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Chapter2:FundamentalsOutline

2.1Definitionofsystem2.2Forcesandbehavior:2.3Systemsandtheirwaveforms2.4Allsystemsaresub-systems,andbigdominatessmall2.5Fluidsandgasesassystems2.6Energy2.7Definingchaos2.8Videosandreferencesaboutchaos2.9Toolsandtechniquesforanalysis

2.1DefinitionofsystemAtitsmostabstractlevelasystemiscomprisedofapluralityofpartslinkedsothebehaviorofonepartaffectsthebehavioroftheotherseitherdirectlyorindirectly,eitherstronglyorweakly.Tobemoreaccurateasystemisasub-setofpartsintheenvironmentthatarelinkedstronglyenoughtobearecognizableentityandbehaveassuchwithinthelargerenvironment.Insomecaseslikeasystemintheformofalivinghumanbeingitsclearwhichpartsareincludedandwhicharen’t.Inaneconomicsystemitbecomesdebatablewhichpartstoincludesinceonesystemliketheautoindustryblendsintoanotherlikethesteelindustry.Fluidsandgasescanalsoformsystems.Moreonthatlater.Figure109illustratesasimpleisolatedsystemusingspringstoconnecttheparts,whichhavemasslikelittlesteelballs.Afrequentsystemsbehaviorisoscillationasthefigureattemptstoillustrate.Thislittlediagramisaveryabstractrepresentationofasystembutonetokeepinmind.Certainlyitappliestosystemscomprisedofdiscreteparts.Howeverthepartsingaseoussystemsliketheatmosphereorfluidicliketheoceansarenotsoeasytovisualizeinthismanner.Sometimesitsnecessarytoconsideraportionofsuchsystems,suchasan“blob”ofhotairorwater,asapartandwatchhowitmovesorchanges.

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Conservingversusdissipativesystems:Conservingsystemsaresystemswhosetotalinternalenergyremainsconstanteventhoughtheenergyassociatedwitheachpartmaybeoscillating.Thatenergyisinputtothesystembysomeinitialoutsideforcethatstretchesthespring-likebondsawayfromtheirequilibriumlength.Dissipativesystemshaveacomplextechnicaldefinitionhttps://en.wikipedia.org/wiki/Dissipative_systembutIseethemassystemswhereenergyisdissipatedbyfrictionorradiationuntilthesystemcomestoamotionlesshaltunlesstheyarekeptinmotionbycontinuedoroccasionalinputsofenergy.Africtionlessclockpendulumisaconservingsystemwhosependulumwillswingforever.Intherealworldit’sadissipativesystembecausefrictiondrainsenergywhileaspringorweightinsertsenough–intheformofescapementpushes-tokeepitmoving.Pushingaswingisthesameidea.Allrealworldsystemsaredissipative,

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somemorethanothers.Thesolarsystemisclosesttoaconservingreal-worldsystemwhileweathersystemslikethunderstormsdissipateenergyrapidly.Incomplexsystemsenergymovementthroughadissipativesystemisaprocesswhereonethingeffectsanotherinachainofeventsuntilalltheenergyescapes.Thehumanbodyisanexample.Foodenergyisinputatoneendofametabolicchemicalprocess–asequenceorcascadeofreactions-whilebodyheatandphysicalexertionsdrainitaway.Suchprocessescanbehavechaotically.DissipativesystemswhichimplementaprocessconstituteawholedimensionofsystembehaviorI’venottimetoexplorehere.2.2Forcesandbehavior:Forcestrength:Therearefourbasicforcesinnature:

GravityWeakbutextendsovercosmicdistancesElectromagneticFairlystrong.Sourceofmagneticattractionandalsosourceofelectrostaticforceswhereoppositechargesattractandlikechargesrepel.Worksattheatomicandmolecularleveltobindatomsintomoleculesandholdsolidstogether.NuclearStrongForceExtremelystrong.Holdsatomicnucleitogether.Powershydrogenfusionbombs.Actsonlyoveraveryshortdistancewithinatomicnuclei.WeakNuclearForceNotrelevantforourpurposes.

Onlygravity,electrostatic,andcentrifugalforcesarerelevantinsystemsdiscussedinthisbook.Centrifugalforceisnotoneofnature’sfundamentalforceshowever.Balancebetweenattractingandrepellingforces:Thephysicsassociatedwithintermolecularforces,namelytheforcesthatholdmoleculestogethertoformliquidsandsolids,underlaythesimplediagramstofollowinFigure111.Theimagebelowshowshowthe“resultant”ornetforcebetweentwomoleculesisthesummationofattractiveelectrostaticforcesandrepulsiveelectrostaticforces,eachofwhichvarieswiththedistancebetweenthemolecules.Electrostaticforcesboildownto“likechargesrepel”and“oppositechargesattract”.Thephysicsbehindtheseforcesgetscomplexbutapparentlytherepulsingforcegetsextremelystrongwhentheelectroncloudssurroundingeachatomgettooclose.http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/paulirep.html.Theserepulsingforcesarewhatmakesitverydifficulttocrushsolidmaterialsorcompressliquids.

