3.0 examples of chaotic systems - richard c. harkness · 3.0 examples of chaotic systems outline...

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3.0ExamplesofchaoticsystemsOutline

3.1Doublependulum3.2MagneticPendulum3.3Lorenzwaterwheel3.4Planets3.5Spring/masssystems3.6Molecules3.7Thermo-syphon3.8Rayleigh-Benardconvection3.9Convectioninearth’smantle3.10Otherexamples

Thischapterdescribessimplereal-worldsystemsthatexhibitvariousformsofbehaviorincludingchaotic.Theyaremostinterestingwhenchaoticsovideosonthewebalmostalwaysshowthemthatway.Thetechnicalliterature,whichexaminesthemathbehindthem,doesnotseemtogiveequalattentiontoeach.TheLorenzequationsandtheLorenzwaterwheel-saidtobehavelikethem-areextensivelycoveredintextbooksandarticles.Thereisremarkablylittleonthedoublependulumandmagneticpendulum.I’vefocusedonthedoublependulumbecauseit’sintuitivelyeasiertounderstandandthere’saniceJavamodelforexploringit.Althoughseveralsystemsaredescribedbelowtheirbehaviorissimilarinfundamentalways.Alloscillate.Mostcanoscillatechaotically.Therearecommonorgenericaspectsoftheirbehavior.Explanationsandvideosofthesearenumerousontheweb.Somegoodonesarecited.BeforegoingfurtherIrecommendthesesophisticatedvideosaboutchaos.Theygiveabroadsemi-technicalfeelforthecomplexityofthissubject.Thefirstshowsalargenumberofbilliardballsallapparentlymovingchaoticallyandtakingdifferentpathsifhitslightlydifferently.Italsoshowsthreeplanetsinmotionatt=10.:https://www.youtube.com/watch?v=c0gDLEHbYCkThesecondissimilarandalsoworthwatching.https://www.youtube.com/watch?v=_xfi0NwoqX83.1DoublependulumThedoublependulumisoneofthesimplestsystemsthatexhibitsvariousformsofbehaviorincludingchaos.It’sextensivelyanalyzedinthisbookbyusingacomputersimulation.

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Theauthor’shome-madedoublependulum,isshownbelow.

ThebestlivedemoofadoublependulumI’vefoundwasmadebyStrogatzanotedexpertinchaostheory.Itsfoundhere:https://www.youtube.com/watch?v=anwl6OZ1UuQ&ebc=ANyPxKpVTVQJGdaxEk1AvyBgCXkKdrV2SKjzJxpQGeBiMpxM3F4p8oCYU4_XDCVkTD1h_nlkssunljJ-FHl56FRZdGc1wGEPWgItnotonlyshowsthependuluminmotionbutalsodemonstratesits“sensitivedependenceoninitialconditions”orSDIC.Thesevideosshowprecisiondoublependulumsthatweresetinmotionwithaveryenergeticpushthatinsertedaconsiderableamountofenergyintothesystem.Whenbotharmsarehighandmovingslowlymostofthisenergyispotentialenergy,butastheydropPEconvertstokineticenergymakingthearmsspinfaster.https://www.youtube.com/watch?v=z3W5aw-VKKAandhttps://www.youtube.com/watch?annotation_id=annotation_632684&feature=iv&src_vid=z3W5aw-VKKA&v=vjVQWG7xUQkWhattolookfor:Theeasiestaspectsofchaoticbehaviortoobservebywatchingthependuluminactionarereversalsinthesmallpendulumsdirectionofrotation,orit’stransitionsbetweenswingingandrotating.Otherparameterssuchasangles,elevation,andratesofrotationarealsochangingbuttheyhappensofasttheyarenotreadilyobserved.Thekeypointisthatallthesechangesinbehaviorhappeninaseeminglyrandommannerwhenthesystemoperateschaotically.Sometimesthesmallpendulumrotatesonceortwicetimesbeforestoppingorchangingdirection.othertimesitrotateslonger.Theintervalsbetweenitgoingoverthetopappearrandom.Whenwatchingthedoublependulumonefeelsthateacharmistryingtodoitsownthing,namelyswinglikeasimplependulumfollowingitsownnaturalrhythm.Unfortunatelytheotherarmisn’tquiteinsyncandcontinuestodisturbthatrhythm.Sometimes,likewell-timedpushesonachild’sswing,armAboostswhatArmBistryingtodonaturally.Othertimesthepushesorpullsareoutofsyncandtheforcesfighteachother.Often–anditsbestobservedwhentheupperarmisswingingback

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andforth-itlooksliketheinnerarmisslingingorwhippingtheouterarmaround.Thuswetendtoseeashortseriesoforderlybackandforthswingswheretheouterarmswingsabitmoreeachtimeuntilitfinallygainsenoughenergytomakeacompleterevolution.Thereisafeelingthatenergyissurgingbackandforth(itis)insomewavelikemanner.Whenrunningatfairlylowenergynotehowtheinnerarmisoftenhangingnearlystraightdownandismotionless–andthuscontainslittleenergy-whentheouterarmswingsoverthetop.Thisindicatestheinnerarmhastransferreditsenergytotheouterarm.Itshardtoseebuttheouterarmspinsfastestwhentheinnerarmislowandalmoststill,meaningitstransferredmostitsenergytotheouterarm.Computersimulationsusepoint-massbobsratherthanmetalarms.Thetrailsleftbythelowerorouterboboftenlooklikethis:

Theimagebelowshowshowthespeedofthetwobobsvariesovertime..Notehowirregularthesewaveformsare.Thesystemwasverylikelychaoticinthisrun.

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Itsusefultolookatlong-termwaveformsaswellasshort-termoneslikeabovebecauseitmaytakearelativelylongtimeforapatternofbehaviortorepeatandallowustoconcludethesystemisperiodic.Orshowitisn’t.Figure28showstwolong-termwaveformsfromoneoftheauthor’ssimulations.Theyrecordhowtheamountofkineticenergyinthelowerbobvariesovertime.KEisofcourseproportionaltospeedsoaplotofspeedwouldlookthesame.Theupperplotwasmadebyreleasingthearmsatalowanglethusimpartingarelativelylowamountofenergyintothesystem.Obviouslythereisaregularrepetitivepatterntothisbehavior.Thefirstpeakatt=10isseenagainatt=30andt=50showingthatthegeneralbehaviorisperiodicwithaperiodofabout20seconds.Anysmalldifferencestotheheightsofthedifferentpeaksarenotobvious,althoughtheyexist.Thelowerwaveformreflectsstronglychaoticbehavior.Calmperiodsarerandomlyinterruptedbyhighspikes.Thisisatypicalchaoticwaveform.Ifothersystemsbehavethiswayitsaysweshouldn’tgetcomplacentduringperiodsofcalmbecausetheymightsuddenlyendwithunpredictableandviolentspikes.

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ThesimulationmodelIusedcansimulateenergylevelshighenoughtomakethesystemchaoticbutnotreallyhighlevelswherethearmsrotaterapidlylikeanairplanepropeller.Withaviolentswingtheauthorcouldprobablyputtenorfiftytimesmoreenergyintohishomemadependulumthanthesimulationmodelcoulddemonstrate.Itsnotclearwhetherthiswouldresultinchaoticorperiodicmotion.Thearmsseemedstaymoreorlessalignedandrotaterapidlylikeanairplanepropellerandthusappearedperiodic.Ontheotherhandtheymayhavewobbledabitandthusbeenchaotic.Clearlyafterslowingabittheseeminglysmoothrotationbrokeupandthearmsbecameviolentlychaotic.

