WelcometoMATH226.3x:NonlinearDifferentialEquations:OrderandChaos.Thissyllabusprovidesageneraldescriptionofthecoursecontent,theschedule,theassessmentsandgrading,andgeneralguidelines.Pleasecheckthesyllabusifyouhaveanyquestionsregardingtheoperationofthiscourse.

Nonlinear Differential Equations

Phenomenaasdiverseasanautomobilessuspensionsystem,theswayingofabridge,andthedampingofaskyscraperaregovernedbydifferentialequations.MATH226xisanintroductiontothemathematicaltheoryofordinarydifferentialequations.Thiscourseadoptsamoderndynamicalsystemsapproachtothesubject.Thatis,equationsareanalyzedusingqualitative,numerical,andifpossible,symbolictechniques.MATH226.3xisthecapstonecourseinthethreeMOOCsequenceofcoursesthattogetherareMATH226x.WewillapplythetheoryandrelevanttechniquesfromMATH226.1xandMATH226.2xtothestudyofnonlinearsystemsofordinarydifferentialequations.Thereisnosystematicapproachthatappliestoallnonlinearsystemsofdifferentialequations,butinthiscoursewepresentahandfuloftechniquesthataddressawiderangeofsystems.Wedonothesitatetosacrificerigorforintuition.WeareabletodiscussadvancedtopicssuchaslinearizationandHamiltoniansystemsbyignoringcertaintechnicaldetailsandbyrestrictingourdiscussiontoautonomoussystemswithtwodependentvariables.Inthefinalmoduleofthecourse,wediscussathreedimensionalsystemthatwasfirststudiedbyEdwardLorenzatMITintheearly1960s.Inthatdiscussion,weintroducesomeofthetermsandconceptsrelatedtothestudyofchaoticdynamicalsystems.Overallourtreatmentin226.3willbelesssystematicthanitwasin226.1and226.2,butwehopethatitwillpersuadeyoutocontinueyourstudyofnonlineardifferentialequationsaftercompletingthiscourse.

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About the Team

PaulBlanchardisprofessorofmathematicsatBostonUniversity.HegrewupinSutton,Massachusetts,USA,andspentthreeundergraduateyearsatBrownUniversity.Duringhissenioryear,hedecidedtohaveanadventureandlearnanewlanguage,sohewasanoccasionalstudentattheUniversityofWarwickinEngland.HereceivedhisPh.D.fromYaleUniversity.Hehastaughtmathematicsformorethanthirtyfiveyears,mostatBostonUniversity.HismainareaofmathematicalresearchiscomplexanalyticdynamicalsystemsandtherelatedpointsetsJuliasetsandtheMandelbrotset.HeisaFellowoftheAmericanMathematicalSociety. Formanyofthelasttwentyyears,hiseffortshavefocused

onmodernizingthetraditionalsophomoreleveldifferentialequationscourse.Thatefforthasresultedinnumerousworkshopsandminicourses.HehasalsoauthoredfiveeditionsofDifferentialEquationswithRobertL.DevaneyandGlenR.Hall.Whenhebecomesexhaustedfixingtheerrorsmadebyhistwocoauthors,heheadsforthegolfcoursetoenjoyadifferenttypeoffrustration.

PatrickCummingsisaPh.D.candidateintheDepartmentofMathematicsandStatisticsatBostonUniversity.Hisresearchinvolvesextendingthetheoryoffinitedimensionaldynamicalsystemstoinfinitedimensionaldynamicalsystemsdefinedbypartialdifferentialequations.PatrickreceivedhisBachelorofArtsdegreeinMathematicsfromMaristCollegein2012.WhileatBostonUniversity,hehasbeenateachingassistantforMA226,theresidentialequivalentofMATH226x.

Course Outline

Module Content Module1:NonlinearSystems ReleasedonThursday,July30at1:00PMEDT

We introduce four important examples of nonlinear systems. These examples illustrate many important concepts and will reappear throughout the course.

Module2:EquilibriumPointAnalysisReleasedonThursday,July30at1:00PMEDT

In MATH226.2x, we were able to understand the solutions of linear systems both qualitatively and analytically. Unfortunately, nonlinear systems are in general much less amenable to the analytic and algebraic techniques that we have developed, but we can use the mathematics of linear systems to

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understand the behavior of solutions of nonlinear systems near their equilibrium points.

Module3:QualitativeAnalysis ReleasedonThursday,August6at1:00PMEDT

The process of linearization discussed in Module 2 gives us a powerful technique for understanding the behavior of solutions of a nonlinear system near an equilibrium point. Unfortunately it provides local information onlyinformation that can be used only near equilibrium points. So far our only general techniques for the study of the behavior of nonlinear systems away from equilibrium points are numerical. In this module we develop qualitative techniques that can be used in combination with linearization and numerics.

