Deterministic chaos - Determinism and predictability - Deterministic chaos and absolute chaos - Logistic map - Fractals - Measuring chaos - Chaos in classical billiards
M. Peressi - UniTS - Laurea Magistrale in Physics Laboratory of Computational Physics - Unit XIII Determinism and predictability Deterministic chaos and absolute chaos Determinism
Determinism indicates that every event is determined by a chain of prior occurrences.
Pierre Simon de Laplace (1749-1827) strongly believed in causal determinism:
“ We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. “ (from: "Essai philosophique sur les probabilites") Predictability
Determinism ≠ predictability The world could be highly predictable, in some senses, and yet not deterministic; and it could be deterministic yet highly unpredictable...
Determinism: related to the nature of the physical system Predictability: related to what we can do (observe, analyze, calculate); to predict something we need: - knowledge of initial conditions - capability of solving exactly the equation of evolution Chaos and determinism a system is chaotic if its trajectory through the configuration space is sensitively dependent on the initial conditions, that is, if very small causes can produce large effects (in meteorology: "butterfly effect") Chaos and determinism
In the last few decades, physicists have become aware that even the systems studied by classical mechanics can behave in an intrinsically unpredictable manner. Although such a system may be perfectly deterministic in principle, its behavior is completely unpredictable in practice. This phenomenon was called deterministic chaos. Deterministic chaos is not randomness
Deterministic chaos is not the same as absolute chaos. Absolute chaos or randomness is when you don't know nothing at all of what will be the next value: it can be any value!
Another important difference is that for deterministic chaos we have a simple law that will produce all the values in the “attractor”. Instead for randomness there is no known recipe to produce past and future values. Chaos and determinism: logistic map; Mandelbrot function and fractals Chapter 6
The Chaotic Motion of Dynamical Systems Chapter 6
The Chaotic Motion of Dynamical c 2001 by Harvey Gould and Jan Tobochnik Systems! 27 March 2001
We study simple deterministic nonlinear models which exhibit complex behavior.
c 2001 by Harvey Gould and Jan Tobochnik ! 27 March 2001
6.1 IntroductionWe study simple deterministic nonlinear models which exhibit complex behavior.
Most natural phenomena6.1 Introareductionintrinsically nonlinear. Weather patterns and the turbulent motion of fluids are everyday examples.ChaosAlthough andwe determinismhave explored some of the properties of nonlinear Most natural phenomena are intrinsically nonlinear. Weather patterns and the turbulent motion physical systems ofinfluidsChapterare everyda5,iy examples.tiDeterministics easierAlthoughto winchaose trohaveduce exploreddescribedsomesome byofof thethepropimpertiesortanof nonlineart concepts in the context of ecologyph. ysicalOursystemsfirst goalin Chapterintrinsicallywill5,ibtie stoeasier NONmotivto LINEARinatetroduceand someequations.analyzeof the imptheortanone-dimensionalt concepts in the difference context of ecology. Our first goalE.g., willdynamicsbe to motiv of atepopulation:and analyze the one-dimensional difference equation equation
x x=4n+1 =4rxrx(1n(1 xxn),), (6.1) (6.1) n+1 n −− n where xn is the ratio of the population in the nth generation to a reference population. We shall see where xn is the ratiothat ofthethedynamicalpopulationproperties inof (the6.1) arenthsurprisinglygenerationintricatetoanda referencehave importanptopulation.implications We shall see for the development of a more general description of nonlinear phenomena. The significance of the that the dynamicalbehapropvior oferties(6.1)is indicatedof (6.1)byarethe follosurprisinglywing quote frominthetricateecologistandRoberthaMavey: important implications for the development of a more general WHICHdescription DYNAMICALof nonlinear BEHAVIOR?phenomena. The significance of the “ ... Its study does not involve as much conceptual sophistication as does elementary behavior of (6.1)is indicatedcalculus. Suchbystudythewouldfollogrwingeatly enrichquotethe fromstudent’stheintuitionecologistabout nonlineRobarertsys-May: tems. Not only in research but also in the everyday world of politics and economics “ ... Its study dowe eswouldnotall involvebe better offasif mormuche peopleconcrealizeeptuald that simplesophisticnonlineationar systemsas dodonotes elementary necessarily possess simple dynamical properties.” calculus. Such study would greatly enrich the student’s intuition about nonlinear sys- The study of chaos is currently very popular, but the phenomena is not new and has been tems. Not onlyof interestin toreseastronomersarch butand alsomathematiciansin the foreverydayabout one hworldundred years.of pMucoliticsh of theandcurrenectonomics we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties.”152
The study of chaos is currently very popular, but the phenomena is not new and has been of interest to astronomers and mathematicians for about one hundred years. Much of the current
152 CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 153 interest is due to the use of the computer as a tool for making empirical observations. We will use the computer in this spirit.