Fig28Regularversuschao2cwaveforms

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SDIC:Whentwoidenticaldoublependulumsarereleasedfromalmostbutnotexactlythesamepositiontheirbehaviorstartsoutbeingnearlyidenticalbutovertimeitdivergesso-atanygiveninstant-thearmsareinverydifferentpositions.Thisshowsupclearlyinthesevideos:https://www.youtube.com/watch?v=MtJLhb9yaPcandhttps://www.youtube.com/watch?v=LfgA2Auyo1AIhavealotmoretosayaboutitsSDICinChapters7,8,and9.Links:Otherniceanimationsofdoublependulum:http://www.clausewitz.com/mobile/chaosdemos.htm#3Bodyhttps://search.yahoo.com/yhs/search?p=double+pendulum&ei=UTF-8&hspart=mozilla&hsimp=yhs-001https://www.youtube.com/watch?v=QXf95_EKS6EThesesitesalsoshowdoublependulumsinaction:https://www.youtube.com/watch?v=zdW6nTNWbkcThisisaprecisionbuiltversionwithverygoodbearings.http://www.clausewitz.com/mobile/chaosdemos.htm#DblPend(labdemobyStrogatz)https://www.youtube.com/watch?v=U39RMUzCjiUhttps://www.youtube.com/watch?v=QXf95_EKS6Ehttps://www.youtube.com/watch?v=_i3WqnejOQkhttps://www.youtube.com/watch?v=WMPOvmozGMcThisversionhasthreearms.http://scienceworld.wolfram.com/physics/DoublePendulum.html fine animation or trace of the bobs http://groups.physics.northwestern.edu/vpl/mechanics/pendulum.htmlJavasimmodelfordoublependulumbutwithfewcontrolsorplotssocan’treplicatewhatDoolingmodelwilldo.Foramusementwatchthevideoatthissitecalled“complexadaptivesystems”.Yes,Ithasnothingtodowiththedoublependulumbutshowsanothertypeofsystemsbehaviorthatisrelatedtofishschooling,birdsflockingandothersuchself-assemblingsystems.http://www.clausewitz.com/mobile/chaosdemos.htm#DblPendThis“pendulumwave”toyexhibitscomplexchangesinbehaviorasfrictionslowlyslowsitsoscillations.Sometimestheballsself-organizeintoniceshapes,atothertimestheydon’t.Anotherexampleofhowcomplexdynamicbehaviorcanbe.See:https://www.youtube.com/watch?v=yVkdfJ9PkRQ

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3.2MagneticPendulumTheauthor’smagneticpendulumisshownbelow.Thebasecontainsthreestrongmagnetsandthere’sasteelnutontheendofthependulum.Thenutisattractedtoeachmagnetbutalsotothecenterpositionbyearth’sgravity.Itsa5bodysystemwhereeachbodyinfluenceseachoftheothersviathegravitationalormagnetforcesbetweenthem.The5bodiesarethebob,earth,andthethreemagnets.

Asimulationmodelwasusedbytheauthortoplotthetraceleftbythependulum.Itcanmodelthreependulumsandsuperimposethetrailleftbyeach.Itfoundat:http://demonstrations.wolfram.com/PendulumWithThreeMagnets/(needinstallCDFPlayer,then“downloaddemoatCDF”andopenwithCDFplayer.Click+nexttotimeslidertogetstart,pauseandstopcontrols)ThescreenshotbelowshowsthetrajectoriesofthreebobsreleasedfromconsiderablydifferentheightssotheystartwithverydifferentamountsofgravitationalPE.TheydifferinmagneticPEaswell.Eachtrajectorystartswherethebobwasreleased.Thepatternofmovementdependsgreatlyontheheightofreleaseorinitialpotentialenergy.Red,releasedclosetoamagnet,hadlittleGPEsoitcouldn’tescapethemagnetic“well”arounditsmagnet.Nonethelessthedistancebetweenitandthemagnetappearstovaryirregularlybecauseitsorbitwasalteredbytheothermagnets.Arguablyitsmovingchaotically.Blueandgreenhaveenoughenergy(inertia)toblastthroughthemagneticwells,whichcontinuallydistorttheirpath.Theymovefromonemagnettoanotherinrandomandprobablychaoticmanner.Sincegreenhasmoreenergyorspeeditscourseislessbentbythemagnetsitpasses.

Itseasytoseehowthesinuouspathfollowedbythependulumbobwouldproducewaveformsforthevariousvariablesthatoscillateupanddownreachinghighsandlowsofdifferentmagnitudeinrandommanner.Thevariablesmightincludespeed,

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angleofthearm,distanceofbobfromthecenter,ordistancebetweenthebobandsomemagnet.

ThemagnetpendulumcomputermodelsIfounddidnotproducewaveformsorphasespaceplots,nordidIfindanyintheliterature.Therearenumerousvideosonwebshowingasystemlikethisinmotion:https://www.youtube.com/watch?v=vFdZ9t4Y5hQhttps://www.youtube.com/watch?v=uIZG2MyEu0Ahttps://www.youtube.com/watch?v=hTe6tRPROjI

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https://www.youtube.com/watch?v=Qe5Enm96MFQ(gotomidpointofvideo)3.3LorenzwaterwheelTheLorenzwaterwheelisanotherclassicexampleofasimplesystemthatbehaveschaotically.Itoperatesperiodically–thatisrotatesinonedirectionataconstantspeedwhenthewaterflowintothewheelisfairlyslow.Athigherflowratesitbehaveschaotically.Itchangesspeed,oscillatesbackandforthfromCWtoCCWrotation,andthenreversesdirectioninaseeminglyrandommanner.Atstillhigherinflowratesitrevertstoconstantunidirectionalrotation.Itsbehaviorhasbeenextensivelystudiedinamathematicalsensebuttherootcauseofchaosinthissystemhasneverintuitivelyexplainedtotheauthor’sknowledge.Therearestatementssayingthingslike:thedifferenceinweightofthewaterontheleftversusrightsideofthewheelexertsatorque,whichwilleithersloworincreaseitsrotationrate.AtanygiveninstanttheLorenzwaterwheelisessentiallyapendulumbecausetheweightofwaterononesidediffersfromthatontheothersidethusformingavirtualpendulumbobwhichwantstoswingtoalowerposition.Itscomplexbecausethevirtualbobismovingaroundtherim,includingswitchingfromonesidetotheotherinawaythat’sverydifficulttovisualize.Itsmassisalsochanging.Thewaterwheelisaso-calleddissipativesystembecauseenergyislostduringitsoperation,bothbywatertakingPEwithitwhenitdripsout,andbyfrictioninthebearingsand/orapartly-appliedbrake.HowevernewwaterinjectedatthetopprovidesacontinuinginputofnewPE.Thetwophotosbelowshowsahome-madeversions.

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Watchthis.It’savideoofalaboratoryversioninoperation,alongwithexcellentexplanationbyStrogatz.Goto:https://www.youtube.com/watch?v=7iNCfNBEJHoThepicbelowofanotherlabwaterwheelistakenfromathesisbyRachelFordice:http://www.reed.edu/physics/faculty/illing/campus/pdf/RachelThesis09.pdfThisthesisincludesagood,easytoread,introtochaosandthewaterwheel.

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Chaoticoscillation:Theimagebelowshowshowthespeedofthewaterwheelchangeswhenitoperateschaotically.Plusvaluesshowspeedinonedirection.Minusvaluesshowspeedintheoppositedirection.Notetherandomreversalsandtherandomnumberofevermoreintenseoscillationsbetweenthem.Itotherwordsthewaterwheelgotcloserandclosertostoppingoneachcycleuntilitreversed.ThisimageisfromathesisbyFordycefoundat:http://www.reed.edu/physics/faculty/illing/campus/pdf/RachelThesis09.pdfThisthesisinvolvedbuildingarealwaterwheelinthelabandseeinghowcloselyitreplicatedtheLorenzequations.ItalsodoesanexcellentjobdevelopingthemathbehindthewaterwheelandmappingtheLorenzequationstothewaterwheelequations.

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ThebehavioroftheLorenzwaterwheelisdescribedbythefamousLorenzequations.Inthemjustthreevariablesareneededtodescribethestatusofthesystem.Thediagrambelowshowshowthesevariableschangeoroscillateovertime.

Thesystemwaschaoticwhenthefollowingchartwasmade.(Source:Ca2,p318.)Notethattheoscillationsgrewsteadilymoreintenseuntilthetracecrossedthecenterlinefromminusvaluestoplusvalues.Crossingthatlinemeantthewater-wheelreverseddirection.

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Thesmalloscillationsinthisplotrepresentrelativelygentlyrocking.Sointhissystemitappearsthatrockingoscillationsbuildsteadilyinintensityuntilsomethingdramatic–inthiscaseachangeinrotationdirection-happens.Thatgrowingoscillationgiveswarningthatthedramaticeventisabouttohappen.Incontrastthewaveformsforthedoublependulumwaveform(SeeFigure28)andthespring/massmolecule–shownlater-shownoobvioussequenceofever-growingoscillationsprecedingthedramaticevents.It’shardtoderiveausefulmessagefromtheseconflictingexamplesofchaoticwaveforms.Inonesystembehaviorseemssomewhatpredictable,intheothersitdoesn’t.It’salsoanexamplewhyitssohardtoidentifygenericbehaviorsthatarecommontoalldynamicsystems.Lorenzbutterfly:Sincetherearejustthreeoscillatingvariablestheycanbeplottedusinga3-Dphasespacediagramasshownbelow.Whenthesystemischaotictheplotlookslikeabutterfly.