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Theredlineinimagebelowshowsoneagainhowthenetforcebetweentwomoleculesvarieswiththedistancebetweenthem.http://www.schoolphysics.co.uk/age16-19/Thermal%20physics/Heat%20energy/text/Forces_between_molecules/index.htmlAttheequilibriumdistance“M”thenetforceiszero.Thisisalsothelowestenergyconfiguration,meaningittakesaninputofenergytoeitherpushthemoleculesclosertogetherorpullthemfurtherapart.Ontheotherhandifthemoleculesstartoneithersideoftheequilibriumtheygiveoffenergy–inanexothermicreaction-astheyapproachthatequilibrium.Ifheatisappliedtotheequilibriumconfigurationthedistancebetweenthemoleculeswouldbegintooscillatearoundthe“M”position.Inthereal-worldthereisalwayssomeheatsomoleculesalwaysoccillateinternally,gentlyatlowtemperature,violentlyathightemperature.Thepotentialenergycurvecanalsobevisualizedasaramporwell.Ifaballwereatthebottomitwouldtakeenergytorollituponeithersideandcreateapendulumlikeoscillation.

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Theshapeofthisredcurvebelowshouldberememberedbecauseitsfundamentaltounderstandingthedynamicsofallsystemswhereseparatepartsareattractedbyonetypeofforcebutrepellediftheygettooclosebyanothertypeofforce.Theresultistheystayacertaindistanceapart–oroscillateaboutit-andformastructureorconfiguration.Thisistruewhethertheforcesbetweenthepartsaresprings,electrostatic,gravitational,orthenuclearstrongforce.Ithinkasimilarsituationisfoundinallothertypesofsystems:ecological,economic,social,etc.Thereissomeforcethatbindspartsintoasystemandanotherwhichkeepsthemfrommergingentirely.Sometimesitprobablyamountstoabalanceofpowerorforcebetweencompetingeconomicandpoliticalorganizations.Thecompromisetheyreachamountstoanequilibrium.Thesystemtheyforminthatequilibriumislikeagiantmolecule.Asocietalmoleculesotospeak.Thefourfundamentalforcesinnaturecanactinsequenceasmassesgetclosertoeachother.Atlongdistancesgravitypullsthingstogetheruntiltheyformasolidballatwhichtimerepulsingintermolecularforcestakeoverandkeeptheballfrombeingcrushedfurther.Planetsaregoodexamples.Iftheballofmatterislargeenough-likeaburnt-outstarofsufficientsize-gravitywilloverwhelmtherepulsiveintermolecularforcesandcrushthestarintoaneutronstar.Somethingcalled“DegenerateNeutronPressure” willresistfurthercollapseatthatpoint.Howeverifthestarwaslargeenoughnoforcecanovercomegravityandthedeadstarwillcollapseintoablackhole.http://minerva.union.edu/vianil/web_stuff2/Election_and_Neutron_Pressure.htm

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Behavior:Behavioriswhatasystemdoes,howitchangesovertime.Changeismeasuredintermsofthepositionorspeedoftheparts,temperature,pressure,orsomeotherattributethatvariesovertime.Insocietalsystemsitmightbegroupmembership,organizationalstructure,size,territory,revenue,marketshare,policyposition,projectsundertaken,andsoforth.Allthesearecalledvariables.Whenaspring/masssystemisinequilibriumallthespringsarerelaxedandthepartsaren’tmoving.Whenonepartisdisturbed,thatispulledasidebysomeoutsideforce,thespringsarestretched.Whenthepartisreleasedthespringstryto

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pullitbacktoitsequilibriumposition.Howeverthepartdoesn’tstoptherebutratherovershootsandthengetspulledbacktheotherway.Thusthepartoscillatesbackandforth,untilandunlessfrictionslowsittoastop.Amainpremiseofthisbookisthatmostreal-worldsystemsbehavethisway,betheymechanicallikethesolarsystem,chemicallikelivingorganisms,orecological,economic,orpolitical.Theactionsofonepartaffectthepartsithasrelationshipswith.Ofcourserealworldsystemsofpracticalinterestarenotconnectedbysprings,butsincepartsdoaffecteachothertheremustbesomeforceorforceequivalentlinkingthem.Usingspringsisaconvenientwaytorepresentthisconnectingforce.Figure111takesacloserlookatthis.Forasystemtoexistthereneedtobetwotypesofforces;onepullingthepartstogethersotheymakeasystemandotherkeepingthemsomedistanceapartsotheyremainseparateparts.Theattractingforceinsolarsystemsisgravityandtheattractingforceinmoleculesandsolidobjectsissomeversionofelectrostaticorelectromagneticforce.Botharenon-linearinthattheirstrengthfallsoffexponentiallywithdistance.Thespringmasssystemsobviouslyareheldtogetherbysprings,whichatthemolecularlevelalsoinvolveelectromagneticforces.Howeveratthemacrolevelmostspringsarelinearmeaningtwiceasmuchforcestretchesthemtwiceasmuch.Iwon’tgetintorepellingforcesnowexcepttosaytheyexist,althoughnotalwaysintheformexpected.Forinstancecentrifugalforceisnotoneofnaturesbasicforcesbutitdoesbalancegravitytokeepplanetsinorbit.

