Chapter 6 Double Pendulum Analysis (Part 1)

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Chapter6DoublePendulumAnalysis(Part1)ABSTRACTforChapters6,7,8and9:Behaviorofthedoublependulumwasexploredbymakingseveralhundredrunswithasimulationmodel.Onlyasmallpartoftheparameterspacewasexploredbutitshowedhowcomplexbehaviorcanbeinaverysimplesystem.Patternsofbehavioratdifferentlevelsoftotalsystemenergyarereported.Atlowtomoderateenergylevelsbobsperformarhythmicdance-likesequenceofmovesthatrepeatsperiodically,howeverwithdifferentsizestepsthuscreatingpatternswhoseshapegraduallychanges.Thisiscalledquasi-periodic.Athighenergylevelsthesystemexperiencessensitivedependenceoninitialconditionsandisthereforechaotic.Waveformswithineachperiodarecomplex.Theabilitytopredictfuturebehaviorwiththesepatternsisdiscussed.Itshypothesizedthedoublependulumisgovernedbya4-dimensionalstrangeattractorsimilarinconcepttotheLorenzbutterfly.It’sfoundthatdifferentphasespacepatternsoccuratthesameenergyleveldependingonthepositionsofthebobswhenreleased.Tightlyconstrainedinstancesofperfectlyperiodicbehaviorwerefoundwithinotherwisequasi-periodicranges.Asharp,energydefined,qualitativechangeinbehaviorwasfoundbetweenbehaviorwheretheboboccasionallygoesoverthetoporspinsandbehaviorwhereitalwaysreversesandfallsback.Thisoccurredonlyabovetheenergylevelwherethesystemfirstbecamechaoticandthereforedoesnotdemarkthethresholdintochaos.Thewayenergymovesfromoneformandonepartofthesystemtoanotheristreated.Therandomoccurrenceofspikesinthemagnitudeofsystemvariablesafterrelativelycalmperiodsishighlightedasperhapsthemostimportantpracticalimplicationofchaos.Suggestionsforfurtherresearchareoffered.

Outline

6.1Researchobjectives6.2Description:6.3TypesofBehavior6.4Doublependulumsimulationmodel

6.1ResearchobjectivesThepurposeofthisChapteristoexploreandcategorizethebehaviorofthedoublependulumoverawiderangeofconditionsandespeciallytoseehowitsbehaviorchangesasafunctionofthetotalenergywithinthesystem.Attemptswerealsomadetoseewhatpracticaldifferenceitmakeswhetherasystemischaoticornot.Akeyobjectivewastodiscovertherootcauseofchaosinthissystem,butitwasunsuccessful.

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TheapproachwastouseaJavasimulationmodel.Theresearchwasdoneintwophasesspacedmonthsapart.ThefirstphaseisdocumentedinChapters7and8andthesecondinChapter9.Thesecondphasewasdonemainlytofindandexploretheboundarybetweenquasi-periodicandchaoticbehavior,whichunfortunatelywasnotdonewellinphaseonedotoaperceivedlackofanyeasywaytodetectoneofthemaincriteriaforchaos,namelysensitivedependenceoninitialconditionsorSDIC.Althoughhundredsofsimulationrunsweremade,manymorewouldbeneededtofullymapthebehaviorofthedoublependulum.HopefullyotherswilltakethisonandalsouseadifferentmodeltoensurethesefindingsarenotbiasedbysomeunknownandunsuspecteddefectinthemodelIused.Aftersearchingthetechnicalliterature–notexhaustivelybutfairlywell-theauthorbelievestheanalysisinthisChapteristhemostthroughexplorationofthedoublependulumdonetodate.Itcontainsnewresearchandwillhopefullysuggestideasforadditionalresearch.Notonlyisthefulloperatingmapofthedoublependulumincompletetherootcauseofchaosinthissystemisnotyetunderstood.Howthedoublependulum–atoysystem-behavesisofnopracticalvaluebyitselfunlessitcanshedlightonhowimportantreal-worldsystemsliketheclimate,ecosystem,economy,andgovernmentbehave.Thusasmallattemptwasmadetoseehowthesefindingsmayapplytotheselargersystems.6.2Description:Thedoublependulumisperhapsthesimplestphysicaldevicethatexhibitschaoticbehavior.Itsoftenmentionedbutapparentlyhasreceivedrelativelylittlesystematicstudy.It’salsosomethingyoucaneasilybuildandexperimentwith.Thefirstphotobelowshowstheauthorspendulum,constructedwithaluminumarmsabout8incheslong.Iusedgoodqualityballbearingspress-fitintocarefullyreamedholestominimizefrictionandwobble.Theextraholeswereexperimentstoseewhichproducedthemostinterestingbehavior,butactuallyneitherthelengthofthearmsortheplacementofholesiscritical.