6.2 A Simple One-Dimensional Map
Many biological populations effectively consist of a single generation with no overlap between successive generations. We might imagine an island with an insect population that breeds in the summer and leaves eggs that hatch the following spring. Because the population growth occurs CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 153 at discreteCHAPTERtimes,6.itTHEis appropriateCHAOTIC MOTIONto modelOFtheDYNAMICALpopulationSYSTEMSgrowth by difference equations153 rather than by differential equations. A simple model of density-independent growth that relates the populationinterestinisgenerationdue to the usenof+1thetocomputerthe populationas a tool forinmakinggenerationempiricaln isobservgivenations.by We will use theinterestcomputeris dueintothisthespirit.use of the computer as a tool for making empirical observations. We will use CHAPTER 6. THEthe computerCHAOTIC MOTIONin this spirit.OF DYNAMICAL SYSTEMS 153 Pn+1 = aPn, (6.2) interest is due to the use of the computer as a tool for making empirical observations. We will use the computerwherein6.26.2thisPn spirit.is theAA SimplepSimpleopulationOne-DimensionalOne-Dimensionalin generation n and a MapisMapa constant. In the following, we assume that the time inCHAPTERterval b6.etwTHEeenCHAgenerationsOTIC MOTIONis unitOFyD, YNAMICALand refer toSYSTEMSn as the time. 153 Many biological populations effectively consist of a single generation with no overlap between 6.2 A SimpleIfMana>y biologicalOne-Dimensional1, each generationpopulations eMapwillffectivbelye aconsisttimesoflargera singlethangenerationthe previouswith no oone.verlap Inbetthisween case (6.2) successivsuccessivinterestee generations.generations.is due to the useWWeeofmighmighthe tcomputert imagineimagineasanana islandtislandool forwithwithmakingananempiricalinsectinsect ppopulationopulationobservations.thatthatWebreedsbreedswill useinin thethe Many biologicalleads tosummersummerpopulationsgeometricalthe computerandandeffleaectivleavvinelyeesgrosthiseggsconsisteggswthspirit.thatthatofandahatchatcsingleanhh generationthetheunbfollofollooundedwingwingwith nospring.spring.poopulation.verlapBecauseBecausebetween Althoughthethe ppopulationopulationthegrogrounwthwthboundedooccursccurs nature of successivgeometricale generations.atCHAPTERat discretediscreteWgroe6.mighTHEtimes,wthtimes,t imagineCHAisititOTICclear,isisanappropriateappropriateMOTIONislandit withisOFremarkDantoYNAMICALtoinsectmomodeldelablepopulationtheSYSTEMSthethatppopulationopulationthatmostbreedsgrogroofinwthwththeus153dobbyy didinotffffererencencintegrateee eequationsquationsourratherratherunderstanding summer and leaves eggs that hatch the following spring. Because the population growth occurs of geometricalthanthan bbyy diCHAPTERdigrofffferenerenwthtialtial6. inTHEequations.equations.toCHAourOTICevMOTIONAerydaA simplesimpleOFy DlivmoYNAMICALmoes.deldelCanofofSYSTEMSdensitdensita banky-indepy-indeppaendenendeny 4%tt153groigronterestwthwth thatthateacrelatesrelatesh yearthetheindefinitely? at discrete times,interestit is6.2appropriateis due toAthe useSimpleto ofmothedelcomputerthe pOne-Dimensionalopulationas a tool grofor makingwth byempiricaldifferencobserve Mapequationsations. Wrathere will use pthepopulationopulationcomputer ininthisin generationgenerationspirit. nn+1+1toto thethe ppopulationopulation inin generationgeneration nn isis givgivenen bbyy than byCandifferenthetialwequations.orld’sinteresthAumanissimpledue to themopopulationusedelofofthedensitcomputery-indepgroas awtendenooal fort tamakinggroconstanwthempiricalthat trelatesobservrateations.theforevWe willer?use population in generation n +1the computerto the pinopulationthis spirit.in generation n is given by CHAPTERMan6.yTHEbiologicalCHAOTICpopulationsMOTION OF DeffYNAMICALectively consistSYSTEMSPn+1 =of aPa singlen, generation153 with no overlap between (6.2) It6.2is naturalA Simpleto One-Dimensionalformulate a moreMaprealisticPn+1 =moaPndel, in which the population is b(6.2)ounded by the successive generations.Pn+1 W=eaPmighn, t imagine an island with an insect(6.2)population that breeds in the finite carrying6.2capacitA Simpley of itsOne-Dimensionalenvironment. AMapsimple