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Figure111aillustratestheinterplayofattractingandrepellingforces.Thelargeballwithanxthroughitatleftisonepartandthesmallblackballistheotherpart.Thegreenlinerepresentsalongrangeattractingforce,which,likegravity,getsweakerthefurtherapartthesetwobodiesare.Theredlinerepresentsarepellingforcethat’smuchstrongeratclosedistancebutfallsoffrapidlywithdistance.Ifthiskindofrepellingforcedidn’texistsolidbodieslikeearthwouldbecrushedbygravityintoatinyspec.Andasblackholestestifyifgravityisstrongenoughburnt-outstarswillbecrushedtominiscule“singularities”.Thedottedlineisthesumoftheseattractingandrepellingforcesandthusthenetforceonabody.Atoneparticulardistancethenetforceiszeroandabodyplacedtherewillremaininequilibrium.Ifitgetscloseritwillbepushedout.Ifitisfurtheroutthenetforcepullsitin.

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Thenextfourdiagramslookatthingsfromanenergyperspective,rememberingthattopushagainstaforcetakesenergy.Figure111”b”thru“e”useaneasytounderstandrampanalogy.Ifapartisoutonthegreensideoftherampitwillrollinanddownhilltowardthelowpoint.Ifitsontheredrepellingsideitwillrollout.Heretheballisshownrollingin.Thisistheessenceofsystemsself-assembly.Partsthatstartfarapartarepulledtogetheruntiltheyreachanequilibriumdistanceapart.Thebig-bangblastedallthematterintheuniversefarapartand–inlocalized“gravitybound”areaslikegalacticclusters-gravityhasbeenpullingitbacktogethereversince.Atsomepointgravitywillbebalancedbyarepellingforceandinwardmovementwillstop.Againwemakeanexceptionforblackholes.Figure111cshowsthatiftheballispulledasideandthenreleaseditwillrolldownonesideanduptheother.Withoutfrictionthiswillresultinendlessbackandfortheoscillation.Arguablyitshowalmostallsystemsactuallybehavetoonedegreeoranother.Itskeytonotethatwhentheballishigh-upishaspotentialenergyorPE.TheforceexertedonitatthatpointacceleratesittowardtheequilibriumpintthusconvertingthePEtokineticenergyastheballspeedsup.Thisistheessentialcorereasonwhysystemsoscillate.Figure111dshowsthebodyatadistancewheretheforcesarebalancedandthusdon’tseektomoveit.Itsatamotionlessequilibrium.Itsfairtosayallsystemstendtoevolvetowardanequilibriumconfigurationwhereallforcesarebalancedandnothingchangesormoves.Howevernotallequilibriumsareequal.Therearerelativelyweakofunstablelocalequilibriumsandifasysteminoneisdisturbedenoughitmightsnapintoadifferentandmorestableconfiguration.Thatwouldbringittoalowerenergylevel.Figure111eshowswhyittakesenergytomoveapartonceit’sinanequilibriumconfiguration.Itslikerollingtheballuphilleitherway.Samethingwhentryingtochangeasystemwhenitsinornearequilibrium.Intherealworlditexplainswhyittakesconsiderableefforttochangemolecules,governmentpolicy,organizationalstructure,orstartarevolution.FinallyFigure111fshowstheequivalentsituationifspringsareapplyingtheattractingandrepellingforces.Obviouslywhenrelaxedtheyholdthingsapartsomedistance,whencompressedtheybecomerepellingforcesandwhenstretchedtheyactasattractingforces.Thuswecanbuildrealphysicalmodelsandcomputermodelsusingspringsthatmimicthebehaviorofsystemswherethepartsareeffectedbygravityandotherforces.Inanysystemwhereonepartinfluencesanothersomespring-likeforceisinvolved.Ifthepartsarerigidlyattacheditsreallytheequivalentofveryverystiffsprings.Thesystemswestudyinthischaptermainlyinvolvegravitationalormagneticforces,butsomeusesprings.Thereisadifferenceinhowtheybehavebecausespringsapplylinearforceswhereasgravityandmagneticforcesarenon-linear.TogeneralizeIoftensay“spring-likeforces”

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whenI’mnotsurewhethertheyarelinearlikesimplespringsornon-linearlikegravity.2.3SystemsandtheirwaveformsFigure104showsthatsystemswithjusttwopartsoscillatedifferentlythansystemswiththreeormoreparts.