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Theimagebelowshowsasimilardesignandalsothatthekeyvariables,angle1and2,aremeasuredfromthevertical.

Actualdoublependulumsareusuallyconstructedfrommetalarmsbutforcomputingsimplicitysimulationmodelsoftenusepointmassescalledbobs.Thefollowingimageshowsthepathsthebobswouldtakeafterbeingreleasedfromafairlyhighposition.Theactofliftingthemtothatpositioninputsenergyintothesystem,andiftherewerenoairorbearingfrictionthatenergywouldberetainedwithinthesystemandthesamegeneralbehaviorwouldcontinueindefinitely.Withfrictionitdecreaseinintensityuntilthereisnoenergyleftandthearmshangstraightdown.

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Thefollowingtimeexposureshowshowthetraceswouldlookafteralongerperiodoftime.Itwasmadeusinglightsattachedtoanactualdoublependulum.http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf

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Thesystem(i.e.thependulum)wasoperatingchaoticallyduringtheperiodthisexposurewastaken.Thegreenlightisatthemainpivotpoint.Theredlightwasattachedatthepivotpointbetweenthetwoarms,andthebluelightwasattachedtotheendoftheouterarm.Notethatthemovementsofthesearmsarephysicallyconstrainedtomoveonlywithinthisspacenomatterhowfastorcomplicatedtheirmovements.Thelargearmwasphysicallytiedtothemainpivotpointsotheonlythingitcoulddowasswingbackandforth,orrotate.Theouterarmwastiedtotheendoftheinnerarmsoitsouterendcouldonlyswingbackandforthrelativetotheredbobandtravelwithinthedonutshapedareaevidentfromtheenvelopeofbluetraces.Thepositionandspeedofthearmsmustatalltimesreflect,indeedcontainormanifest,alltheenergywithinthesystem,whichremainsconstantthroughouttherunifthesystemisfrictionless.Evenifthepartsinthissystembehavechaoticallytheycan’tjustflyawayanyoldway.Thefactthatthemovementswereconfinedisimportantbecausethemovingpartsinmanychaoticsystemsareconfinedeitherbyphysicalbarrierslikewallsor