model of density-dependent growth is interestwhereissummerduePton theisandtheuse ofleaptheopulationvecomputers eggs thatasina togenerationohatcl for hmakingthe folloempiricaln andwingobservaspring.isations.a constanBecauseWe willt.useInthethepopulationfollowing,growwthe assumeoccurs that where Pn is thethewhereManpcomputeropulationy biologicalPnin inthisis generationpthespirit.opulationspopulationneffectivand elya inisconsistagenerationconstanof a t.singleIn nthegenerationandfolloawing,iswithawconstannoe assumeoverlapt.thatbeIntweenthe following, we assume that successivthe timee generations.intervalWbeemightweent imaginegenerationsan island withis unitan insecty, andpopulationreferthatto breedsn as inthethetime. the time intervalthebetwtimeateendiscretegenerationsinMantervy biologicaltimes,al bisetunititwpopulationseenisy,appropriateandgenerationsrefereffectivtoelyntoconsistasismotheunitdeloftime.aythe,singleandpopulationgenerationrefer towithgron wthasno othevberlapy time.dibffeterweenence equations rather summer and successivleaves eggse generations.that hatchWthee mighfollotwingimaginespring.anPislandnBecause+1with=theanPnpinsectopulation(a populationbPgronwth)that. occursbreeds in the (6.3) If a>1, each generationthanIf a>by di1,willffereneacbehtiala generationtimesequations.larger thanwillA simplethebe previousa timesmodelone.largerof densitIn thisthany-indepcasethe(6.2endenprevious) t growthone.thatIn relatesthis casethe (6.2) 6.2at discreteAIf Simplea>times,summer1,it iseacandOne-Dimensionalappropriatehleavgenerationes eggsto thatmodelhatcthewillh thepMapopulationbfolloe awingtimesgrospring.wth blargeryBecausedifferencthanthe−e epquationsopulationthe ratherpreviousgrowth occursone. In this case (6.2) leads to geometricalthanleadsbpgroyopulationtodiwthfferengeometricalatanddiscretetial anequations.in times,generationunboundeditgroAis appropriatewthsimplepnopulation.+1andmodeltotoanmoofthedeldensitunAlthoughpthebopulationoundedy-indeppopulationtheendenuninpgroopulation.tbgenerationwthgrooundedwthby dithatffnatureerencrelatesnAlthougheisequationsofgivtheen ratherbythe unbounded nature of geometricalEquationgroManwthleadspgeometricalopulationy biologicalis (clear,to6.3ingeometricalthan)igenerationpitopulationsissgrobyremarknonlineardiwthfferenne+1ffabletialisectivgrotoclear,equations.elythatthewthconsistpdueopulationmostandit Aisofoftosimpleanaremarkusinsinglethegenerationundomobgenerationnotdelableoundedpresenceofinndensittegratethatis withgivy-indeppenopulation.mostournobofyendenounderstandingverlaptheoft grousbewthtquadraticAlthoughwdoeenthatnotrelatesintegratethethetermunbouroundedinunderstandingPn.natureTheoflinear term of geometricalsuccessivgrowthe ingenerations.to ourpopulationeverydaWe inmighygenerationlivt imaginees. Cann +1anaislandbankto thewithpapopulationy an4%insectinPinterestngeneration+1population=eacaPh nythatear,is givbreedsindefinitely?en by in the (6.2) represengeometricalts the naturalgrowth isgroclear,wthitofPnis+1theremark= aPpn,opulation;able that mostthe ofquadraticus do not(6.2)termintegraterepresenour understandingts a reduction of this Can the world’ssummerofhumangeometricalandpleaopulationves eggsgrothatgrowthwhatcathinatheconstantofolloourwingtevratespring.erydaforevBecauseyer?lives.theCanpopulationa bankgrowthpaoyccurs4% interest each year indefinitely? at discrete times, it is appropriate to model the populationPngro+1 wth= aPbny, difference equations rather (6.2) naturalofwhereCangrogeometricalwherePthewthis thewPorld’spcaused,opulationisgrothehwthumanpinopulationforgenerationintopexample,opulationourninandevgenerationerydaa isgroba yconstanywlivoanvtes.ercrot.anda InconstanCanthea wdingisfolloaabankwing,tconstanrateorwpae forevassumet.byyIn4%theer?thethatinterestfollospreadwing,eachofweydisease.earassumeindefinitely?that It is naturalthantobyformdinfferenulatetialnaequations.more realisticA simplemomodeldelinofwhicdensith y-indepthe populationendent growthis bthatoundedrelatesbythethe the time intervwhereal bePtwneenis thegenerationspopulationisinunitgenerationy, and refern andto na isasatheconstantime.t. In the following, we assume that finite