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Inatwo-partsystemifonepartispulledasideorsetinmotionitpushesorpullsontheothercausingittomoveinasinusoidaloscillationthat,withoutfrictionorcollision,willcontinueindefinitely.Figure104introducesthebasicconcepts.Attopleftweseethreedifferenttwo-partsystems.Theupperhasamasssittingonaspringsuchthatgravitypullsitdownwhilespringpressurepushesitup.When

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displacedfromequilibriumitwillbounceupanddownandanyvariablewilltraceaperfectsinewaveasshownonright.Withoutfrictioneachwavewillbethesameasthosebeforeandfuturebehaviorcaneasilybepredicted.Iftheweightislifteditwillintroducepotentialenergyintothesystemjustasliftingaweightoffthefloorwould.Ifitispusheddownitwillgainspringpotentialenergy.Thusattimezeroonleftthesystemhassomeleveloftotalpotentialenergy“E”.Whentheweightisreleasedthemasswillaccelerateconvertingpotentialenergy(PE)intokineticenergy(KE).Withoutfrictionthelawofconservationofenergymeansthetotalenergymuststayconstantbutitcanexchangebetweendifferentforms,namelyPEorKE.TheplotofanyvariableinthissystemwhetheritsPE,KE,speed,orthedisplacementofthemasswillformasignwave.Theplotshowsthatasdisplacementorpotentialenergyincreases,speedorkineticenergyfalls.Thevaluesmustalwaysstayinanenvelopeasdeterminedbythetotalenergyinthesystem.Wewillseethatthisapplieswhetherthemotionisperiodicasitishereorchaotic.Variablesdovarybutonlywithinbounds,evenwhenthatvariationischaotic.ThelowerpartofFigure104showswaveformsassociatedwithsystemshavingthreeormoreparts.Thepartsdon’toscillatewithasimplesinusoidalmotion.Rathertheirmovementsaremorecomplex,possiblychaotic,becauseeachdisturbsalltheothers.Thekeypointaboutsystemswiththreeormorepartsisthatthepartsrarelymoveinaperfectlyrepeatableorperiodicmannerasthechartattemptstoillustrate.Thepeaksandvalleysintheirwaveformareconstantlychanging.Inadditiontheenergyinthesystemmovesfromparttopartbecausethepushingpartslowsdownandtransfersenergytothepartbeingpushedandaccelerated.Figure104attemptstoshowhowtheenergyineachpartchangesovertime.Andthatitusuallychangesrandomlyorchaotically.2.4Allsystemsaresub-systems,andbigdominatessmallThesetwopointswerealreadymentionedinChapter1butinsertedherejustforareminder.AsdescribedinChapter1nosystemexistsintotalisolation.Computermodelsoftenassumetheydobutit’snottrueintherealworld.Thebehaviorofarealworldsystemisalwaysaffectedtosomedegreebyeventsinitsenvironment,includingthebehaviorofothersystemstowhichitiscoupled,weaklyorstrongly.Thereisahierarchyofsystemsrangingfromtheverypowerfultotheveryweak.Themostpowerfulsystemofpracticalconcernisthesolarsystemwhosebehavioraffectsalmosteverythingonearth,butitcan’tbearguedthatwhathappensonearthhasanynoticeableeffectonearthsorbit.“Small”systems,evenonesaslargeastheeconomyrideatopandareinfluencedbylargerunderlyingsystemslikeocean