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attachments.Othersystemslikethesolarsystemareconfinedbytheamountofenergyinthesystemwhichlimitshowfarapartthepartscantravel.Thetotalenergy–potentialpluskinetic-inthissystemmustbeexpressedintheheightandspeedofthetwoarms.Inotherwordsifonemeasuredtheheightofthearmsabovethelowestpointtheycouldreach(withinthesystem,notrelativetothefloororearthscenter)onecouldcomputetheirgravitationalpotentialenergyatanypointintime.Hangingstraightdownitszero.Knowingtheirspeedwouldallowonetocomputetheirkineticenergy.ThesumofGPEandKEwouldremainconstantinafrictionlesssystem.Howeveritoscillatesbetweenitsdifferentforms,namelyPEandKE,anditoscillatesbackandforthbetweenthetwodifferentbobsorarms.Occasionallyalloralmostallthetotalenergyinthesystemcanmomentarilyconcentrateinjustoneboborarm.Thisisakeypoint.Thebestwaytoseechaosinactionistoshowarealoneinoperation.Trythissite:http://video.mit.edu/watch/double-pendulum-6392/Thissimulationvideoshowsoneoftheabovetracesbeingdrawn:http://scienceworld.wolfram.com/physics/DoublePendulum.htmlHereisanothergooddblpendvideo:http://isites.harvard.edu/icb/icb.do?keyword=k16940&pageid=icb.page80864&pageContentId=icb.pagecontent278143&state=maximize&view=view.do&viewParam_name=indepth.html#a_icb_pagecontent278143Whattowatchfor:Theeasiestaspectsofchaoticbehaviortoobservebywatchingthependuluminactionarereversalsinthesmallpendulumsdirectionofrotation,orit’stransitionsbetweenswingingandrotating.Otherparameterssuchasangles,elevation,andratesofrotationarealsochangingbuttheyhappensofasttheyarenotreadilyobserved.Thekeypointisthatallthesechangesinbehaviorhappeninaseeminglyrandommannerwhenthesystemoperateschaotically.SometimesthesmallpendulumrotatesXtimesbeforechangesdirectionorstopsrotating,othertimesitsYtimes.Whenwatchingthedoublependulumonefeelsthateacharmistryingtodoitsownthing,namelyswinglikeasimplependulumfollowingitsownnaturalrhythm.Unfortunatelytheotherarmisn’tquiteinsyncandcontinuestodisturbthatrhythm.Sometimes,likewell-timedpushesonachild’sswing,armAboostswhatArmBistryingtodonaturally.Othertimesthepushesorpullsareoutofsyncandtheforcesfighteachother.Often–anditsbestobservedwhentheupperarmisswingingbackandforth-itlooksliketheinnerarmisslingingorwhippingtheouterarmaround.Thuswetendtoseeashortseriesoforderlybackandforthswingswheretheouterarmswingsabitmoreeachtimeuntilitfinallygainsenoughenergytomakeacompleterevolution.Thereisafeelingthatenergyissurgingbackandforth(itis)insomewavelikemanner.Theideathattheintensityofonepartsoscillationis

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pumpedupbytugsorpullsfromanotherpartuntilthereisadrasticchangeinbehaviorseemskeytounderstandingchaos.Whenrunningatfairlylowenergynotehowtheinnerarmisoftenhangingnearlystraightdownandismotionless–andthuscontainslittleenergy-whentheouterarmswingsoverthetop.Thisindicatestheinnerarmhastransferreditsenergytotheouterarm.Behaviorofauthorsownpendulum:IusedmyowndoublependulumtoexperimentwithhigherenergylevelsthantheJavasimulationsreportedthroughoutthisbookcouldaccommodate.

***insertavideoofitoperatingatlowenergylikemyiPhotoclip#15Iinputthesehighenergiesbyholdingthetwoarmsinlineandswingingthemrapidlyaroundsotheylookedlikeawhirlingairplanepropeller.Thistookfarmoreforceonmypartthansimplyholdingthemverticalbeforerelease.ThisindicatingIwasdefinitelyimputingmoreenergyandthatthesystemhadtocontainormanifestthatenergyassoonasIreleasedit.ItwasmostlyintheformofKEduetorapidrotation.Inthiscasetheinitialwhirlingmotionwasquiteconsistentandorderlyshowingthisparticularsystemcanbehaveinansomewhatperiodicmannerathighenergylevel.Itcouldnotbedeterminedifthishighenergy,propeller-like,behaviorwaschaoticornotsinceminorwobblesinthearmscouldnotbedetectedifindeedtheyexisted.Howeverastherotationslowedduetofrictionbehaviordegeneratedintoamoreviolentchaoticmotionthananyseeninthesimulations.Asfurtherenergywaslosttheouterpendulumnolongermadeoccasionalrevolutionsandeventuallyendedupswinginggentlybackandforthnearitslowpoint.