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This“Lorenzbutterfly”isperhapsthemosticonicimageofchaos.Itappearsinvirtuallyeverypopulararticleorbookaboutchaosplusmosttextbooks.Technicallyitscalledastrangeattractor.Itssaidthelinescanneverintersectbutratheranynewtracemustfitbetweentheolderones.That’spossiblebecausethereareaninfinitenumberofnumbersbetweensayzeroandone.Expertsgetintofractalswhentreatingthissubject,butthatcomplexityisoutofscopeforourpurposes.Whenthewaterinflowtothewaterwheelislowitdoesn’tnecessarityoscillatechaotically.Afewgoodcomputersimulationsshowingthemovingdottraceoutthebutterflyarefoundat:

ThisisoneofthebetteruTubesshowingthemovingdottracingthebutterflyanditalsoshowshowthisrelatestothewaveforms:https://www.youtube.com/watch?v=6i57udsPKmsHerearesomeothers:https://www.youtube.com/watch?v=8z_tSVeEFTAhttps://www.youtube.com/watch?v=gOFrT-DGStIhttps://www.youtube.com/watch?v=TfnKzaGguaEThis utube shows the three waveforms and the attractor all developing at the same time. It also shows SDIC as the waveforms that initially lie atop each other later diverge. https://www.youtube.com/watch?v=6I-sa4mSkgo

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Thevariables–asshowinthebutterfly-interactinacoordinatedmannerandtheymustatalltimesbesomewherealongtheline,alsocalledatrajectoryortrace.Whenthesystemischaoticthemovingdotcirclesaroundafewtimesinonewingthentraversesovertocircleintheother.Whenthetraceswitchesfromonewingtotheotheritindicatesthatthewaterwheelhaschangeddirection.Thisbehaviordoesnotappearverychaotic(althoughitactuallyis)sincethetracegoesroundandroundinanorderlyandpredictablemanner.What’srandomiswhenitsuddenlyleavesonewingforanother.Thathappensonrandomintervals.Anothersignofchaosisthatthevariablesneverreturntothesameexactvaluestheyhadbefore.Inotherwordsthelinesinthebutterflynevercross.Eachnewlinemustfitbetweenpriorlines.Expertsexplainthisintermsoffractals.I’llomitthat.It’swellcoveredelsewhereandoutofscopeforourpurposes.

Non-chaoticoscillation:Thesystemhasvarioustypesofbehaviordependingonhowvariousparametersareset.AccordingtoStrogatzthosebehaviorscanvaryinaverycomplexmanner.(Ca2,p.330+)ThefollowingfourimagesshowhowthebehavioroftheLorenzequationandtheactualwaterwheelchangesasthe“drivingforce”isincreased.Ibelievethe“drivingforce”isthedifferencebetweentheratewaterisaddedatthetopandtherateitdripsout,aswellasfrictionappliedbyabrake.IneithercaseIbelieveitsproportionaltothetotalenergyinthesystem.Thefirstimageshowsthesystemoscillatingperfectlyperiodicallysuchthateachcrestinthewaveformhasthesameheight.Thismeansthewheelisrockingbackandforthanequalangleeachtime.Thisiscalledperiod-1oscillationanditsobviousbecausethetracefollowsasinglelinetimeaftertime.Whenthedrivingforcereachesacriticalvaluethesinglelinebeginsratherquicklytoseparateintotwo,meaningthateveryotherpeakinthewaveformhasexactlythesameheight.Thisisperiod-2oscillation.Thethirdimageshowsperiod-4oscillation.Thefinalimageshowsthesystemoperatingchaotically.That’sobvioussinceovertimethepatternfillsintheentireavailablearea.Notwopeaksareeverexactlythesameheight,althoughtheymaycomeveryclose.Notwopatternsareeverexactlythesame.ThewaveformofanyvariablelikeXisaperiodic.Theseimagesweremadebytheauthorusingasimulationmodelat:http://amath.colorado.edu/faculty/juanga/3DAttractors.html

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SDIC:Theimagebelow,alsofromtheFordycethesis,showshowsensitivethedownstreamwaveformistoverysmalldifferencesintheinitialconditionswhenthesystemischaotic.Thegreentracewasmadewithoneoftheinitialconditionsatsomespecificvalue.Thebluetraceresultedwhenthatconditionwaschangedjustatinybit.(Seewordingbelowthecharttoseehowmuch)Source:http://www.reed.edu/physics/faculty/illing/campus/pdf/RachelThesis09.pdfIfthiswerea20-dayweatherforecastoftemperatureforsayKansascityitsobviousthatasmallerrorwhenprogrammingtheforecastmodelwouldnotaffecttheaccuracyofthenear-termforecastbutthedifferentwaveformsonrightshowitwouldgreatlyaffectthelong-termforecast.Wewouldn’tknowwhichwasright,thebluelineorthegreenline.

Otherlinks:This informative video relates the Lorenz attractor to the weather: https://www.youtube.com/watch?v=SlwEt5QhAGY http://nullprogram.com/ChaosWheel/ shows amount of water in each cup buts that’s all ThefollowinguTubevideosshowLorenzwaterwheelsinoperation.https://www.youtube.com/watch?v=zhOBibeW5J0good,hasplasticcupshttps://www.youtube.com/watch?v=7A_rl-DAmUEbetter,bikewheel

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http://www.siliconsolar.com/building-a-lorenz-water-wheel/https://www.youtube.com/watch?v=SlwEt5QhAGYSophisticatedvideolinkinganimationofwaterwheelandLorenzequationstoweather.AlsodemonstratesSDIC.Herearesomesimulationmodelsyoucanexperimentwith:http://www.aw-bc.com/ide/idefiles/media/JavaTools/lrnzphsp.html shows circulation, has phase space plot and waveforms over a few seconds.: http://amath.colorado.edu/faculty/juanga/3DAttractors.htmlDoesn’thaveadequatecontrolsorplotsThisisaneatJavasimulationofwaterwheel.Youcanchangetheparameters.Accepttherisktostartit,thenitrunsfine:http://people.web.psi.ch/gassmann/waterwheel/Wwheel1.HTMLhttp://demonstrations.wolfram.com/LorenzsWaterWheel/ lacks adequate controls and plots. (wouldn’t run smoothly) http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html this applet plots the butterfly with two traces to demo SDIC. Treats equation only, not waterwheel. 3.4PlanetsAlthoughtheplanetsinoursolarsystemhavealmostcircularorbitstheydointeractgravitationallyanddisturbeachothersorbitsjustabit.Itappearsthesolarsystemmaybechaotic.

“Chaosisubiquitousincelestialmechanics.Inthesolarsystem,chaoticinteractionswiththeplanetsshapethestructureoftheminorbodies,includingtheasteroidbeltandtheKuiperbelt(Lecar2001)”http://w.astro.berkeley.edu/~echiang/students/thesis.pdf

AbookbyPetersoncalledChaosintheSolarSystemseemstohavethesameopinionalthoughIcouldn’tfindwhereitsaidsoexplicitly.(Ca3)Thebestintroductiontochaosinplanetarybehavioristoviewtwovideosshowinghowthreegravitationallyattractingbodiesmoveinspace.See:https://www.youtube.com/watch?v=VX9IdCnNWJIandhttps://vimeo.com/11993047The3-Bodyproblem:Mathematicianshavestudiedthecomplexandnotfullyunderstooddynamicsofsimplelittle3-bodysystems.Apparentlytheyhaveconcludedthatallsuchinteractionsinvolving3ormorebodiesarechaotic.Thefollowingquotesattesttothis:

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“Second,ingeneralforN>2,theN-bodyproblemischaotic”https://en.wikipedia.org/wiki/N-body_problem“thethree-bodyproblemgenerallybecomesanalyticallyunsolvable,thatis,thereexistnogeneralformulasthatdescribethemotionandpermitthecalculationofpositionsandvelocitiesofthebodiesfromarbitraryinitialconditions.”http://butikov.faculty.ifmo.ru/Projects/Collection.html“SincePoincarétherehasbeenalotofworkonprovingthatsuchchaoticbehavioroccursinthethree-bodyproblem.”http://www.math.umn.edu/~rmoeckel/presentations/PoincareTalk.pdf

Thescreenshotbelowisfromananimationvideoshowingthemotionofasmallplanetaroundtwostars.http://www.clausewitz.com/mobile/chaosdemos.htm#DblPendandhttp://www.clausewitz.com/mobile/chaosdemos.htm#3Body

Undercertainpreciseconditions,wherethebodiesmoveinsynchronization,thissystemcanbehaveperfectlyperiodically.Thistechnicalsitecontainsanimationsofthosespecialsituations:http://www.scholarpedia.org/article/Three_body_problemByimplicationallotherconfigurationsarechaotic,aswouldbetrueforanysystemwithmorethanthreebodies.