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currentcirculationandclimate,whichinturnrideatopandarethusinfluencedinthelongtermbychangesinearthsorbitcausedbyinteractionwithotherplanets.Thebigpartswithinasystemaffectthesmallpartsmorethanviceversa.ForinstanceJupiter’sbehaviorgravitationallyaffectsearthfar,farmorethanthereverse.Andclearlythesunsbehavior,especiallysolarstorms,affectsearthbutitshardtoimaginehowearthcouldaffectthesuninanymeaningfulway.Governmenthasfarmoreaffectonanindividualcorporationthanthereverse.Onereasontomentionthisisthatthepartsintoysystemslikethedoublependulumtendtobemoreorlessthesamesize.Notdifferentbyanorder(s)ofmagnitude.2.5FluidsandgasesassystemsThisbookdealsalmostentirelywithmechanicalsystemscomprisedofafewdiscretepartsconnectedbyspring-likeforcesintosomespatialconfigurationlikeadoublependulum,solarsystem,ormolecule.Howeversystemscanbemadefromfluidsandgases.Werefertocirculatingoceancurrentsassystemsaswellastheatmosphere.Thesesystemsbehavechaoticallytoagreaterorlesserextentandwillbereferencedfromtimetotimebutnotexploredindepth.Itsnoteasytoseehowthesamephysicallawsapplytotinymechanicalsystemswithafewpartsandalsotofluidsandgaseswithuntoldtrillionsofparts,butitseemsthatafluidorgasisessentiallyahugespring/masssystemwherethevibrationandmovementofeachmoleculeeffectseveryother.Ifwelookjustatthemicroscalewhereelectromagneticforcesgovernthevibrationwithinmoleculeswetendtooverlookthegravitationalforcesthatalsoaffectgroupsofmoleculesatthemacrolevel.Thuswhenamoleculeofwaterisheatedtheintra-molecularspringsvibratehardermakingthemoleculeeffectivelylarger,andthuslessdense.Gravityactsonablobofless-densewatermakingitriseup.Thusonthemacroscaleweseeconvectioncurrentsintheoceanandatmosphere,andconsidertheirconfigurationsassystemsintheirownright.Wehaveahierarchyofsystemsfromtheverysmalltotheverylargeandabehaviorateachlevel.Ihaven’ttimetoreallyfocusonthebehavioroffluidicsystemsinthisbook.2.6EnergyConservationof:Ifthereisnofrictionorotherwaytothrowoffenergythenthetotallevelofenergyinasystemremainsconstant.Thelawaboutconservationofenergyrequiresthatitallstaywithinthesystemalthoughitcanoscillatebetweenitspotentialandkineticformsandmovefromplacetoplaceorparttopart.Real-worldsystemsalmostalwaysshedenergythrufriction,whichturnstoheat,andthenradiatesaway.Gravitywavesalsoshedenergyfromplanetaryorstellarsystems,albeitveryslowly.AlmostallthesimulationsIdiscussinthisbook

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conserveenergysotheycontainthesametotalenergythroughout.Thisisrealisticenoughforourpurposes.Systemasenergystoragedevice:Inmanynaturalsystemsmadewithpartshavingmass–asopposedtosocietalsystems-energyisstoredinthesystemwhenpartsaremovingandthushavekineticenergy,orwhenthespring-likeforcesareeitherstretchedorcompressed.Thisgivesthempotentialenergy.Agrapefruitsittingonthecounterhaspotentialenergybecauseliftingittherestretcheditsgravitationalbondwithearth.Energyindissipativesystems:Systemsareeitherconservativeordissipative.Inaconservativesystemthereisnofrictionorotherwayforenergytobleedoffsothetotalenergyinthesystemremainsconstant.Dissipativesystemsdissipateenergyastheymove.EnergyisaddedtotheLorenzwaterwheelbypouringwaterinatthetop.Itdissipatesorbleedsoffwhenthewaterleaksfromthecupsatalowerlevel.IntheRayleigh-Benardconvectioncellsitsaddedasheatalongthebottomanddissipatedatthecoolertop.AssumingtheenergybleedsoffatthesamerateitsaddedIbelievethatthereisstillacertainfixedamountofenergyinadissipativesystemsandthepartsmustmovesoastoexpressorcontainit.Energychangesformandmoveswithinthesystem:ThebackandforthconversionofenergyfromitspotentialformtoitskineticformiswellrecognizedbutstrangelyenoughI’venotseenthefactitmovesfromplacetoplaceorparttopartmentionedintheliterature.Idiscussthisalotlaterbutfornowsufficeittosaythatinadoublependulumthearmspushandpulloneachother.Thepushingarmgivessomeofitsenergytotheonebeingpushedsoenergymoves,actuallyoscillatesbetweenthedifferentparts.2.7DefiningchaosCasualdefinition:Thecasualdefinitionofchaosisthatutterconfusionexistsandanythingcanhappen.Scientistshaveaverydifferentdefinition.Tothem,chaosisarandomlookingoscillationthatstayswithinboundsandfollowsthelawsofphysics.Scientificdefinition:AccordingtoStrogatzthereisnouniversallyaccepteddefinitionofchaos.howeverheoffersthefollowing.

“Chaosisaperiodiclong-termbehaviorinadeterministicsystemthatexhibitssensitivedependenceoninitialconditions”(Ca2,p.323)

Notetheword“aperiodic”.DuringmyearlyresearchIfellvictimtothinkingthatanybehaviorthat’saperiodicisbydefinitionchaotic,whileignoringhispointthatitalsoneedstoshowSDIC.Bothtestsshouldbeapplied,andI’vesincedoneso.

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HereiswhatWikipediasaysbywayofdefinition:

“Although no universally accepted mathematical definition of chaos exists, a commonly used definition originally formulated by Robert L. Devaney says that, for a dynamical system to be classified as chaotic, it must have these properties:[12]

1. it must be sensitive to initial conditions 2. it must be topologically mixing 3. it must have dense periodic orbits

In some cases, the last two properties in the above have been shown to actually imply sensitivity to initial conditions.[13][14] In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.” https://en.wikipedia.org/wiki/Chaos_theory

Idon’tclearlyunderstandwhat’smeantbytopologicallymixingordenseperiodicorbitsbutIthinktheyrefertophasespaceplotsandthatthedoublependulumandothertoysystemsdiscussedinthisbooksatisfythosecriteria.