Alinktovideosofmypendulumdoingthisisfoundat*******((needtoincludeoflinktomyclip#17whereitspinslikeapropellerandclip#18wheretheouterarmspinswhiletheinneronedoesalso.Thistookevenmoreenergy.))

Ialsotriedaddingevenmoreenergybycausingtheouterarmtorotaterapidlyrelativetotheinnerarmwhileswingingthewholeassemblyintorapidrotationaroundthefixedpivot.Thistookquiteabitofphysicalarmenergyonmypart.Theresultingactionwasfastandviolentwiththewholeassemblyrotating.Howevertheouterarmquicklystartedbehaving“chaotically”bywhirlingintermittently.Itthendegeneratedintothenormalchaoticbehaviorseeninmostvideos.Eventhoughthistrialverylikelyinducedmoreenergythantheprioritsbehaviorwasneverorderly,orsoitseemed.Youhavenowobservedactualrealworldbehavior.IntheanalysissectionofthischapterIwillattempttodrawfurtherinsightsintothebehaviorofthedoublependulum.

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6.3TypesofBehaviorTherearefourvariablesthatchangeasthedoublependulummoves:theangleofeachbobrelativetoitsstraightdownposition,whichisproportionaltotheheightofthebobanditspotentialenergy,andthekineticenergyofeachbob,whichisproportionaltoitsspeed.Behaviorishowthesevariableschangeandinteract.Basicallythebobsorarmswanttomovelikesimplependulums,buttheyaren’tusuallysynchronizedsotheypushandpulloneachother.Inotherwordstheydisturbeachother.Behaviorwillbedescribedinthreewaysduringthesesimulations.First,bywatchingthetraceinspaceleftbythebobsintheupperplot.Watchingitonecanalmostvisualizehowtheyalternatelyaccelerateorretardeachotherandslingeachotherabout.Thatdancecanbeseen.Secondweplotawaveformshowinghowoneofthevariables,usuallyangle2,changesovertime.Welookforevidencethatthewaveformduringsometimeintervalorperiodwillrepeattimeaftertimeduringsubsequenttimeintervalsofthesameduration.ThirdwelookatthepatternproducedbyplottingonevariablelikeKE2againstanotherlikeangle1.Doesthatpatternrepeattimeaftertimesooneplotliesatoptheother?Thisplotisatwodimensionalviewofwhatisreallya4-dimensionalphasespaceportrait.Itcanbecomparedwith2-Dviewsofthe3-DLorenzbutterfly.Periodicbehavior:Acommonthemeintheliteratureassertsthatatleastsomedynamicalsystemsbehaveperiodicallywhenfunctioningatlowenergy,butbecomechaoticathighenergy.Whenoperatingperiodically–inthetruesenseofthatword-thebehaviorduringoneperiodoftimeisexactlyrepeatedduringthesecondandallsubsequentequallengthperiodsoftime.Ifwetrackedsomevariablelikethespeedofonearminthependulumitwouldtracesomesortofwaveformandthatwaveformwouldrepeat.Everypeakwouldbethesameheight,everyvalleytheexactsamedepth.Inotherwordsthevalueswouldrepeat.Definitionsofperiodicandaperiodicdonotspecifywhetherabehaviororwaveformmustbeexactlythesametobecalledperiodicorwhetherapproximatelythesameisgoodenough.Thisisafaultintheliteratureandsomeauthorsseemcasualabouttheterm.Thiscreatesaproblemintryingtodescribehowthedoublependulumactuallybehavesandrequiredtheauthortocreatetheterms“patternperiodic”(whichIusedearly-on)and“perfectlyperiodic”whichIusedlater.Thetechnicalliteratureoftenusestheterm“quasi-periodic”tolabelabehaviorthatissomewherebetweenperfectlyperiodicandchaotic.Iuseittoo.Herearesomerelateddefinitions:

Periodic:

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Mathematics.(ofafunction)havingagraphthatrepeatsafterafixedinterval(period)oftheindependentvariable.From:http://dictionary.reference.com/browse/periodic

Aperiodic:Theoppositeofaperiodicsignalisanaperiodicsignal.Anaperiodicfunctionneverrepeats,althoughtechnicallyanaperiodicfunctioncanbeconsideredlikeaperiodicfunctionwithaninfiniteperiod.From:https://en.wikibooks.org/wiki/Signals_and_Systems/Aperiodic_Signals2.Physics.oforpertainingtovibrationsoroscillationswithnoapparentperiod.From:http://www.thefreedictionary.com/aperiodic1:ofirregularoccurrence:notperiodic<aperiodicfloods>From:http://www.merriam-webster.com/dictionary/aperiodic

Imainlyusedplotsoftwovariables,KE2versusa1,tojudgewhetheragivenbehaviorwasperfectlyperiodicorquasi-periodic.WhenplottedtogethertheyformwhatIcalla“partialphasespaceplot”.Itwouldtakefour-dimensionstoplotacompletephasespaceplot.IftheKE2/a1patternwasretracedexactlytimeaftertimesoastoproduceasinglelineitindicatesthatthebehaviorwasperfectlyperiodic.Itdoesn’tmatterhowcomplexthatpatternit,justthatitrepeatexactly.Ifitdidnotrepeatexactly,justapproximately,Ilabeleditquasi-periodic.Belowarethreeslidescomparingperfectlyperiodic,quasi-periodicandchaoticbehavior.Slide95showsperfectlyperiodicbehaviorwhichrequiresthetracetofollowathinlineasitdrawsthepatterntimeaftertime.Notethesymmetryintheplotatright.Notealsothatthebehaviorduringanygivenperiodcanbecomplexcomprisedofdifferentheightandshapedpeaks.Ilikenittoadancewithasequenceofstepsthatrepeatexactlytimeaftertime.

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Slide96showswhatquasi-periodicbehaviorlookslike.Theimagesatleftshowhowthepartialphasespaceplotdevelopsovertime.Thepatternseemstodrift.Thesamegeneraldancemovesrepeatbutthestepsarenotthesamesizeeachtime.Inotherwordsthevaluesareslightlydifferenteachtimealthoughthegeneralpatternstaysmuchthesame.Sometimesthevaluesdriftsoastofilltheentireenvelopeasshown.Becausethepatternsdon’toverlayexactly,northevaluesrepeatexactly,thisisnotwhatoneshouldcallperiodicbehavior.Quasi-periodicistermIuse.Thetwotopimagesatrightshowhowthequasi-periodicwaveform

Slide95Perfectlyperiodicbehavior

Pa2ernrepeatsaboutevery13secondsoverlayingexactlyduringthis150second-longrun.Behavior(“dance”)duringanyoneperiodcanbecomplex.

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looksovertime.Thereisoftensymmetryinthewaveformapparentasitdrawstherightandlefthandpartsofthepattern.Awaveformthatshowsagenerallyrepeatingbehaviorformsinlong-termplots.Thebottomrightshowsacommonformofquasi-periodicbehaviorwherethetracesremainwithinwell-definedbands.Itistrickytojudgethelengthofa“period”usingtheke2/a1plotswhenthesystemisquasi-periodic.Afterfindingthatthesystemwasnotperfectlyperiodicevenatverylowenergylevelslikesettinga1to0.1degreeitwasjudgedthatthedoublependulum,atleasttheseconfigurations,isnotconsistentlyperiodicatlowenergylevelsanddonotthereforeexperienceperioddoublingasdoestheiconiclogisticsequation.Thuslittleeffortwasspenttryingtoestimateperiods.Itishoweverperfectlyperiodicatsomeparticularenergylevelsandquasi-periodicaboveandbelowthoselevels.