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ThisN-bodysimulationvideoshowssmallbodiesmovingerraticallyandcollidingastheyself-assembleintoaplanetorstar.Thisissomewhatunrealisticinthattheyaren’tembeddedinaswirlingdiscofgas.https://en.wikipedia.org/wiki/N-body_simulationDiscussionsofthethreebodyproblemarefoundat:https://en.wikipedia.org/wiki/Three-body_problemandhttp://www.sciencemag.org/news/2013/03/physicists-discover-whopping-13-new-solutions-three-body-problemandhttp://www.ams.org/samplings/feature-column/fcarc-orbits1Formoreonthethree-bodyproblemsee:(Ca7,p.85+)andahostofwebsites.NOTE:Asanon-expertIremainconfusedaboutwhetherallsystemswiththreeormorepartsconnectedbynon-linearforcesarealwayschaotic,orjustchaoticiftheirinternalenergylevelishighenough.Thestatementsaboveseemtosaytheyarealwayschaoticbutdon’tsaysoexplicitly.Ontheotherhandthe3-part(earthplustwobobs)double-pendulumisn’tchaoticatlowenergylevels,andIdon’tthinktheLorenzwaterwheeliseither.Asimulation:Althoughtheplanetsinoursolarsystemareinnearlycircularorbitsnowandbehaveinanalmostperfectlyperiodicmannernowtheywereassembledbycollisionsofplanetesimalsthatweredefinitelynotbehavingsonicelywhenthesolarsystemformed.Theirorbitsweremoreellipticalandtheydisturbedeachothergravitationally,especiallyduringcloseencounters.Indeedchaosisfoundthroughoutthesolarsystem.(ca3)Scientistshaveusedsimulationmodelsrunonsupercomputerstoforecastthemovementsofplanetsoverthenextfewbillionyears.Somerunsshowthatsomemaycollideinabillionyearsorso,especiallyMercurywithEarth.SDICandotheruncertaintiesmakeitimpossibletosayforsure.ThescreenshotbelowisfromarunmadebytheauthorusingasimulationmodelcalledMySolarSystem.https://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.htmlItsbeenmuchusedandappreciatedbytheauthor.This3-bodyrunshowsthecomplexpathstakenbytwosmallbodiesinellipticalorbitsaroundastar.Thetwobodiescontinuallydistorteachothersorbits,especiallyduringcloseencounterswheregravitationalattractionbetweenthemisstrongest.Herebluehasjustbeenejectedfromthesolarsystembyacloseencounterwithred.Thistransferredsomeofredskineticenergytoblue.Ifeachbodywerealoneitwouldoscillateperfectlyperiodicallyaroundthestarbutwhenthetwobodiesattracttheoscillationsbecomecoupledanddisturbeachother,thusleadingtocomplexwaveformsforvariableslikedistancefromthestarandvelocity.Mostrunsofthissimulationproducecollisionsuntilonlyoneplanetisleft.

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Variableslikethespeedofoneplanet,anditsdistancefromthestar,mayvarychaoticallybecausethisisan“N-Bodysystem”whereNis3.

Anentertainingandenlighteningclassdemousingarubbersheettoshowhowplanetsorbitisfoundat:https://www.youtube.com/watch?v=MTY1Kje0yLgThefollowingtwoscreenshotsarealsofromMySolarSystem.https://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.htmlItriedtogettheseplanetstomoreorlessfolloweachotherinanorbitaroundtheircommoncenter.Thefirstscreenshotshowstheirinitialpositions.

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Howeverafterashorttimetheplanetshadfollowedthepathsindicatedbelow.ApparentlythepotentialenergybetweenthethreebodieswashighinitiallybutinthefirstencountersomeofthePEbetweenredandyellowwasusedtothrowthered/yellowpairfurtherawayfromblue.Intheprocessredandyellowgotcloser.WhenbluecamebackforasecondencounterevenmoreofthePEbetweenredandyellowwasusedtoejectbluetotherightandred/yellowtotheleft.Thisbehaviorissomewhatcounterintuitive.Onewouldexpectgravitytopulltheplanetscloserratherthanendupsendingthemapart.Indeedmayconfigurationslikethiscausethemtocollide.However,aftersomeexperimentationwithpositionsandspeed,thisparticularrunproducedsomethingmoreinteresting.BasicallytherewasPEbetweenthethreeplanetsatthebeginning.AftersomeinteractiontwobodiesdidendupcloserthusreducingthePEbetweenthem,butthatPEwasusedtosendbluefurtheraway.Itsnecessarytorealizethatgravityfallsoffwiththesquareofdistancetoseehowthismakessense.Thismotionisarguablychaotic.Virtuallyallthesesimulationsresultincollisionsandorejectionsafterashorttime.Itisveryhardtogetthethreeplanetsintostableorbits.Onecouldenvisionasituationwheretheyalltracedacloverleaflikepatternindefinitelybutapparentlythatsituationisveryunstable,likeaneggonend.

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Verysophisticatedsimulationmodelsofthesolarsystemhavebeenconstructedandrunonsupercomputersinordertopredictthelong-termstabilityofthesolarsystem.Theyshowasmallprobability–about1%-thatMercurywillcollidewithVenusinthedistantfuture,andthiscouldthencauseVenusandEarthtocollide.http://www.scholarpedia.org/article/Stability_of_the_solar_systemThissuggeststhatthesolarsystemhasnotyetreacheditslowestenergy,moststable,configuration.Pleasenotethatwhenoneofthebodiesina3-bodysystemisfarlargerthantheothersitchangesthesituationintowhat’scalleda“restrictedthreebodyproblem.Thisapparentlymakestheorbitsofthesmallbodiesmorepredictablebymaking“simplifications”inthemath.Withsuchsimplificationsthesundominatesbehavioroftheplanetsandkeepsthemfrominteractingaswildlyastheydointhe3-bodysimulationsImadeaboveusingbodiesaboutthesamesize.Sowhat?:TojumpwayaheadinthisstoryandsaywhyIthinkthisstuffmattersIthinkitslikelythatanysystemwith3ormoreinteractingbodiesisalmostalwaysoperatingmoreorlesschaoticallyiftheforceslinkingthebodiesarenon-linear.I

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haven’tseenthissaiddirectlyinthetechnicalliterature,butthenagaintheliteraturedoesn’taddressmanyofthepracticalimplicationsofwhattheoristswriteabout.IsuspectthatN-bodytheoryappliestomanylargesystemsofpracticalimport,forinstanceecological,andeconomicsystems.Theyhavemanypartsthatobviouslyexertforcesoneachother.Perhapsthoseforcesarenon-linearasmanyexpertsseemtobelieve.Ifsothereisatheoreticalbasisforsuggestingthesesystemsareintrinsicallychaotic.3.5Spring/masssystems

3.5.1TheNeumannmoleculesimulatorThebestspringmasssimulationmodelformypurposesisonedevelopedbyErikNeumannandfoundat:http://www.myphysicslab.com/springs/molecule/molecule6-en.htmlItmodelssixmassesconnectedbylinearsprings.It’ssetinmotionbypullingasideanymassusingthemouse.Ifdampingisturnedonthemagnitudeoftheoscillationswilldiedownandthesystemwillcometorestinoneofseveralequilibriumconfigurations.Withnodampingtheoscillationswillcontinueundiminishedindefinitely.Hecallsthisa“molecule”andindeeditdoesapproximatetheinternalbehaviorofone.Butit’sactuallyagenericspring/massmodel.Ifoundthisupdatedmodellateinthegamewhilefinalizingthisbook.Ithinkitnicelydemonstratesthatenergycanmomentarilyconcentrateinoronepartinamulti-partdynamicsystemandtherebydisturbitgreatly.OrshouldIsaystresssomepartsomuchitdoessomethingextreme.Again,Iamthinkinghowthistypeofthingmighthelpexplainextremeweatherevents,revolutions,stockmarketcrashes,andspikesseeninagreatnumberofotherreal-worldsystems.Partoftheuserinterfaceappearsinthescreenshotbelow.Onecancustomizevariousplotsliketheoneatleft.Duringasimulationrunthemovingmassesappearatright.