Here’sanotherfromMathInsight:

“Adynamicalsystemexhibitschaosifithassolutionsthatappeartobequiterandomandthesolutionsexhibitsensitivedependenceoninitialconditions.”http://mathinsight.org/definition/chaos

OneofthebestplainEnglishdescriptionsofchaosI’vefound.Includesdiscussionofitsdefinition:http://plato.stanford.edu/entries/chaos/#DefChaLyapunovExponent:TheLyapunovexponentisawidelyusedtestforjudgingwhetherasystemischaoticornot.ItbasicallymeasureshowsensitivethesystemistosmalldifferencesininitialconditionsorSDIC.Inotherwordsitmeasureshowfastthewaveformsdiverge.Youcanseethissimplybycomparingthem,buttheLyapunovexponentisatechnicalmeasure.

“TheSystemissaidtobehavechaoticallyiftheLyapunovexponentispositive,whileaLyapunovexponentlessthanorequaltozerodenotesnon-chaoticbehavior.”http://psi.nbi.dk/@psi/wiki/The%20Double%20Pendulum/files/projekt_2013-14_RON_EH_BTN.pdfBelowisaplotofthatexponentforthedoublependulum.

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Accordingtothistestthedoublependulumbecomeschaoticwhenthependulumisreleasedwitha1=a2=0.7radians(40degrees).Thiswasapparentlycomputedforadoublependulumwithsolidarmswhereasallmysimulationsusedpointmasses,whichIcallbobs.Itsnotclearthatthesameanglesthatmarkthethresholdtochaoswithsolidarmsapplytopointmassbobs.Iassumedtheydidn’t.OneproblemwithusingtheLyapunovexponentisthatonemustknowtheequationsofmotionforthesystemandenoughmathtocomputeit.TheSDICtest:Itstrueandveryeasytodemonstrate–assumingyouhavetherightcomputermodels-thattoysystemexhibitsensitivedependenceoninitialconditionsorSDIC.SDICmeansthatasmalldifferenceintheinitialconditionofachaoticsystemwillmagnifyovertimeandmakeaverylargedifferenceinitsfuturecondition.Someexperimentershavebuilttwoidenticaldoublependulumsandreleasedthematthesametimealbeitatslightlydifferentanglestoseehowthewaveformsdiverge.Othershavedoneessentiallythesamethingusingcomputermodels.Thefirstfigurebelowshowsthewaveforms–generatedbyacomputermodel-ofthreedoublependulumsreleasedatalmostthesamelow10degreeangle.Theydifferedonlyby0.01radians(onehalfdegree).Thewaveforeachpendulumisplottedinred,blueorgreen.With10degreereleasetheyhadverylittleenergyinthisrun.Eachblockshowsadifferentvariable.Forinstancetheupperleftplotsthe

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angleoftheupperarm.It’sabithardtoseebuttheblue,redandgreenwaveformsdifferedlittleinthissimulation.Atfirstglancetheyseemperiodicbutcloserinspectionshowstheheightsofthepeaksareslowlydrifting.Myexperimentsshowedthesamethingatlowenergy.Namelythisisnotperfectlyperiodicbehavior,ratheritsquasi-periodic.

Theimagebelowshowshowthered,greenandbluewaveformsdivergethenthearmsarereleasedatamuchhigher105degreeanglegivingthesystemconsiderablymoreenergyandmakingitchaotic.Thesewaveformsshowwhythelong-termbehaviorofchaoticsystemscan’tbepredicted.Whichlineisright,theredone,theblueone,orthegreenone?Ontheotherhandthethreelineslieclosetighterforawhileindicatingthatshort-termforecastsarepossiblewithagoodmodel.Thisisthereasonitsrelativelyeasyto

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predictwhetheritwillrainthreedaysfromnow,butnot10ormoredaysout.Wecanofcourseciteprobabilitiesbasedonhistoricrecords.

Eyeballtest:UnfortunatelyIdidn’thaveaccesstoamodelthatcouldplotwaveformsfortwoormoredoublependulumsatthesametimesoIwasunabletomakeplotslikethoseaboveandusethemtosayifaparticularrunwaschaoticornot.Iwashoweverabletomaketwoseparaterunswithslightlydifferentinitialconditionsandcomparethemvisually.Iusedthattechniqueextensivelyinmyanalysisofthedoublependulum.DetailsappearinChapters7,8and9.Systemsbehaviorincludingchaoticbehaviorisdeterministic.Thismeansthatthewaythesystemistodayisthesoledeterminantofhowitwillevolveinthefuture.Startingwithitscurrentsituationthewayitwillchangeorevolvewillobeysthelawsofphysics.Nothinginthebehaviorofthesesystemshappensbychanceor