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Slide97showswhatchaoticplotslooklike.Notethatgivenenoughtimesomeformsofquasi-periodicbehaviorfilltheentireenvelopewithtracesjustlikechaosdoes.

Slide96Quasi-periodicbehavior

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Figure90illustratesthedifferencebetweenthewaveformofsimpleperiodicbehaviorandthecomplexwaveformsapplicabletothedoublependulum.Ifbehaviorisperfectlyperiodicanygivenpeak,liketheonemarked“A”willbeexactlythesameheightduringeveryperiod.

Slide97Chao%cbehavior

Le.-handplotsshowhowpar%alphasespaceplotdevelopedover%metofillen%reenvelop.Right-handplotsshowhowKE2waveformlookedoverdifferent%meintervals.Calmsmarkedingreen,spikesinred.

Runsmadewithdefaultvalues:L1,L2,M1,M2=1.a2=0.drag=0

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Ifasystemhasapatternofbehaviorthatrepeatsfromoneperiodtoanotherthenitfuturebehavioranytimeinthefuturecanbepredicted.Ifitwereperfectlyperiodicitsexactvaluesatanyfuturetimecouldbepredicted,assumingweknewtheshapeofthewaveformbasedonhistoricalobservationandknewwherewewereonthatwaveform.Ifit’snearlyperiodicorquasi-periodicthenwemaybeabletopredictthepatternofbehavioraroundsomefuturetimebutnotexactvalues.

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Perioddoubling:Whetherornotthedoublependulumexperiencesperioddoublingcouldnotbeabsolutelyconfirmedordeniedusingthesesimulations,althoughitseemsveryunlikely.Firstisthequestionofwhatperiod-2behaviorwouldlooklike.Woulditbethesamepatternorwaveform–asitwasinthelogisticsequation-butshiftedbackandforthinvalue?Ifsothetracewouldpresumablytraceonenicepatternonthefirstpass,anoffsetpatternonthesecondpass,andthenreturnonthethirdpasstotraceapatternexactlyontopofthefirstone.Thiswasneverobserved.Alternatelyperhapsthepatterninperiod-2behaviorwouldbeverydifferenteverywaveformduringeveryothercycle.Thiswasnotobservedeither.Finally,asnotedaboveperiod-1behavior,thatisaperfectlyperiodicwaveformorperfectlyrepeatingphasespaceplot,wasnotfoundatlowenergy.Inotherwordstherewasnoperiod-1behavior.Dramaticevents:Whenthelogisticsequationwasoperatingchaoticallythepopulationwouldsometimeszoomtotheupperlimitandthencrashtonearextinction.Icallthisadramaticevent.LikewisethetraceintheiconicstrangeattractorfortheLorenzequationwouldsuddenlyandrandomlymovefromonewingofthebutterflytotheother.AndtheLorenzwaterwheelwillsuddenlyreverseorstartspinning.Thesearedramaticqualitativechangesinbehavior.Thedoublependulumalsoexperiencessuchdramaticeventswhenchaotic.Atrelativelylowenergytheupperbobswingsbackandforthandthelowerbobswingsbackandforth.Nothingdramatic.Thatsamegeneralbehaviorrepeatstimeaftertime.Ontheotherhandathighenergythelowerbobrandomlyswingsrightoverthetopratherthanfallingback.Ilabelthesegoingover-the-top(OTT)eventsasdramaticeventssincetheymarkadistinctchangeinbehavior.ItturnsoutthatthepresenceorabsenceofsuchdramaticOTTeventsisnotacriterionfordecidingwhetherthesystemischaoticornot.Itdoesnotmarkthedemarkorthresholdbetweenquasi-periodicandchaoticbehaviorinthedoublependulum.ThesystemcanbecomechaoticwithlessenergythanittakestoenableOTTeventstooccur.Howeveriftheydooccurthesystemischaotic,atleastintheterritoryI’veexplored.Mostpopularvideosaboutthedoublependulumonlyshowthisgoing-over-the-toporspinningbehaviorbecauseitsquitedramatic,andofcourselookschaoticinanyone’seyes.6.4DoublependulumsimulationmodelIntroductiontosimulationmodel:BehaviorofthedoublependulumwasexploredusingaJavasimulationmodelwrittenbyProfessorDoolingattheUniversityofNorthCarolinaatPembroke.It’sfoundat:http://www2.uncp.edu/home/dooling/applets/double_pen.files/tom/models/doublepen.html.ThesecondmodelIusedafewtimeswaswrittenbyEricNewmanandisfoundat:http://www.myphysicslab.com/dbl_pendulum.html.