carryingpcapacitopulationCanthetheyItinofistimegenerationitswnaturalorld’seninvironmentervnh+1alumantotobt.formethetAwpeenpsimpleopulationulategenerationsmoaindelmoregenerationgroof densitwisrealisticunitanty-depisaygiv,constanandenendenmobydelrefert grotinratetowthwhicnisforevashthetheer?time.population is bounded by the It isIfcona>v1,theenieneactimeh generationinttervtoal rescalebewilltweenbegenerationsa timesthe largerpisopulationunitthany, andthereferprevioustobnyasone.lettingthe time.In thisPcasen =((6.2)a/b)xn and rewriting (6.3)as finite carryingIf a>capacit1, each ygenerationof its environmenwill be at.timesA simplelarger mothandeltheof previousdensity-depone.endenIn thist grocasewth(6.2is ) leads toItgeometricalis naturalIf a>growth1,toPneac+1formandh =generationanPulatenun(Pabnounded+1willabP= moreaPnb)p.neopulation.,a timesrealisticlargerAlthoughthanmothedelthepreviousuninboundedwhicone.h(6.2)nature(6.3)Inthethisofpcaseopulation(6.2) is bounded by the geometricalleadsgroleadstowthgeometricaltoisgeometricalclear, it is groremarkgrowthwthableand−andthatan unanmostboundedunofboundeduspopulation.do not pinopulation.tegrateAlthoughourtheunderstandingAlthoughunbounded naturethe unof bounded nature of wherefiniteP iscarryingthe populationcapacitin generationy of itsn andenavironmenis a constant. InAthesimplefollowing,mowedelassumeof thatdensity-dependent growth is Equation (6.3)iofs geometricalnonlinearn geometricalgroduewthtointhetogroourwthpresenceeviserydaclear,yoflivit theises.remarkCanquadraticaablebankxthatnpaterm+1ymost4%=inofinterestaxPus .donTheeac(1nothinylineartegrateearxindefinitely?n)termour. understanding (6.4) the timegeometricalinterval betweengrogenerationswth is clear,is unity,itandisreferremarktoPnnasable+1the=time.thatPn(mostna bPofnus). do not integrate our understanding (6.3) represents the naturalCan thegroworld’swthof geometricalhofumanthe populationopulation;growth ingrotothewouratquadraticevaerydaconstany livttermes.rateCanforevrepresena banker? patsya−4%reductioninterest− eacofhthisyear indefinitely? If a>of geometrical1, eacCanh generationthe world’sgrowillwthhumanbeinpatoopulationtimesourlargerevgroerydawthanat aytheconstanlivpreviouses.t Canrateone.foreva banker?In thispacasey 4(%6.2)interest each year indefinitely? natural growth caused,It is fornaturalexample,to formbulatey overcroa morewdingrealisticor bmoy delthePinspreadnwhic+1 h=ofthePdisease.pnopulation(a bPisn)b.ounded by the (6.3) leadsEquationto geometrical(6.3gro)iwths andnonlinearan unboundedduepopulation.to the presenceAlthough theofunbtheoundedquadraticnature of term in Pn. The linear term The replacemenfinite carryingCan thecapacittwItorld’soisfynaturaloPf itsnhumanentobyvironmenformxpulatenopulationt.cahangesAmoresimplerealisticgromowthedelmoaoftdeladensitsystemconstanin whicy-deph−theendent ofratepopulationtunitsgroforevwth er?isisusedboundedtoby thedefine the various parameters. It is convgeometricalenienrepresent to rescalegrotswthfinitethetheiscarryingclear,naturalpopulationitcapacitis remarkgroyboywthablef itslettingenthatofvironmenmostPthen =(oft.population;a/busA simpledo)xnnotandmointegratedelrewritingtheof densitourquadraticunderstandingy-dep(6.3)aendens ttermgrowthrepresenis ts a reduction of this To writeof geometrical(6.4)igrowthn theinto ourformeveryda(y 6.1lives.),Canwaebankdefinepay 4% theinterestparametereach year indefinitely?r = a/4 and obtain EquationIt(is6.3natural)is nonlinearto formulatePndue+1 =aPtoAnmore(a thesimplebPrealisticnpresence). mo 1Dofdelthein mapwhicquadratich the p(6.3)opulationterm in isPnb.oundedThe linearby theterm Cannaturalthe world’sgrohumanwthpopulationcaused,x gro=foraxw atexample,(1a constanx ).tPraten−+1by=forevoPvnercro(er?a bPwdingn). or by the(6.4)spread of(6.3)disease. represenfinitetscarryingthe naturalcapacitn+1 groy wthofn itsofennthevironmenpopulation;t. A− simplethe moquadraticdel of densittermy-deprepresenendenttsgroa wthreductionis of this EquationIt is natural(6.3)itos formnonlinearulate aduemoretorealisticthe presence−model ofin thewhicquadratich the populationterm inisPbounded. The linearby the term It is Equationconvenien(6.3)it stnonlinearo rescaleduetheto thepopulationpresence of thebyquadraticlettingntermP in =(P . a/bThe)linearx andterm