Module4:HamiltonianSystems ReleasedonThursday,August6at1:00PMEDT

Nonlinearsystemsofdifferentialequationsarealmostimpossibletosolveexplicitly.Thesolutioncurvesofsystemsbehaveinmanydifferentways,andtherearenoqualitativetechniquesthatareguaranteedtoworkinallcases.Fortunatelytherearecertainspecialtypesofnonlinearsystemsthatariseofteninphysicalsystemsandforwhichtherearespecialtechniquesthatenableustogainsomeunderstandingofthephaseportrait.Inthismodulewediscussoneofthemostimportantofthesespecialtypes.

Module5:DissipativeandGradientSystems ReleasedonThursday,August13at1:00PMEDT

The Hamiltonian systems discussed in Module 4 are idealized systems. In this module we discuss systems for which there is a quantity that dissipates over time.

Module6:ChaosandtheLorenzAttractor ReleasedonThursday,August20at1:00PMEDT

In1963EdwardLorenzpublishedapaperthatwouldeventuallyhaveaprofoundeffectonthemathematicalanalysisofnonlinearequations.WeendMATH226xwithadiscussionofthesystemthatnowcarrieshisname.Wewillalsoattempttoconveyitsplaceinthedevelopmentofthemathematicsthatunderlieschaostheory.

FinalExam ReleasedonThursday,August27at1:00PMEDT DueonThursday,September3at1:00PMEDT

Thisexamwilltestalltopicspresentedinthiscourseandwillbeworth60%ofyouroverallgrade.

EndofCourse Monday,September3at1:00PMEDT

Thecourseofficiallyendsatthistime.Thecontentwillstillbeavailableafterthecoursecloses,butthoseseekingacertificatemustachieveanoverallgradeof50%bythisdate.

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Assessments and Grading Eachmoduleconsistsofaseriesofvideosinterspacedwithbriefexercisesdesignedtohelpyouassessyourunderstandingofthematerialdiscussedinthevideo.Thesecontentcheckexerciseswillbeworth10%ofyouroverallgrade. Attheendofeachmoduletherewillbeanexercisesetthatwillprovidemoredetailedpracticewiththeconceptspresentedinthemodule.Theseexercisesetswillbeworth30%ofyouroverallgrade. ThefinalexamforthecoursewillbereleasedonThursday,August27at1:00PM(EDT).Itwillcoverallofthematerialdiscussedinallsixmodules.Toreceivecredit,youmustsubmityouranswersbySeptember3at1PM(EDT).Thefinalexamwillbeworth60%ofyouroverallgrade. Thedeadlineforallassessmentswillbetheendofthecourse,thatis,September3at1PM(EDT).Youmaydelaycompletionofthecontentcheckexercisesandexercisesetsuntiltheendofthecoursewhilestillgettingcredit.However,westronglyrecommendthatyoucompleteallexercisesasyougo.

Discussion Forum Guidelines Wehopethatyoufindthediscussionforumstobeausefulcomponentofthiscourse.Theyaremeanttobeanareawherethestudentscaninteractwitheachother,askquestions,ortalktothecoursestaff.Wegreatlyencourageyoutousetheseforumsonaregularbasis. Wesupportandencouragetheuseoftheforumtodiscussoraskquestionsaboutexercisesandconsequentlytheirsolutions.Wewillnotdeletequestionsordiscussionsthatcontainsolutionshowever,wedoaskthatyoudonotabusetheforumsasawaytoshareanswerstoexercises. Weaskthatyoudonotpostcommentsthatarederogatory,defamatory,orinanywayattackotherstudents.Becourteousandshowthesamerespectyouhopetoreceive.Discussionforummoderatorswilldeletepoststhatarerude,inappropriate,orofftopic.Commenterswhorepeatedlyabusethispublicforumwillberemovedfromthecourse. Thereisafeatureinthediscussionforumsthatallowsyoutoselectfromtwoposttypes,QuestionandDiscussion.TheQuestiontypeismeantforspecificissueswiththeplatformorwithcontent,andtheDiscussiontypeismeanttoshareideasandstartconversation.Pleasekeepthisdistinctioninmindwhenpostingtothediscussionforum.

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FAQ Q:ShouldIemailtheprofessororanypersonsinvolvedwiththiscoursedirectly? A:No.Ifyoufeeltheneedtocontactthecoursestaffinvolvedinthiscourse,pleasedosothroughtheDiscussionForum. Q:DoIneedtobuyanypersonalmaterialstotakethiscourse? A:No.Youdonotneedtopurchasetextbooksoranymaterialstoaidyouincompletingthecourse. Q:I'venevertakenanedXcoursebeforeandthisisconfusing.WhatdoIdo? A:ThereisaprecourseedXwalkthroughthatbeginnerscanwatch.ItexplainsindetailhowtousetheedXplatform.Forfurtherinformation,pleasevisitthedemoedXcourse. Q:Ifoundamistakeinthecourse.WheredoIreportit? A:OntheWikipage,thereisaspecificsectionforErrata.Youcangothere,editthepage,andpostinformationconcerninganyerrorsorissuesyouhavefound.Wewilltrytofixthemassoonaspossible. Q:HowdoIlearnmoreaboutthemathematicsdiscussedinModulex? A:Manyofthemodulesdiscusstopicsthatcanbestudiedinmuchmoredetail.Ifyoufindatopicespeciallyinterestingandwouldliketoknowmore,thenpleasepostaquestiononthediscussionforum.Ifweknowofagoodreferenceorresource,thenwewillpostitonthewiki.