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Thescreenshotbelowshowshowthevelocityofmass#1intheYdirectionvariedduringarelativelyshorttimeperiod.Itshowednorepetitiveaspectandlookedchaotic.

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Thisisn’tofficiallychaos:Overalongertimespanthewaveformlookedlikethis.

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Itcertainlylookschaoticbecauseitcloselyresemblesthewaveformofthedoublependulumwhenitschaotic.Thekeythingtonoticeisthatthespeedofmass#1occasional,andseeminglyrandomly,spikedinvalue.JustbeforeeachspikeMass#1hadbeendisplacedfarfromitsequilibriumpositionsothespringsattachedtoithadbeenstretchedputtingagreatdealofforceonmass#1andgivingitalargeamountofpotentialenergy.Thespikeoccurswhenthatpotentialenergyhasbeenconvertedtokineticenergyintheformofspeed.

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Accordingtotheliterature,springmasssystemswithlinearspringscan’toscillatechaoticallyandthewaveformsrepresent“superposition”,whichmeansthevariouswavessimplyaddatopeachother,sometimescancelingsometimesreinforcing.Neverthelessthisresultispotentiallyimportantbecauseitshowsthatspikesinbehaviorcanoccurinsystemsthataren’tofficiallychaoticandwhichinvolvelinearforces.Idon’tknowifexpertsinotherfieldshavedeterminedwhethertheforcesintheirsystemsarelinearornot.Thissuggestsitdoesn’tmatter.Theybothseemtoproducewaveformswithoccasionalandseeminglyrandomspikes.NowIreverttorunsmadepreviouslywithanearlierversionofthismodel.Thisscreenshotshowsthemoleculeinoneofitsequilibriumconfigurationspriortoarun.

Whenonemassispulledasideandreleasedthesystembeginstooscillatewithmassesmovingbackandforthupanddowninrandomfashion.Thescreenshotwastakenwhentheyweremovingviolently.Notehowfarthespringstoredhavebeenstretched.Redhasconsiderablepotentialenergyinthisposition.Thestretchedspringswillrapidlyaccelerateitbacktowardtheothers.(Ifredwereacorporationthissuggestsithasresources-energyofsomesort-andtheforcesexertedonitwillcauseittoactinsomemanner.)

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Thewaveformbelowshowshowthespeedofmass#0changedovertime.Expertsassertthatthatspring/masssystemswithlinearspringscan’tbecomechaoticbutthewaveformcertainlylooksthatwayinthisparticularplot.Whetherofficiallychaoticornotthisisarealsystemanditproducesashort-termwaveformsimilartothatproducedbythedoublependulum,aknownchaoticsystem.Takenotethatsometimesmass#0oscillatesmildlyforawhileandthensuddenlymovesviolently.ThisisacharacteristicofgreatimportanceIFitappliesinlargenaturalandsocietalsystems.Thecalmperiodisbracketedbybluelines.

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Thepartialphasespaceplotbelowsupportsthenotionthatthissystemisn’toscillatinginarepeatableorperiodicmanner,althoughthereisnoproofthiscomplexmessypathwouldnoteventuallyduplicatetimeaftertimeifthesimulationranlonger.Thishighlightsanotherissuewhendealingwithchaos.Howlongmustoneobserveasystemtobesureitsbehaviorwillneverrepeat,becauseifiteverdoesitsperiodicnotchaotic.Onecouldarguethatifitdoesn’trepeatinareasonabletimeitschaoticforallpracticalpurposes.Thisishowthepartialphasespaceplotlooks.It’sapartialplotbecauseafullphasespaceplotmusthaveasmanydimensionsastherearevariablesinthesystem.Inthiscaseanygivenatomhassixvariables,namelyitspositionandvelocityineachofthreedimensions.Obviouslywecanplotamaximumofthree,butpartial2-Dplotsaresufficienttoshowifasystemisperiodicornot.

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ThescreenshotsbelowarefromthelatestversionofthisSuddensystemreconfiguration:This“Molecule”modeldemonstratesanotherfundamentalsystemsbehaviorlittlediscussedinthisbookbecauseI’vebeenfocusedsomuchonoscillation.Itsthefactthatsystemscanhaveseveraldifferentstableequilibriumconfigurations,andifdisturbedcouldendupmovingfromonetoanother.Inotherwordsoneconfigurationcouldbedestroyedandthepartsre-assembleintoanother.Averycrudewaytovisualizethisistoimaginedroppingabout6marblesintoaneggcarton.Theywouldallsettleintopocketsandformsometypeofstablepatternanalogoustowayatomsarearrangedinamolecule.Howeverifthecartonwereshakenenoughtheywouldpopoutofthosepocketsandsettleintoothers.Themoleculemodelpicturedaboveactuallyhasuptoeightstableconfigurationsdependingonhowstiffthespringsare.Iffrictionaldampingisactivatedinthesimulationtheatomsinthis“molecule”cancometorestinanyoneofthemafterbeingdisturbed.Inthismodeltheconfigurationsdifferintermsofwheretheredatomisrelativetothegreenatomandsoforth.Thenotionthatmanysomeimportantreal-worldsystemscouldbeinfragileequilibriumsandsusceptibletoradicalchangegivenasmallperturbationseemsveryimportant.ItsarguablythecaseontheeveofpoliticalrevolutionsliketheFrenchandArabspringrevolutions.Onechallengeisknowinghowclosewearetosuchatippingpoint.IwishIhadtimetosaymoreaboutthistopic.3.5.2FabricsimulationsConsiderableefforthasgoneintodevelopingcomputersimulationsoffabrictoshowhowitdrapesaroundotherobjectsinarealisticfashion.TheseareeasytofindonuTube.Thesemodelsareessentiallyspring/massnetworksthathavebeenrefinedtobehavelikerealfabrics.Theyareusefulforourpurposesbecausetheycanillustratehowdisturbingonepartofalargemulti-partsystemcansendwavesofchangethatsurgebackandforthinthesystemeventuallydisturbingalltheotherparts.Theymakethosewavesofchangemorevisiblethanthemoleculesimulationscando.Thescreenshotbelowisfromasimpleversionofafabricmodelandcanbefoundat:https://www.chromeexperiments.com/experiment/2d-cloth-simulationorhttp://andrew-hoyer.com/experiments/cloth/The“fabric”ishangingmotionlessfromthreepointsinthisscreenshot.

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Theadvantageofthismodelisthattheusercangrabonedot(amass)withthemouseandpullitasidetostarttheaction.Theimagebelowistakenfromavideoofaspringmasssystemofthetypeusedforsimulatingfabris.ItwasoneofaseriespostedbyPaulNathan.See:https://www.youtube.com/watch?v=ib1vmRDs8Vw&t=18sAndhttps://www.youtube.com/watch?v=qOvb3WLAX0EThisscreenshotcapturesawaveofchangeradiatingoutfromaplacewherethenetworkwasdisturbed.

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Hereisasimilaronebuttheusercan’tanimateit:https://www.youtube.com/watch?v=S5Rhx_NUBGcExtras:ScreenshotbelowisfromacubeshapedspringmassmodelavailableattheWolframsite.Itbeginsoscillatingwhenoneofthemassesisdisplacedbuthasn’ttheplotsorcontrolsneededforanalyticalwork.

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AnotherInteractiveClothSimulation,byMatthiasWloka:http://www.paulsprojects.net/opengl/cloth/cloth.htmlHereisanotherspringmassfabricyoucanstartawaveinbymovinganyplacewiththemouse:http://jlongster.com/s/lljs-cloth/

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3.6MoleculesMoleculesconsistofpartsconnectedbyspring-likeelectromagneticforcessoitnotsurprisingthatthepartsoscillatewhenthemoleculescontainenergyintheformofheatandmanifestitbyvirtueofmovingparts,justasthedoublependulumdoes.Ibeginwithaquote:

“In a turn of events that would have astonished anyone but N.Bohr, we now know that chaotic trajectories identical to those that govern the motion comets, asteroids, and spacecraft are traversed on the atomic scale by highly excited Rydberg electrons. This almost perfect parallel between the governing equations of atomic physics and celestial mechanics implies that the transport mechanism for these two situations is virtually identical...” http://www.iitk.ac.in/directions/directions_dec07/3jan~SRIHARI.pdf