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luck.Waveformsdoappeartovaryrandomly,andtheyareoftendescribedthatway,butitsjustphysicsdoingitsthing.2.8VideosandreferencesaboutchaosAnice4-minuteintroductiontochaosbyJamesGleichisfoundat:http://www.clausewitz.com/mobile/chaosdemos.htm#DblPendCompellinglabdemousingrubbersheettoshowsolarsystemdynamics:https://www.youtube.com/watch?v=MTY1Kje0yLgSophisticateddiscussioninplainEnglishaboutthedefinitionofchaoswithextensivereferencesat:http://plato.stanford.edu/entries/chaos/…Areadablediscussionofchaosincludeddoffingoscillator.Phasespace,etc.:https://books.google.com/books?id=l2E4ciBQ9qEC&pg=PA117&lpg=PA117&dq=chaos+spring+mass&source=bl&ots=7HJuhOrR4X&sig=n2i2PQJO5HLIQhbDdTFLDz0ko1M&hl=en&sa=X&ved=0ahUKEwiXyoyWtdfJAhUQ0mMKHV7zBd4Q6AEINDAD#v=onepage&q=chaos%20spring%20mass&f=falseThisisgoodonclimateandchaos:http://www.realclimate.org/index.php/archives/2005/11/chaos-and-climate/andsoisthis:https://www.aip.org/history/climate/chaos.htm2.9ToolsandtechniquesforanalysisTherearesimplemechanicalandelectricalsystemswhosereal-worldbehaviorcanbeobserved,andwatchingthemisinstructive.Howevercomputersimulationmodelsarefarmoreusefulsinceonecanaccuratelyvarycertainparametersandmakeplotsshowinghowthevariableschangeinresponse.SeveralJavamodelseasilyobtainedonthewebhavebeenusedbytheauthortogreatbenefit.AmodelofthedoublependulumcreatedbyDr.Doolinghasbeenespeciallyhelpful.http://www2.uncp.edu/home/dooling/applets/double_pen.files/tom/models/doublepen.html.Orhttp://www2.uncp.edu/home/dooling/applets/double_pen.files/tom/models/doublepen.htmlWithmodelsonecanplothowvariableschangesovertimebutsometimesitsmoreinstructivetoshowhowonevariableischangingrelativetohowanother.IftherearetwovariablesthevalueofoneisplottedontheXaxisandtheotherontheYaxissowegeta2-dimensionalplot.Threevariablesproducea3-Dplot.Thesearecalled“phasespace”portraitsorplots.Iftherearemorethanthreevariablesthephase

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spaceplotwouldrequire4ormoredimensions.That’simpossible;howeverevenplottingtwoorthreeisoftensufficient.Itsofteneasiertoseehowperiodic,quasi-periodicorchaoticasystemisbymakingphasespaceplotsthatitisbylookingatthewaveforms.Thevalueofallthevariablesatanyinstantshowsupasadotinphasespacebutasthevariableschangethedotmovesleavingatraceortrajectorybehindit.Ifthedottracesthesamepatterntimeaftertimethesystemisoscillatingperiodically.Ifnotitseitherquasi-periodicorchaotic.Ihighlyrecommendspendingsometimewiththesesimulationmodelsasnothinggivesafeelforasystemsbehaviorbetterthanwatchingitinaction.TheyareoftenwritteninJavaandeasilydownloadedfromthewebaftergettingyoursecuritysettingstoacceptthem.ThescreenshotbelowshowstheuserinterfacetoDr.Dooling’smodelofthedoublependulum.Iuseditextensively.Userscanadjustanyoftheparameters.Whenbothbobshangmotionlessandstraightdownthereiszeroenergyinthesystem.Beforeinitiatingaruntheboborbobsareliftedtosomeangleinordertoinsertpotentialenergyintothesystem.UsuallyIjustvariedangle1toseehowthesystembehavedatdifferentenergylevels.Thesescreenshotsshowtheuserinterfaceandsomeoftheplotsthatcanbegenerated.Theupperleftdiagramshowsthependulumbobsandthelastfewsecondsofthetraceleftbehindthemastheymove.Inthiscasethelowerleftplotshowsawaveformdepictinghowthekineticenergyofthebluebobchangedovertime.Inthisruntheredbobwasreleasedhighsothesystemhadalargeamountofenergy.Thismadeitchaoticandenabledthebluebobtoswingoverthetoporspinonoccasion.Hereitsjustgoneoverthetop.Thewaveformishighlyirregularbutwehaven’twatcheditlongenoughorseenenoughotherdatayettoconcludewhetheritsperiodic,quasi-periodicorchaotic.

