TheMalkus - PDF document

TheMalkus–LorenzwaterwheelrevisitedLeslieE.MatsonDepartmentofPhysics,UniversityofOregon,Eugene,Oregon97403Received28February2006;accepted16August2007TheMalkus–Lorenzwaterwheelisanalyzedinamoredirectway.TwonewdimensionlessparametersassociatedwithpropertiesofthewaterwheelareusedinplaceofthetraditionalLorenzparameters,whichrelateonlytoLorenz’suidmodel.Theprimaryresultisaperformancemapinthenewdimensionlessparameterspace,whichshowswherethemajorbehaviortypesoccurandassociatedbifurcations.Asliceacrossthismapisexaminedbyabifurcationplotthatshowsdetailsofthetransitionsbetweenvarioustypesofbehavior.Anexampleofa12-cupwaterwheelmodelisprovided,anditsbehavioriscomparedoverawiderangeofparametersontheperformancemapto Thequantity1/ V.THEWATERWHEELBEHAVIORMAPisamapindicatingthebehavioroftheMalkuswaterwheel.TheA,B,C,andDbifurcationsareshownascompletedcontours,extendedtoothervaluesof,thusboundingtheregionsforthevariousbehaviorthatwehavenoted.ThelocationsoftheparametersproducingFigs.areindicatedbydots.Thehorizontaldashedlineisanex-amplewhereonlytheaxlefrictionischanged.Thecurveddashedlineisanexamplewhereonlytheinputwaterratechanges,with=0.0001attheupperright,0.0307atthecircle,and10.0atthelowerleft.Forinnitethecurvewouldendat=0.02182,approximatelyontheBcurve.Thexedparametersforthiscurveare=0.25,=0.025,=15°,=0.01,and=0.0695.A.Periodicorbitexamplesalsohasvedotsalong=0.02thatlocatetheparametersfortheveperiodicorbitsshowninFig..Fig-showstheinitialpathofthewater’scenterofmassforthependulumcase,startingwithallthewateratalmostthetopofthewheel.TheplotsinFigs.showen-largedplotsofthenalstableorbits.Thesymmetricpendu-lumorbitisfoundeverywherewithintheEcurve.Bor-deringtheEcurvearenonsymmetricpendulumversions,andsimilarorbitsthatrepeatonlyaftercompletingmorethanonecycle,calledperiod-doubledorbits.Asalways,otherinitialconditionscanlieinotherbasinsofattraction,leadingtootherbehavior.ThecurvesF,G,andHinFig.locateportionsoftheregions,someextremelynarrow,wherethe“pumpkin,”“one-two,”and“happyface”orbitsofFig.canbefound.InthependulumcaseinFig.,theorbitloopsabout,then,andrepeats.InthepumpkincaseFig.,theorbitmakestwoloopsabout,twoloopsabout,andrepeats.Forthenonsymmetricone-twocaseinFig.,theorbitloopsonceaboutandthentwiceabout.ThehappyfaceinFig.hasthreeloopsaboutandthenthreeaboutTheloopingisinthreedimensions,sothatsomeloopsarenotwellshownintheBetweentheIandJcontoursthetrajectoryconvergestoastablesymmetricorbitinwhichthetopbucketllsuntilthewheel,slightlyoffbalance,fallstotheleftortherightonalternatecycles,almoststopsduetoaxlefriction,onlytollthenewtopbucket,asinthe“llandfall”caseshowninB.BifurcationdiagramsThespiderlikeFig.showsacrosssectionoftheFig.mapalongthe=0.01line,showingbifurcationswherethebehaviorchanges.AttheleftandrightedgesofFig.singlelineswhereunidirectionalmotionoccurs,withsomeswitchingbetweenthetwodirectionsofmotion,resultingindottedlines.ThesymmetricpendulumwithsixextremaasinoccursbetweenthetwoEmarkers.AttheEbifur-cationwhere0.11,thetrajectoryswitchesfromasym-metricpendulumtoanasymmetricpendulummotion,thendifculttoseeinFig.throughperiod-doublingandnoisyperiodicityintochaos.Atmostvaluesofinthechaoticregion,0.130.3,thedotsfortheseeFig.overlaptodrawacontinuousline,withspacesbetweenoutliersthatwouldbellediniftherunlengthwereincreased.Thereareaninnitenumberofgapsinthechaoticregionwherestableorbitsoccur,suchasthepumpkinatF,butmostareextremelynarrowsocannotbeseen.Thesym-metricorbitinFig.isseeninFig.withintheI-to-JandJ-to-Iregions.BetweenthetwoJlabelsfor0.640.86lieapairofasymmetricorbits,eachconsistingofonesideofthellandfallcase.Duetoswitchingbetweenthetwoorbitsasisvaried,thetwolinesforeachorbitare Fig.8.Waterwheelmapforthewaterwheelparameters.AbovetheAcurvethewheelalwayscomestoastop.BetweentheAandCbifurcationcurvestheonlystablebehaviorissteadyunidirectionalmotion.BetweentheCandDcurves,steadyturning,periodicmotion,orchaosresultsdependingontheinitialconditions.BelowtheDcurvethereareonlyperiodicorbitsorchaos.PeriodicorbitexamplesaresymmetricpendulummotionwithintheEcontour,a“llandfall”motionbetweenIandJ,andothersinaninnitenumberofnarrowstripssuchasthepartialcurvesF,G,andH,whichlieinthechaoticregion.ThehorizontaldashedlineshowswherethebifurcationplotofFig.islocated,withonlychanging.Thecurveddashedlineisanexamplewithonlytheinputwaterrate Fig.9.Examplesofstableperiodicorbits.Themotionofthewater’scenterofmassintheplaneofthewheelisshownforvestableorbits,allwith=0.02,locatedatvedotsonFig.Plotoftheinitialtransientbehaviorleadingtotheperiodicpendulumstableorbit.Thenalpendulumorbitandtheothernalorbitsareshownenlargedin.Thestartingpointsareallcloseto11191119Am.J.Phys.,Vol.75,No.12,December2007LeslieE.Matson