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Bothsimulationsusebobssuspendedon“strings”asopposedtothemetalarmsusedinmanyreal-worldlaboratoryversions.Bobsorarmsbehaveaboutthesameway.Digitalcomputermodelscannotanddonotpreciselyreplicaterealworldbehavior.Thereisapossibilitythesemodelscontainsomecomputingartifactthatcausesparametervaluestoslowlydriftwhentheyshouldn’t.ForinstancethetotalenergylevelintheNewmanmodelslowlydropsalthoughthesimulationissupposedlyfrictionless.Thismustbekeptinmindwhenviewingplotswherethetracecompletesonepatternandthendriftstodrawasimilarbutoffsetpattern.Itcouldbeacomputingartifact.Neverthelessit’sfeltthatmostifnotalldriftintheDoolingmodelreflectsactualbehaviorandnotcomputingartifactssincetherewasnodriftinseveralruns.Ifthemodelweredriftingtherewouldn’tbesuchexceptions.Bothmodelsareeasilyrunonahomecomputeranditshighlyrecommendedyoudososincethereisnosubstituteforwatchingthebobsmoveinslowspeedandwatchingthewaveformsandpatternsevolveovertime.Notethatalldynamicvariables-suchasangle,position,PE,orKEinteractinthesedynamicsystems.Ifonebehavesperiodically,orchaotically,theotherswillalldolikewise.Thusitsnotthatimportantwhichoneoronesareplottedtodeterminewhetherasystemisperiodicorchaotic.Thekeythingistoobservethewaveformsandpatternsandtheextenttowhichtheyrepeatperiodically.OneminordeficiencywiththeDoolingmodelisthatthelengthsandmassescannotbesetaccuratelywiththeslideranywherebetweenitsfarrightandfarleftpositions.ForinstanceLength2canpresumablybesetaccuratelyto0.5byslidingtheslidertoitsfarleftandaccuratelyto1.0byslidingitfarright.Howeverbetweentheseextremesthenumericalvalueshowndoesnotchangeaccuratelyastheslidermovessinceitsroundedoff.Thelengthofthearmdoeschangeinthemodelasthesliderischangedbutthevaluesetintheboxisroundedoffsodoesnotrepresentanexactvalue.Thusonlytheonlyaccuratewaytousethemodelistohavethesliderfarrightorfarleftsoforinstancem2iseither0.5or1.0.Thereareseveralthingswewanttoexplore:

1) Arethemovementsofthedoublependulumeverperiodicmeaningthatacertainpatternofbehaviorrepeatsoverandoveragain.Ifsowhatdoesbehaviorduringthisperiodlooklike?WesawperiodicbehaviorintheLogisticsequationatlow“r”valuesandwonderifthedoublependulumalsobehavesperiodicallyatlowenergylevels.

2) Whatisthenatureofbehavior,especiallywithinoneperiod?3) Doesperioddoublingoccur?ItdidintheLogisticsequationandexpertssay

itoccursinothersystemsaswell.4) Howdoesenergyinthissystemmovebetweenitsdifferentforms?5) Whatdoeschaoticbehaviorlooklikeinthedoublependulum?