Therelevantforces:Computermodelsofmoleculardynamicsapparentlyrangeinsophistication.Inthesimpleonesatomsareapparentlyattachedtoeachotherbylinearspringsandoscillateharmonicallyaroundtheirequilibriumposition(seegreenlinesinimagebelow)whereasinrealitytheforcesaren’tlinearasshownbythebluelinesintheMorsePotentialdiagrambelow.https://en.wikipedia.org/wiki/Morse_potential.TheMorsepotential,molecularoscillation,the“moleculardance”,andtheirrelationtochaosisnicelyexplainedat:http://www.iitk.ac.in/directions/directions_dec07/3jan~SRIHARI.pdf

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Thisseemstomeantwothings.One,sincetherealforcesarenon-linearthemotionsinmoleculeswith3ormoreatomsareprobablychaotic.Second,thebluelinesshowthatifamolecule’senergyisincreasedenough,forexamplebyraisingitstemperature,itwillvibratesoharditwilldissociateorflyapart,thusallowingchemicalreactionstoproceed.InamoregenericsensethisisprobablythebasicphysicsbehindthesystemsbehaviorIcall“suddenreconfiguration”.Initanexistingsystembreaksapartbecausethebondsholdingittogetherhavebeenstretchedtoomuch.Afterbreakingfreethepartsreassembleintoadifferentsystem.Inthiscaseadifferentmolecule.Insomeofthesimulationsshownbelowitisn’tclearwhetherthemodelsusedsimplelinearsprings(easertodealwithwhenmodeling)ornon-linearMorsePotentialforces.Andsomeseemtoillustratejustonemodeofvibrationatatimewhereasinrealityallwouldprobablybecoupledandchaotic.Notethatthereareavarietyofatomicbondsthatrangeinstrengthfromstrongcovalentbondstoweakhydrogenbonds.Presumablythestrongbondsformfirstandremainwhiletheweakerbondsformlast.Oscillation:Atomswithinmoleculesoscillate.Thefollowingscreenshotisfromananimationofavibratingbenzenemolecule.https://search.yahoo.com/yhs/search?p=vibration+of+molecules+utube&ei=UTF-8&hspart=mozilla&hsimp=yhs-001orathttps://www.youtube.com/watch?v=HuSbLBDagdc

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Screenshotbelowisfromanimationofvibratingmolecules:http://www2.ess.ucla.edu/~schauble/MoleculeHTML/SF6_html/SF6_page.html

Watchthisforsure:Screenshotbelowisfromacompellingvideoofapeptidemoleculeinthefinalstagesofself-assemblyfromtwosmallermolecules.Ibelievethisiswhat’shappening:Environmentalheatinjectsenergyandkeepstheatomsoscillating.Wavesofoscillationalternatelystretchandrelaxbondsbetweenthestronglybondedatomsatcorecausingtheentireconfiguration,moleculeorsystemtoflexchaotically.Thisbringsoutlyingatomstogethermomentarilysotheyform

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weakhydrogenbonds,andthussub-optimalmolecules.Thesedon’tsurvivethenextchaotictwistofthemolecules.ThroughtrialanderrorenoughweakbondsformtoeventuallycreateastablePeptidemolecule.https://video.search.yahoo.com/video/play?p=youtube+Simulation+Molecule+Formation&vid=6123f88bce545dc14a13008b76a6d18b&turl=http%3A%2F%2Ftse4.mm.bing.net%2Fth%3Fid%3DOVP.V5a5b65a0197b98175963d40f7ec3fed6%26pid%3D15.1%26h%3D145%26w%3D300%26c%3D7%26rs%3D1&rurl=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DB4sS1iIp-Qk&tit=Molecular+Dynamics+Simulation+of+self+assembling+Peptide+in+gromacs&c=17&h=145&w=300&l=337&sigr=11bl0m7j8&sigt=123br2jv7&sigi=1311kacei&ct=p&age=1406748225&fr2=p%3As%2Cv%3Av&b=391&fr=yhs-mozilla-001&hsimp=yhs-001&hspart=mozilla&tt=b

Presumablytheatomshavefallen–attheendofthissimulation-tothelowestenergy,mosttightlybondedstatepossible,oratleastonestableenoughtosurviveatthistemperature.Chaoticenergymovementwithinthesystemmaybekeytoself-assemblyinothersystemsaswell.Asenergymovesarounditconcentratesmomentarilyoncertainpartsstretchingtheirbondsbeyondthebreakingpointandfreeingthemtomove

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aroundtoformother,potentiallymorestable,configurations.Thusanassemblingsystemmightexplore–andevenrestforawhile-inseveraldifferentconfigurationsbeforefindingthemoststableandenduringone.Oscillationischaotic:Atleastsomeexperts“intramolecular dynamics” assertthatthevibrationwithinmoleculesischaotic.Seeforexample:http://store.elsevier.com/Nonlinearity-and-Chaos-in-Molecular-Vibrations/Guozhen-Wu/isbn-9780444519061/and:

“…we now know that chaotic trajectories identical to those that govern the motion of comets, asteroids, and spacecraft are traversed on the atomic scale by highly excited Rydberg electrons. This almost perfect parallel between the governing equations of atomic physics and celestial mechanics implies that the transport mechanism for these two situations is virtually identical... The orbits used to design space missions thus also determine theionization rates of atoms and chemical-reaction rates of molecules!” http://www.iitk.ac.in/directions/directions_dec07/3jan~SRIHARI.pdf this tech paper includes much more about oscillation within molecules)

Moleculesaregoodlittlesystemstoexamine.IthinkmostofthebasicsystemsbehaviorslistedinChapter1applytomolecules.Ifsohowtheyself-assembleintoincreasinglymorecomplexversions,andhowtheybehavemaybeanalogoustowhathappensinothersystemsthataffectourdailylives.Alookathowsimplemoleculesbehaveisausefulstepinextendingourunderstandingofsystemdynamicsfromverysimplemechanicalsystemslikethedoublependulumuptowardthemuchmorecomplexmolecularsystemsthatself-assembleintolivingorganisms.Itallcomesdowntopartslinkedorbondedtogetherintoasystembyspring-likeforces.Ifthosebondsarestretchedtoofartheybreakallowingnewsystemstoform.Thishappensacrossabroadrangeofsystemsstretchingfromatomsandmoleculestosolarsystemsandgalaxies.LaterinthebookIhopetoshowthatitalsoappliestosocietalsystems.Ischaoticoscillationnecessarytoconcentratepassionenoughtopasscontroversiallegislationorcauserevolutions?There’smoreaboutmolecules in various places includingChapter12. Otherlinks:Wikipediahasanimationofmolecularvibrationmodesplusotherdetails:https://en.wikipedia.org/wiki/Molecular_vibration

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Lectureonmolecularvibrationdiscussingenergylevels:https://www.youtube.com/watch?v=DJI518yTr2cAnimationshowinghowmolecularbondsarestretchedandcompressed:https://en.wikipedia.org/wiki/Molecular_vibrationNicelysemi-technicalexplanationofhowpotentialandkineticenergychangeasatomsvibrate.PlotscurveofPEandKE,andattractiveandrepulsiveforcesasafunctionofdistance.https://www.youtube.com/watch?v=Wl5QHeS2UXEThisveryclearlecturetreatsbondstrengthandenergyneededtobreakit:https://www.youtube.com/watch?v=I9jd1Ew_YGU&ebc=ANyPxKoqzaVjxJaWWAApc3GCwrV25dzhktewYBQIVTPvshRz4sBpn-91bqDWk_QjiYaKx9m3OLde7n83nBwadHzaR7ZSz-u16gThisexplainssomeoftheintermolecularforcesandhowtheyfalloffwithdistance.https://en.wikipedia.org/wiki/Molecular_mechanicsThisexplainsintermolecularforces:(butitsnottooclear)https://www.youtube.com/watch?v=S8QsLUO_tgQOtheranimationsofvibratingmolecules:https://www.youtube.com/watch?v=HuSbLBDagdchttps://www.youtube.com/watch?v=iy-8rguvGnMhttps://www.youtube.com/watch?v=3RqEIr8NtMI&index=2&list=RDHuSbLBDagdc

3.7Thermo-syphonThethermo-syphonisadonutshapedtubecontainingaliquid,whichiscomprisedofagreatnumberofpartssoitcanbeasystemofsorts.Whenheatedonthebottomtheliquidwillbegincirculatingonewayortheother.Dependingonhowfastheatisappliedtheliquidcancirculatesteadilyinonedirection–periodicbehavior-orswitchrandomlyfromCWtoCCWrotation,meaningitsbehavingchaotically.Thethermo-syphonexampleisinsertedtoshowthatdynamicbehaviorcanbemadeinotherthansimplemechanicalsystems.Fluidicsystemslikethisthermo-syphon,magmaconvectioncellsintheearthsmantle,andatmosphericcellscanoscillateinvariouswaysincludingchaotically.AuTubevideoofthethermo-syphoninoperationisfoundat:https://www.youtube.com/watch?v=ofzPGPjPv6gAnothersimilarvideoisfoundat:https://www.youtube.com/watch?v=Vbni-7veJ-cNotehowindividualparticles

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inthisfluiddon’tallmovetogether.Sometimestheygoindifferentdirectionsorspeedswithinthegeneraloverallcurrent.Imagesbelowarefrom:PredictingflowreversalsinchaoticnaturalconvectionusingdataassimilationbyHARRIS,RIDOUANE,HITTandDANFORTH,February2012.at:www.uvm.edu/~cdanfort/.../harris-tellus-2012.pdforintheirearlierpaperat:http://www.uvm.edu/~cdanfort/research/harris-pre-draft-2009.pdf

Thesesimulationsnapshotstakenatdifferenttimesshowhowhotportionsofliquid(red)intrudeintocoldliquid(blue)makingthisacomplex3-Dphenomena.Theflowratewithinthesedonutswaschaoticasshowninthewaveformbelow.