Time Zones Anoteabouttimereferences:TimewillbereportedbycoursestaffasEasternDaylightTime,NorthAmerica(EDT).AnytimeslistedbyedX,suchasduedateslistedonthecoursesite,willbereportedinUniversalTimeCoordinated(UTC).Thecoursestaffwillmakeeveryefforttomaketimesandtimezonesasclearaspossible.Therearevarioustimezoneconvertersonthewebsuchashttp://www.timeanddate.com/worldclock/converter.html.

Honor Code TheedXplatformassumesacertainlevelofdecorumandresponsibilityfromthosetakingthiscourse.PleasereviewtheedXHonorCode,whichisreproducedbelow. ByenrollinginanedXcourse,IagreethatIwill:

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Completeallmidtermsandfinalexamswithmyownworkandonlymyownwork.Iwillnotsubmittheworkofanyotherperson.

Maintainonlyoneuseraccountandnotletanyoneelseusemyusernameand/orpassword.

Notengageinanyactivitythatwoulddishonestlyimprovemyresults,orimproveorhurttheresultsofothers.

Notpostanswerstoproblemsthatarebeingusedtoassessstudentperformance. UnlessotherwiseindicatedbytheinstructorofanedXcourse,learnersonedXareencouragedto:

Collaboratewithothersonthelecturevideos,exercises,homeworkandlabs. Discusswithothersgeneralconceptsandmaterialsineachcourse. PresentideasandwrittenworktofellowedXlearnersorothersforcommentor

criticism.

Credits and Acknowledgements Aswithanymajoreffort,thiscoursewouldnotbepossiblewithoutlargecontributionsfrommanysources.Wewouldliketoextendaspecialthankstothevariousteamswhohaveputinuncountablehoursofworktohelpcreatethiscourse.Specifically,wewanttothankthefollowingpeopleandorganizationsthathavecontributedalargeamountofefforttomakethiscoursebecomeareality:RomyRuukel,TimBrenner,VanessaRuanoformanagingthisprocessandbeingresponsibleforeveryaspectofthemakingofthiscourseJoeDwyerforeditingtheannotatedslidevideosthatappearinthiscourseandforfilmingandeditingtheaboutvideoKellanReckforhisgraphicsintheaboutvideoCourtneyTeixeirawhodrewtheimagesonthetitlecardsAndrewAbrahamsonandAdamBrillaofBUsMetropolitanCollegewhohelpeduswithourtabletcaptureintheirmediaroomKacieClearyofBUsInformationServicesandTechnologywhohelpeduswithtabletcaptureinMugarMemorialLibraryDanielShankforaccuracycheckingProfessorJohnPolkingofRiceUniversityforlettingususehisprogrampplaneinthiscourseMathWorksforprovidinglicensesforMATLABduringthecourseJohnKotwicki,BrandonArmstrong,andespeciallyErinByrneofMathWorksfortheirassistancewithMATLABHubertHohnwhoworkedwithusdesigningandimplementingDETools,softwarethatweusewhenweteachdifferentialequationsCengageLearningforprovidingpartialsupportduringthedevelopmentofDEToolsandtheDigitalLearningInitiativeandtheDepartmentofMathematicsandStatisticsatBostonUniversityforsupportingPaulBlanchardandPatrickCummingsduringthedevelopmentofthiscourse. ThiscoursewouldnothavebeenpossibleiftheNationalScienceFoundationhadnotpartiallyfundedtheBostonUniversityDifferentialEquationsProjectfrom1993to1998.

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ManyundergraduateandgraduatestudentshaveworkedontheBUDifferentialEquationsProjectovertheyears:GarethRoberts,AlexKasman,BrianPersaud,MelissaVellela,SamKaplan,BillBasener,SebastianMarotta,StephanieR.Jones,AdrianVajiac,DanielCuzzocreo,DuffCampbell,LeeDeville,J.DougWright,DanLook,NuriaFagella,NickBenes,AdrianIovita,KinyaOno,andBeverlySteinhoff. PaulBlanchardwouldespeciallyliketothankhiscolleaguesandcoauthors,RobertL.DevaneyandGlenR.Hall,formanyyearsofenjoyablecollaborationonthedevelopmentofmaterialsusedtoteachdifferentialequations.

Terms of Service Forfurtherinformation,pleasereviewtheedXTermsofService(https://www.edx.org/edxtermsservice).

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