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Usuallyitseasiertodetermineifasystemisbehavingperfectlyperiodically,quasi-periodically,orchaoticallybyplottingonevariableagainstanotherasshowninthescreenshotsbelow.Thisproducesa2-Dviewintowhatisreallya4-D“phasespaceportrait”becausethedoublependulumhasfourvariablesPE1,PE2,KE1andKE2.Thefirstscreenshotbelowthe2-Dorpartialphasespaceplotforasystemthatwasoscillatingperfectlyperiodicallyorveryclosetosame.Thetracegoesroundandroundretracingexactlythesamepatterntimeaftertime,oneatoptheother.Givenwherealltheotherparameters–linebobmassandarmlength-weresetthereleaseanglea1hadtobeexactlyrighttogetperfectlyperiodicoperation.Inthiscaseit

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was51.44degrees.Thisvaluewasfoundbytrialanderrorstartingwitharunthatjustseemedclose.

Belowweshowwhatquasi-periodicbehaviorlookslike.Thetracemakesthesamegeneralpatternoverandover–itwasstoppedforthisscreenshotafterfourtimes-butthevaluesareslightlydifferenteachtimesothepatternsareoffset.Theselineswillcontinuetodriftuntiltheyfill-intheentireenvelope.Thesizeoftheenvelopeislimitedbytheenergyinthesystem.Inthiscaseangle1canneverexceedthe51degreesitwasreleasedat.Notethatanglea1wasalmostthesameastheanglethatmadetherunaboveperfectlyperiodic.Theperfectlyperiodicrunwasfoundbyfinetuningangle1bytrialanderror.

Thefollowingthreescreenshotsfromanotherrunshowhowthevaluesdriftovertimeduringquasi-periodicoperation.Thefirstwastakenabout13secondsintotherun,thesecondwastakenafter150secondsandthelastafter300seconds.Ijudgedthisasquasi-periodicbecauseitwasobviousthatthegeneralpatternofbehaviorrepeatedtimeaftertimebutsincethepatternsdidnotoverlypreciselythiswasnot

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perfectlyperiodicbehavior.Othermorecasualobservershaveapparentlycalledthisbehavior“periodic”,whichisalooseinterpretationofthatterm.

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Iftheaboverunhadrunlongenoughthetracewouldhaveprobablyfilledintheentireenvelopeasitdidinotherquasi-periodicruns.Theredlinesshowtherangeoverwhichvariablea1changed.Iusuallysaythevaluesorpatternsdrift.Itsobviousthatsomeamountofpredictionispossiblewhenthesystemisoperatingquasi-periodically.Belowisanexampleofchaoticbehavior.Thetracefollowsarandomroute.Thereisnoconsistentpattern.Ifletrunlongertheselineswouldalsofill-intheentireenvelope.

Youshouldnowhavesomeideahowthesetoolsareappliedtobetterunderstandthebehaviorofthissystem.Wediscusstheusefulinsightslater.

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Ifonevariableinasystemischangingitmeansalltheothersarechanginginsimilarmanner.Inotherwordswecouldplotangle1,angle2,ke1orke2andtheywouldallhavesimilarwaveformsovertime.Itshandytoknowthatwhenviewingsimulations.Havingsaidthat,plottingtwovariablesinapartialphasespaceplotmakesiteasiertodetermineifasystemisperfectlyperiodic,quasi-periodicorchaotic.Thestageisnowsettohaveacloserlook.AftershowingpicturesorlinkingtovideosofsomedynamicsystemsinChapter3,Chapters4and5willfollowbylistingwhatIfeelarethemostimportantfindingsaboutsystemsbehaviorthatI’vebecomeawareof.Eachfindingorconclusionwillincludeplots,screenshots,orotherdatatosupportitsoitsnotjustspeculation.Chapters6,7,8,9and10describeindetailsimulationrunsI’vemadeinordertodiscoverhowthedoublependulumandmagneticpendulumbehave.Chapters11and12attempttorelatewhatweknowabouttoysystemstoseverallargereal-worldsystems.ExpertslikeStrogatzwritethattheapplicationofchaostheorytocomplexsystemsislargely“unexploredterritory”.(Ca2,p.10)Makingthatjumpinvolvessomespeculationonmypart.That’spartlybecauseIhaven’tfoundsimulationmodelstohelpbridgethegap.Inotherwordsmodelsthathavemorepartsthanthesmalltoysystemsstudiedsofarand/oraredesignedforseriousanalysis.Forinstancethespring/masssimulationsI’vefoundwithabout6partsdon’tproduceadequatewaveformandphasespaceplotsand/ortheydon’tallowinitialconditionstobesetaccurately.WhatIthinkwouldbemosthelpfulisaspring/massmodelthatcouldhandleuptosay20massesallconnectedbysprings.Theusershouldbeabletoselectthenumberofmasses,insertspringswhereverhechoses,makethespringslinearornon-linear,producealltherequisiteplots,beabletoplottheforceonanymass,tracktheenergyofeachmass,beabletotestforSDIC,etc.InChapter11I’vedescribedsuchamodelinmoredetail.