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Chaoslikethisthatoccursinliquidsorgasesiscalled“spatiotemporal”chaosbecausevalueslikespeedortemperaturevarychaoticallyfromplacetoplacewithintheliquidorgasaswellasovertime.Seep.5at:http://www.uvm.edu/~cdanfort/research/harris-pre-draft-2009.pdfLiketheLorenzwaterwheelthisisanotherdissipativesystemwhichshedsenergy.Inthiscaseitsheatwhichescapesthroughthewalls.Newenergyisaddedbyheatingthewateratthebottom,whichmakesitlessdensethanwateratthetop.ThismakesthetopwaterrelativelymoreheavyandthusincreasingitsPE.Thisheavymassthenwantstofallfromitsunstablepositionandindoingsocausesthefluidtobegincirculating.Thisisanotherexampleofhowthephysicalconstraintsoftheapparatusdictatehowtherelevantforcescanmovetheparts.Thependulumismechanicallyconstrainedsothatwhilegravitypullsitstraightdownthearmforcesittomoveinanarc.HerethedonuttubeforcesthebulkofliquidtocirculateeitherCWorCCW.Thethermo-syphonseemsanalogoustoacross-sectionofanoceancurrentsuchastheelNinocurrent.HowevertheelNinocurrentisheatedfromthetopnotbottomanditscirculationisdrivenbywindsnotitsinternaldynamics.It’skindoftoobad,sinceitwouldhavebeennicetofindarealworldcurrentthatbehaveslikethethermo-syphon.Ifsowecouldhaveseenitschaoticbehaviorifitwerealreadychaotic,orspeculatedabouthowitmightbecomechaotic–withimportantsideeffects-ifheatedenoughbyglobalwarming.

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Itwasnicetofindthefollowingquoteinthetechnicalliterature.ItshowsI’mnottheonlyonetryingtorelatetoysystembehaviortothosebigreal-worldsystemswecareabout.

“The methods we use to forecast the toy model are also similar to the methods used for global geophysical systems. Both require state estimation to find the IC from which to generate forecasts. Also, when forecasts are made in either system, climatology and dynamically accessible regimes are often more important than specific behavior: the occurrence of flow reversals for the thermosyphon; periodic behavior such as the El Nino Southern Oscillation and statistics such as globally and regionally averaged temperatures and their effects on rainfall, ice cover, etc. for climate. Each of these is a statistic that must be post-processed from the model output. To meet these global challenges, many tools are needed in the modeling toolbox. These techniques may also be useful for forecasting socio-technological systems, which are rapidly gaining importance as drivers of human behavior. In this way, toy models can provide us with insights that are applicable to the important scientific problems of today.” From: http://www.uvm.edu/~cdanfort/research/harris-tellus-2012.pdf

3.8Rayleigh-Benardconvection3.8.1GeneralDiscretepartsystemsorganizeintofluidandgassystems:Althoughsystemscomprisedofdiscretepartsmayseemworldsapartfromsystemsmadefromfluidsandgasestheymayinfactberelated.Atthemicrolevelwehaveindividualatomsthatarediscretepartsinsmallsystemscalledmolecules.Electrostaticforcesgoverntheirbehavior.WhatwelearnaboutsmallN-bodysystemslikethedoubleandmagneticpendulums,andsmallspring/masssystemsisusefulinunderstandingtheirbehavior.Howeverwhenwarmatomsoscillatetheytakeupmorevolume.Countlessbillionsofthem,allgettingwarmerandtakingmorevolume,cananddoself-organizeintowarm,buoyantandrisingplumes.Theseindividualplumesfurtherself-organizeintonicelyformedconvectioncells,ordegenerateintochaoticturbulence.Atthismacro-levelgravitynotelectrostaticforcesgovernbehavior.

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Thisisaintriguingideatofollowasitmayseamlesslyintegratethebehaviorofsimpletoysystemshavingdiscretepartswiththatoflargefluidandgassystemslikeoceancurrentsandatmosphericflows.I’verunoutoftimetopursuethislate-occurringidea.Behaviorvs.energy:Ihaven’thadtimetostudyfluidbehaviorwellenoughtomakemanyvalidgeneralizations.Ideally,onewouldmakeaseriesofsimulationrunsforagivensystem(i.e.typeofcontainerandfluid)withsuccessivelyhigherenergylevelstomaptheirbehavior.Ittakessupercomputerstorunsomeofthesefluiddynamicssimulations.Ihaven’ttheresourcesortimetodothatsowhatfollowsarescreenshotsandvideosfromadisparatevarietyofsimulations.Imustreadbetweenthelinessotospeaktodrawanyconclusions.Ithinkitsgenerallytruethatbehaviordoeschangewiththelevelofenergyinthesefluidicsystems.Atlowenergiesthefluidstendtoself-organizeintoreasonablywellformedconvectioncurrentsorrolls.Athigherenergiesinstabilitiesseemtogrowbetweenfluidsflowingatdifferentspeedsandtheseturnintoeddies.Asenergyisincreasedstillmoreeddiessubdivideintowhat’scalledturbulentflow.Forhowturbulencedevelopssee:https://www.youtube.com/watch?v=e1TbkLIDWysandhttps://www.youtube.com/watch?v=GlTcRhh3gYcAsIsayelsewherethebehaviorofadynamicsystemmustmanifestorcontaintheenergywithinit.Agentleconvectionrollcanapparentlymanifestalowamountofenergy,apparentlyasKE,duetoitsslowlyrotatingmass.Attheotherextremeittakesahugeamountofenergytocreatetheturbulentwakebehindajetengine.WiththosethoughtsinmindIsimplypresentsomeexamplesofsimulatedfluidflowwithinlaboratorycontainers.TherehasbeenagreatdealofresearchonthetopicunderthegeneralnameofRayleigh-Benardconvection.3.8.2Two-dimensionalfluidbehaviorARayleigh-Benardcellisanexperimentalapparatusconsistingofacontainerfilledwithliquid.Whenheatedevenlyacrossthebottomplumesofrisingliquidformandcanself-organizeintoafewlargeconvectioncurrentsorcells.Atlowenergythefluidtendstocirculatesmoothlyformingtwo,orafew,convectioncells,butathighenergytheflowbecomeschaoticandturbulent.Thissystemisincludedjusttoshowhowcomplexbehaviorcanbeinasystemwithbillionsofpartswhentheinsertionofenergycausesthemtomove.Thescreenshotbelowcomesfromasimulationoffluidflowinathinmostlytwo-dimensionalcell.Coldblueplumesofwateraredescendingfromabovewhilehot

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redonesrisefrombelow.Thisparticularrunwasveryturbulentapparentlybecausethebottomplatewasquitehot,buttowardtheenditdideventuallyself-organizetosomeextentintotworoughlyformedcounter-rotatingcirculationcells.Source:https://www.youtube.com/watch?v=OM0l2YPVMf8orhttps://www.youtube.com/watch?v=OM0l2YPVMf8&t=9s

Inthesimulationbelowhotredplumesformandrisealongtheuniformlyheatedbottomplatewhilecoldplumesformanddescendfromtheevenlychilledcoldtopplate.

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Thisscreenshotwastakensomewhatlateraftertheplumeshadself-organizedintocounter-rotatingconvectioncells.