rewriting (6.3)as The replacemenfiniterepresent ofcarryingP tsbythecapacitxnaturalchangesy ofgroitswthenthevironmenofsystemthe pt.opulation;ofA simpleunitsxthenmoused+1delquadratictoof=densitdefinef(termy-depxntherepresen)=4endenvarioust tsgrorxawthparameters.reductionnn(1is n ofxthisn).n (6.5) naturaln grorepresennwthtscaused,the naturalforgrowthexample,of the population;by overcrothe quadraticwdingtermorrepresenby thets aspreadreduction ofof thisdisease. natural growth caused, for example, by overcrowding orPbny+1the=spreadPn(aof disease.bPn). − (6.3) To write (6.4)in the form (natural6.1), wgroe definewth caused,the parameterfor example, rby=ova/ercro4 andwdingobtainor by the−spread of disease. It is convenient to rescalePn+1 = Pthen(a pbPopulationn).x =baxy letting(1 xP). =((6.3)a/b)x and rewriting (6.3)as (6.4) It is convenient to rescale the population by−letting Pnn+1=(a/b)xnnand rewritingn n (6.3)as n The rescaled formIt is con(6.5venien) t hasto rescalethethe pdesirableopulation by lettingfeaturePn =(a/b)thatxn and rewritingits dynamics(6.3)as are determined by a single Equation (6.3x )i=s fnonlinear(x )=4rxdue(1 tox the). presence of the− quadratic(6.5)term in Pn. The linear term Equation (6.3)is nonlinearn+1due to then presencen of−thenquadratic term in Pn. The linear term represents the natural growth of the population;xn+1 =theaxnquadratic(1 xn).term represents a reduction of this (6.4) controlTheparameterrepresenreplacements thert.naturaloNotef Pn bygrothatxwthn cifofhangesxthenxn+1−>pxopulation;=then1,ax+1n(1systemx=n+1axxn)the.n(1willofquadraticunitsxbne).negativusedtermtorepresendefinee. T(6.4)otstheaavoidreductionvariousthisparameters.ofunphthis(6.4)ysical feature, The rescaled formnatural(6.5growth) hascaused,the desirablefor example,featureby overcrothatwdingits ordynamicsby the spreadare− ofdetermineddisease.− by a single TheTo replacemenwritenaturalThe(6.4tgrooreplacemenf)iwthPnnbythecaused,xnt ocformfhangesP byfor(thex6.1example,csystemhanges), wetheofdefinebunitssystemy ovusedercrotheof unitstowdingparameterdefineusedthetoordefinevbariousyrthethe=parameters.spreadva/arious4 andparameters.of obtaindisease. controlweparameterimposeItr.isNoteconthevenienthatconditionst tifo rescalexn > 1,thexpnopulation+1nthatwillnbxbeynegativlettingis restrictedPe. =(Toa/bav)oidx andthistorewritingtheunphysicalin(6.3terv)afeature,s al 0 x 1 and 0 The sequence of values x0, x1, x2, is called the trajectory or the orbit.To check your The sequenceTheofsequencevalues x0of, xv1alues, x2,x , xis, calledx , theis trcalledajectorythe ortrajethectoryorbitor.Ttheo ···corbitheck.Tyouro check your CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS ···0 1understanding,2 154 suppose that the initial value of x or seed is x =0.5 and r =0.2. Use a understanding, suppose that the initial value of x0···or seed is x0 =0.5 and r =0.2. Use a0 0 understanding, suppose that the initial value of x0 or seed is x0 =0.5 and r =0.2. Use a calculator to show that the trajectory is x1 =0.2,x2 =0.128,x3 =0.089293,... In Figure 6.1 the calculator to shoCHAPTERw that the6. THEtrajectoryCHAOTICis xMOTION1 =0.2,xOF2 =0DYNAMICAL.128,x3 =0SYSTEMS.089293,... In Figure 6.1154the calculator to show thatThethe trajectory logisticfirstisthirtx1 y=0iterations.2,x2 =0 ofmap.128(6.5,x)3are=0sho. 089293wn for,...twoInvaluesFigureof6.1r. the The sequence of values x0, x1,firstx2, thirtisycallediterationsthe trofaje(6.5ctory) areor shothewnorbitfor.Ttwoocvhecaluesk yourof r. ··· first thirty iterations of (6.5) are shown for two values of r. understanding, suppose that the initial value of x0 or seed is x0 =0.5 and r =0.2. Use a The sequence of valuesxxn0,+1x1, x2, rxis calledn the−trajexctoryn or the orbit.To check your calculator to show that the trajectory is x =0CHAPTER.2,x =0.6.128THE,x =0CHA.089293OTIC,...MOTIONIn Figure=4···OF6.1DtheYNAMICAL(11.0 SYSTEMS) 1.0154 1 1.02understanding,1.03 suppose that the initial value1.0 of x0 or 1.0seed is x0 =0.5 and r =0.2. Use a first thirty iterations of (6.5) are shown for two valuescalculatorof r. to show that the trajectory0 ≤ xis ≤x1 =01;.2,x 02 x x 0.0 0.0 0.0 0.0 0.0 0.00 15 30 0 15 30 0.5 0.5 n n 0.0 0.0 0 0.00 15 n 15 30 n 0 300.0 0 