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6) Doesobservationofpastbehaviorofferanyhelpinpredictingfuturebehavior,bothwhenthesystemisoperatingperiodicallyandwhenitisoperatingchaotically?

Otherthingsofpotentialnotecanbegleanedformthesesimulations.Forinstancethewayvariableschangeisneverabruptorinstantaneousbutratherhappensinafluidlikemanner.Thisisbecausethependulumbobshaveinertiaandittakestimetoacceleratethem.Ittakestimetoconvertpotentialenergyintomovementandkineticenergy.ThemainthreadIwantedtoexplorewashowthetotalamountofenergywithinthesystemeffectsitsbehavior.Thusaseriesofrunsweremadewitheachhavingmoreenergythantheprior.Thiswasachievedbyreleasingthependulumatalowangle,namely5degrees,forthefirstrun.Angle1wasthenincreaseduntilthelastrunsweremadeat170degrees.Thebluebobwasalmostalwaysstartedata2=0,thatishangingstraightdown.Detailsofmodelruns:Allrunsassumedafrictionlesspendulumsothetotalenergylevelinthesystemremainedconstantduringtherun.Theoreticallywecouldtrackanyvariable–likea1,a2,KE1orKE2-overtimetolearnifthesystemisoperatingperiodicallyornot.Howeverwewouldneedtocutandpasteonewaveformatopanothertoseethatclearly.Itseasiertoplotonevariablevs.anotherovertime.Iftheyformapatternthatrepeatsexactlytimeovertimethesystemisperfectlyperiodic.Thepatternmakeduringthesecond,thirdandallsubsequentperiodswillliedirectlyatopthepatternproducedduringthefirstperiodofoperation.GenerallyI’vechosentoplotKE2versusangle1(calledtheta1inthemodel).KE2isproportionaltothespeedoftheouterbob.I’vealsoincludedafewplotsofangle1vs.angle2.Theyconveythesameinformationbutarenotaseasytointerpret.Thependulumarmsorbobsdoacomplexdanceasitwere.Oftenfollowingthesamegeneralpatternbutusuallytakinglargerandlargerorsmallerandsmallerstepseachtime.Asystemwithevenmoremovingpartswouldprobablyhaveanevenmorecomplicateddance.Itsnotnecessarytopaymuchattentiontoallthedifferentparametersbeingmeasuredinthesesimulationsortheirvalues.Wecansimplyfocusoftheshapesofthepatternsorwaveformstogetagoodfeelforhowthedoublependulumbehaves.Notehowtheparameters,whichrepresenttheinitialconditionsorconstants,areset.Mostaredefaultsettingsbutmass2wasgenerallysetto0.5inphase1and1.0inphasetwo.LaterIemphasizethatchangingthevalueofanyparameter,likemass2,canaltertheresultsdramatically.Inshortchangingthesize,shape,massoranyotherphysicalpartofasystemwillchangeitsbehaviorjustlikechangingthe

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lengthofaclockpendulumdoes.Achangeinoneoftheseso-calledconstantsmightshiftitfromperiodictochaoticoperation.Whyisthisworthmention?LetssupposeweweremodelingthecirculationoftheGulfstream.Possiblyachangeinoneoftheconstants,likeaveragewatertemperatureorsalinity,orevenmaybewaterdepthoversomesubsurfaceobstructionwouldchangethegeneralpatternofcirculationfromperiodictochaotic,orfromonepatternofcirculationtoanother.Couldsomethingthatishappingtipitoverthethresholdintochaos?WeknowacceleratingmeltingoftheGreenlandicesheetischangingthesalinityofthenorthAtlantic.Ihavenospecificknowledgeaboutthissituationandthisispurelyahypotheticalandoversimplifiednotion.Wesimplynotethatthebehaviorofasystemisconstrainedbyitsphysicalbarriersordimensionsaswellasbytotalenergy.Itmustoperatewithinthoselimits.Changethoselimitsandthebehaviorwillchangeinresponse.