Itsimportanttonotethatindividualplumesemergingallalongthebottomusuallymergeintoafewmajorstreamsmovingup,inthiscaseoneachside.Nicelyformedconvectionrollsseemunlikelyinnaturesinceitseemsconditionsmustbejustrighttoproducethem.Icouldnotgetnicecongestioncellstoforminahome-made,stove-topRayleighBenardcell.Thissuggeststhatneatbehavior-suchaswellformedconvectionrolls-requiresalltheparameterstobeperfect(viscosity,temp,cellshape)andthusperfectbehaviorisprobablyrareinnature.Cloudsforexampleareoccasionallyseeninlongwellformedbandsorrolls,butit’srare.Earthsmantle:Thereisevidencethattectonicplatesaredrivenaroundinrandomfashionbyveryslowmovingcurrentsinearth’smoltenmantle.See:http://escweb.wr.usgs.gov/share/mooney/2006_GJI_activetectonics.pdfSocietalsystems:Howplumesformandbehaveis,Ithink,quitesignificantbecauseit’sanabstractmetaphorforhowsocialandpoliticalmovementsform.Certainlyit’sastretchbutconsiderthefollowing.Supposethefluidalongthebottom

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representedalargenumberofindividualsinsomesocietyandtheheatbelowwasanalogoustosomethingforcingorenticingthemtodosomethingtomeetsomepersonalgoal.Somethinglikefindingasolutionforsomeunmetneed.Theneedmightbespiritualorreligious.Itmightbehungerorthewishforpoliticalfreedom.ThinkArabspring.TheRBsimulationsuggeststhatwhentheheat(unhappinesswiththestatusquo)hasrisensufficientlysomeone–apersonslightlymoreunhappythanhisneighbors-willbethefirsttotakeactionandformanewsocial,religiousorpoliticalmovement.Ifseveralfolksdothisitwouldresultinmanysmallplume-likemovementsrisingfromthebottom.Eachusingenergytopushagainstthestatusquo.Howeverthefluid(individuals)immediatelyadjacentanalreadyformingplumewon’tstarttheirownplumeormovement,insteadtheymovesidewaystojoinanalreadyformingone.Why,simplybecauseitseasierandtakeslessenergytobeafollowerthanaleader.Overtimethemanysmallplumes(religiousorpoliticalmovements)tendtomergeintojustafew.Thishashappenedinreligionandinpolitics.ThemainCatholicandProtestantchurchesthatemergedfromthemergerofmanyreligioussub-groupsduringtheReformation,andtheDemocratsandRepublicans,cometomind.Thereisperhapsamorescientificrelationbetweentheplumesinthisfluidandsocialmovements,buthopefullyI’mdrawnacompellinganalogythatinvitesmoreresearch.Isuspectitallcomesdowntothings–betheywatermoleculesorpeople-takingthepathofleastresistance,whichofcourserequireslessenergy.3.8.3Three-dimensionalfluidbehaviorThescreenshotbelowisperhapsmyfavoritesimulationvideo.Itsfoundat:https://www.youtube.com/watch?v=pmLmfciox50&list=UUraRQTwh5EYkCvzelOrHRlg&index=11Ihighlyrecommendwatchingitandseveralmoretogetafeelforspatio-temporalbehavior.Notehowisolatedplumesformfirstthenmorphintorelativelystableupanddowncurrents.Latertheseoscillateuntiltheyinterferewitheachotherandbecomeunstable.Noinformationisgivenastowhetherornotenergyischangingduringthisrun.

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Thissimisexcellentinshowinghowsteadyconvectionrollsformandlookinathreedimensionalcontainer.Watchfortherisinggreencurrentsincenterandoneeachside.Theactionstartsatabout=t=40.https://www.youtube.com/watch?v=buskqZlPdvIThescreenshotbelowisfromanothersimthatshowshowisolatedplumesselforganizeintolargerstructures.https://www.youtube.com/watch?v=eb8PzSmZqRM

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Thisvideoshowssimilarstructuresbutthistimeinanactuallabcontainerthusprovingthesimulationsrepresentreality.https://www.youtube.com/watch?v=nPHwnS3_xlYExtras:Thescreenshotbelowisfromaveryinterestingsimthatisdefinitelyworthwatching.Itshowshowsmall,dispersedplumesself-organizeintoarelativelylarge,stableconvectionflowwherehotfluidascendsintwocornersandcoldbluefluiddescendsintheoppositecorners.Laterenergyseemstooscillatearoundinthissystemuntilitconcentratesenoughtobreakthatpatternapartandformanotherwiththislargecenterplume.Laterittoobreaksup.ItteachesseveralthingsandisfurtherdescribedinChapter4.3.https://www.youtube.com/watch?v=cMGVltlKmr0&index=9&list=UUraRQTwh5EYkCvzelOrHRlg

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Other:https://www.youtube.com/watch?v=pmLmfciox50&list=UUraRQTwh5EYkCvzelOrHRlg&index=11andhttps://www.youtube.com/watch?v=buskqZlPdvIOtherRaleigh-Benardvideosarefoundat:https://www.youtube.com/watch?v=s30nE_hRv2Mhttps://www.youtube.com/watch?v=ECpNSjUG5yU&list=UUraRQTwh5EYkCvzelOrHRlgShowshowcomplexfluidmovementscanbecomeandhowunstableoccasionalrotatingcellscanbehttps://www.youtube.com/watch?v=kJnE12dJ9ichttps://www.youtube.com/watch?v=fZviNVPZjDQ&list=UUraRQTwh5EYkCvzelOrHRlg&index=7https://www.youtube.com/watch?v=r0pYCo9Vx5chttps://www.youtube.com/watch?v=nluyQThAcmMhttps://www.youtube.com/watch?v=r0pYCo9Vx5chttps://www.youtube.com/watch?v=5ApSJe4FaLIformsunstablerollshttps://www.youtube.com/watch?v=buskqZlPdvIcomplex3Dsimulationhttps://www.youtube.com/watch?v=nPHwnS3_xlYrealworldlabdemoofRBrollsinapanPaperaboutSpatiotemporalChaosinRayleigh-BénardConvection:http://www.cmp.caltech.edu/~mcc/Talks/BeijingIACM_06.pdf

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3.9Convectioninearth’smantleThesevideosshowhowconvectioncurrentsintheearthsmantledriveplatetectonics:https://www.youtube.com/watch?v=ryrXAGY1dmEandhttps://www.youtube.com/watch?v=Kpoko_l34ZE

“TheconvectionoftheEarth'smantleisachaoticprocess(inthesenseoffluiddynamics),whichisthoughttobeanintegralpartofthemotionofplates.Asthemoltenrockmovesupanddownchangingpressuresandtemperaturescauseitschemicalcompositiontochangeandproducemanydifferentminerals.Arelativelysimpleprocessbuildingcomplexity.https://en.wikipedia.org/wiki/Mantle_%28geology%29#Earth.27s_mantle

Belowarediagramsofconvectioncellsandplumesinearth’smantle.Source:https://en.wikipedia.org/wiki/Mantle_convection

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Thesevideosshowhowconvectioncurrentsintheearthsmantledriveplatetectonics:https://www.youtube.com/watch?v=ryrXAGY1dmEandhttps://www.youtube.com/watch?v=Kpoko_l34ZE3.10OtherexamplesThefollowingsitesleadtootherexamplesofchaoticbehavior.

Hereisverynicevideoofthethreebodyproblemwhichshowsthechaoticpathsofthreebodieswhichgravitationallyattracteachother:https://vimeo.com/11993047Thisisanotherneat3bodyanimation.Youdon’tneedtouseJavatoseeit:http://www.clausewitz.com/mobile/chaosdemos.htm#3BodyThisentertainingvideoshowstrafficchaosonthirdworldstreets(scrolldowntoplaythevideosegmentcalledAComplexAdaptiveSystem)at:http://www.clausewitz.com/mobile/chaosdemos.htm#3Body

Thissimvideohastwospringmasshollowballscolliding.http://scholar.harvard.edu/tlan/meshed-mass-springAuthorcouldmakethemodelIwantfromthis.CHECKITOUTMOREGoodreadabledescriptionof2ormoremasses/springs.http://www.math24.net/mass-spring-system.htmlsays2mass3springsystemislinear.Usefulinfo.

Wrap-up:Thisconcludesaquicklookatsomesystemsthatbehavechaoticallyundertherightconditions,plussomespringmasssystemsthatcan’tofficiallybechaoticbutneverthelessshowsomeofthesomebehaviors.InthenextchapterIattempttosummarizewhatIfeelarethemostimportantaspectsoftheirbehavior.Thesethingshopefullyapplyacrossawiderangeofsystems.Ipresentevidenceofthis,butnotproof.*****endofChapter3***