15 n 15 30 n 30 0 15 30 0 15 (a) 30 (b) 0 15 30 0 15 30 n n n (a) n (a) (b) (b) (a) (b) (a) (b) Figure 6.1: (a) Time series for r =0.2 and x0 =0.6. Note that the stable fixed point is x =0. (b) Figure 6.1: (a) Time series for r =0.2 and x0 =0.6. Note that the stable fixed point is x =0. (b) Figure 6.1: (a)FigureTime6.1:series(a) forTimer series=0.2forandr =0x0.2Time=0and.x6.0series=0Note.6. forNotethatr that=0the .thestable7 andstablefixedx0fixed=0pp.oinoin1. tNoteiissxx=0=0the. (b).initial(b) transient behavior. The lines between the Time series for r =0.7 and x =0.1. Note the initial transient behavior. The lines between the Figure 6.1: (a) Time series for r =0.Time2 and seriesx0 =0for.6.TimeNoter =0series0.0that.7 andforthe(rxstable=00 =0.7 andfixed.1. xNote0 p=00oin.t1.thepisNoteoinxinitial=0tstheare.) (b)initialtransien0.0a guidetransienttobehatthebehavior.eyvior.e. TheThelineslines bbeetwtweeneenthethe Time series for r =0.7 and x0 =0.1. Note the pinitialoinptsointransientsarearea aguideguidet behatovior.thetheeyTheeye.e. lines between the points are a guide to the0 eye. 15 n 30 0 ∗ 15 n 30 points are a guide to the eye. (*): condition (f(x))max ≤ 1 ⇒ r ≤ 1; x =fixed point ≤ 1 ⇒ r>0 The following class IterateMap computes the trajectory of the logistic map (6.5). The following class IterateMap(a) computes the trajectory of the logistic(b) map (6.5). The followingTheclassfolloIterateMapwing class IterateMapcomputes thecomputestrajectorythe oftrathejectorylogisticof themaplogistic(6.5). map (6.5). The following class IterateMap computes the tra//jectoryupdatedof3/6/01,the logistic7:54mappm (6.5).// updated 3/6/01, 7:54 pm Figure 6.1: (a) Time series for r =0.2 and x0 =0.6. Note that the stable fixed point is x =0. (b) // packageupdatededu.clarku.sip.chapter6;3/6/01, 7:54 pm package edu.clarku.sip.chapter6; // updatedTime3/6/01,series for 7:54r =0.pm7 and x0 =0.1. Note the initial transient behavior. The lines between the // updated 3/6/01, 7:54 pm packageimport edu.clarku.sip.chapter6;edu.clarku.sip.plot.*; import edu.clarku.sip.plot.*; package edu.clarku.sip.chapter6;packagepoinedu.clarku.sip.chapter6;ts are a guide to the eye. importimportedu.clarku.sip.plot.*;edu.clarku.sip.templates.*;import edu.clarku.sip.templates.*; import edu.clarku.sip.plot.*; import edu.clarku.sip.plot.*; import edu.clarku.sip.templates.*; import edu.clarku.sip.templates.*;import edu.clarku.sip.templates.*;public class IterateMap implements Model The{ following class IterateMap computespublictheclasstrajectoryIterateMapof the logisticimplementsmap (6.5Model). publicprivateclassdoubleIterateMapr; implements{ Model public class IterateMap implementspublicModel//classupdatedIterateMap3/6/01, implements7:54 pm Model { { private double x; private double r; { package edu.clarku.sip.chapter6;private int iterations; private double r; private double r; private double x; privateimportdoubleedu.clarku.sip.plot.*;privater; Control myControl = new SControl(this); private double x; private double x; private int iterations; privateimportdoubleedu.clarku.sip.templates.*;privatex; Plot plot; private int iterations; private int iterations; private Control myControl = new SControl(this); private int iterations; private Control myControl = new SControl(this);privatepublic ControlIterateMap()myControl = newprivateSControl(this);Plot plot; public class IterateMap implements Model private Plot plot; private Controlprivate{ myControlPlot plot;= new SControl(this); { private Plot plot;plot = new Plot("iterations","x","Iteratepublic IterateMap()Map"); private double r; public IterateMap() public} IterateMap() { private double x; { public IterateMap() plot = new Plot("iterations","x","Iterate Map"); private{ publicint voiditerations;reset() plot = new Plot("iterations","x","Iterate{ plotMap");= new Plot("iterations","x","Iterate} Map"); } plotprivate= new Plot("iterations","x","IterateControl myControl = new SControl(this);Map"); private} Plot plot; } public void reset() public void reset() publicpublicIterateMap()void reset() public {void reset() plot = new Plot("iterations","x","Iterate Map"); } public void reset() CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 155 { CHAPTERmyControl.setValue("r",6. THE CHAOTIC MOTION0.2OF); DYNAMICAL SYSTEMS 160 myControl.setValue("x", 0.6); CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 155 myControl.setValue("iterations",1.0 50); } CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 155 { myControl.setValue("r", 0.2 {); myControl.setValue("x",public0.6);void calculate() { myControl.setValue("r", 0.2 ); myControl.setValue("iterations",myControl.setValue("x",50); 0.6); } r= myControl.getValue("r");myControl.setValue("iterations", 50); x=myControl.getValue("x");CHAPTER} y 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 155 public void calculate() // method getValue returns double { CHAPTER 6. THE{ CHAOTIC MOTION OF DYNAMICAL SYSTEMS 204 iterationspublic0.5=void(int)calculate()myControl.getValue("iterations"); r= myControl.getValue("r"); myControl.setValue("r", 0.2 ); plot.deleteAllPoints();{ myControl.setValue("x", 0.6); x=myControl.getValue("x"); r=myControl.setValue("iterations",myControl.getValue("r");50); 2. Set c1 for= 10,(intc3 =0i .005,= 0;andi If we compare (6.66b)Ifandwe compare(6.67), w(6.66be obtain) and (6.67), we obtain "n+1/"n = f "(x∗)".n+1/"n = f "(x∗). (6.68) (6.68) If f "(x∗) > 1, the traIfjectoryf "(x∗) will> 1, divtheergetrajectoryfrom willx∗ sincediverge"from x∗>since" ."nThe+1 >opp"nosite. Theis opptrueosite is true | | | | | n+1| | n|| | | | for f "(x∗) < 1. Hence,forthef "(xlo∗cal) xn n →∞ . r=1/4 r The logistic map CHAPTER 6. THE CHAxOTICn+1MOTION=4OFrxDYNAMICALn(1 − SYSTEMSxn) 157 1.0 zoom on the bifurcation diagram 0.5 iterated values of x 0.0 0.7 0.8 0.9 1.0 r 0.75 0.8 0.85 0.9 0.95 1 (a=4r) r Figure 6.2: Bifurcation diagram of the logistic map. For each value of r, the iterated values of xn are plotted after the first 1000 iterations are discarded. Note the transition from periodic to chaotic behavior and the narrow windows of periodic behavior within the region of chaos. public Bifurcate() { plot = new Plot("r", "x", "Bifurcate"); } public void calculate() { r=myControl.getValue("r"); rmax = myControl.getValue("rmax"); dr = myControl.getValue("dr"); ntransient = (int) myControl.getValue("ntransient"); nplot = (int) myControl.getValue("nplot"); x=0.5; nvalues = (int)((rmax - r)/dr); xmax = 1; CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 160 CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 160 1.0 1.0 y y CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 160 0.5 0.5 1.0 CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 160 1.0 They logistic map CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 160 xn+1 =4rxn(1 − xn) 0.0 0.51.0 0.0 0.0 y 0.5 1.0 0.0 0.5 1.0 x x r 0.5 Figure 6.3: Graphical representation of the iterationy of the logistic map (6.5) with rFigure=0.76.3:andGraphical representation of the iteration of the logistic map (6.5) with r =0.7 and = 0) = 4 x0 =0.9. Note that the graphical solution con0.5verges x to the fixed point x∗ 0.643. x0 =0.9. Note that the graphical solution converges to the fixed point x∗ 0.643. ! ( y ≈ ≈ This graphical method is illustrated in0.0Figure 6.3 for r =0.7 and x0 =0.9. If wThise begingraphical method is illustrated in Figure 6.3 for r =0.7 and x0 =0.9. If we begin with any x0 (except x0 =0and x0 = 1), 0.0con0.0tinued iterations will0.5converge to thewithfixedan1.0pyoinxt (except x =0and x = 1), continued iterations will converge to the fixed point x 0 0 0 x∗ 0.642857. Repeat the procedure shown in0.00.0Figure 6.3 by hand0.5 and convince y1.0ourselfx that0.642857.you Repeat the procedure shown in Figure 6.3 by hand and convince yourself that you ≈ 0.0 0.5 x 1.0 ∗ understand the graphical solution of the iterated values of the map.x For this value ofunderstandr, ≈the fixedthe graphical solution of the iterated values of the map. For this value of r, the fixed FigureFigure6.3:6.3:GraphicalGraphical represenrepresentationtationof theofiterationthe iterationof the logisticof themaplogistic(6.5) withmapr =0(.6.57 and) with r =0.7 and point is stable (an attractor of perioFigured 6.3:1).GraphicalIn conrepresentrast,tationnoof matterthe iterationhoof thew logisticclosemapx0(6.5is) withto ther =0.fixed7 and point point is stable (an attractor of period 1). In contrast, no matter how close x0 is to the fixed point x0 =0x0 .