4.4:TheVanishingPeriodThreeRegion 38

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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

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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

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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

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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 n .Thenby thetriangleinequality, d T m x ;T n x d T m x ;T m )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 x + ::: + d T n +1 x ;T n x .12 r m )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 + r m )]TJ/F39 7.9701 Tf 6.587 0 Td [(2 + :: + r n d T x ;x .13 But r m )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 + r m )]TJ/F39 7.9701 Tf 6.587 0 Td [(2 + :: + r n canbemadearbitrarilybymaking n sucientlylarge, since 1 j =1 r j converges.Therefore, f T n x g 1 n =1 isCauchyand 9 x 2 M suchthat x =lim n !1 T n x 42

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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

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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
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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

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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

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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
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Thelastpropositionalsosaysthattheperiodicpointsof F C ,where C = ; 0, where j j < 1 3 ,isdensein S 1 .Thuswehavethat F C mustbechaoticforsuchvalues of C 48

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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

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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

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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

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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

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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

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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

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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

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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

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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

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19 title Counterr 20 set gca ,'FontSize',20 21 pause 22 end 58

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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


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