=09. .Note9.x =0Note.9.thatthatNote thatthe thegraphicalgraphicalgraphicalsolutionsolutionsolutionconconvergesvergesconto thetovergesthefixedfixedpointot pxthe∗oint0.fixedx643.∗ 0.p643.oint x∗ 0.643. at x =0, the iterates diverge away0 from it, and this fixed point is unstable. ≈ ≈ ≈ Note: the graphical intersection between y(x) and the diagonal gives the fixed pointat , xbut=0 it is, notthe iterates diverge away from it, and this fixed point is unstable. How can we explain the qualitativesufficientdifference to determinebetw eenwhetherthe it isfixed stablep oinor unstablet at x =0and x∗ = This graphical method is illustrated in Figure 6.3 for r =0.7 and x0 =0.9. If we begin This graphical method is illustrated in Figure 6.3 for r =0.7 and x0 =0.9. IfHowwe bcanegin we explain the qualitative difference between the fixed point at x =0and x∗ = 0.642857 for r =0.7? The localwithslopany xe0 (exceptof thex0 =0curvandex0 y= 1),=conftin(xued) iterationsdetermineswill convergetheto thedistancefixed point moved with any x0 (except x0 =0and x0 = 1), continued iterations will converge to the fixed point This graphicalx∗ 0.642857. Repmethoeat thedproiscedureillustratedshown in Figurein 6.3Figureby hand and6.3conforvinceryourself=00that.7.642857andyou x0for=0r.9.=0If.7?we bTheeginlocal slope of the curve y = f(x) determines the distance moved ≈ horizontally each time fwithis iterated.xan∗ y0.x642857.understand(exceptARepsloptheeatgraphicalxethesteep=0prosolutioncedureerandofthantheshox iteratedwn=45in1),◦Figurevaluesleadsconof6.3tinthetobymap.uedahandFvoiterationsaluer andthis vconalueofvinceofxrwill,furthertheyourselffixedconvthataergewayyouto the fixed point ≈ 0 0 0 horizontally each time f is iterated. A slope steeper than 45◦ leads to a value of x further away from its initial value. Hence,understandthe pcriterionoint theis stablegraphical(anforattractorthesolutionofstabilitperioof dthe1). yiteratedIn oconftrast,a vfixedaluesno matterofptheoinhowmap.tcloseisxF0thatoisr tothisthethevfixedaluemagnitudepofointr, the fixed x∗ 0.642857.at x =0,Repthe iterateseat thedivergeproawacedurey from it, andshothiswnfixedin pFigureoint is unstable.6.3 by hand and convince yourself that you ≈point is stable (an attractor of period 1). In contrast, no matter how close x0 is fromto the fixedits initialpoint value. Hence, the criterion for the stability of a fixed point is that the magnitude of the slope at the fixedunderstandpoinatt xm=0ust, theHobewiteratesgraphicalcanlesswethandivexplainergesolutionthe45awqualitativ◦ay. fromThatofeit,ditheffandis,erenceiteratedifthisbedftfixedween(xthe)pv/dxoinaluesfixedt isxpunstable.=oinofxt∗theatisx =0map.lessandthanxF∗ o=r thisunitvyalue, of r, the fixed 0.642857 for r =0.7? The local slope of the curve| y = f(x) determines| the distance moved of the slope at the fixed point must be less than 45◦. That is, if df (x)/dx x=x∗ is less than unity, then x∗ is stable; converselypoin,tifis dfHostable(wxcan)/dx(anwe explainattractoristhegreaterqualitativof periothaneddi1).fferenceunitIn yconb,ethentwtrast,een thex∗noisfixedmatterunstable.point ahot xwInsp=0closeandectionxx∗is=to the fixed point horizontally xeac=hxtime∗ f is iterated. A slope steeper than 45◦ leads to a value of x further away 0 | | 0.642857| fromforitsrinitial=0| .v7?alue.TheHence,localthe criterionslope oforf thethe curvstabilite yyof=a fixedf(x)poindeterminest is that thethemagnitudedistance moved of f(x)in Figure 6.3 shoatwsx =0that, thex =0iteratesis unstablediverge abwecauseay fromtheit, andslopthise offixedf(x)apoint xt i=0s unstable.thenis xgreater∗ is stable; conversely, if df (x)/dx x=x∗ is greater than unity, then x∗ is unstable. Inspection of the slope at the fixed point must be less than 45 . That is, if df (x)/dx is less than unity, horizontally each time f is iterated. A slope steep◦ er than 45◦ leadsxto=x∗ a value of x further away | | then x is stable; conversely, if df (x)/dx is greater than unit|y, then x | is unstable. Inspofectionf(x)in Figure 6.3 shows that x =0is unstable because the slope of f(x)at x =0is greater than unity. In contrast, thefromHomagnitudewitscaninitial∗wevalue.ofexplaintheHence,slopthetheequalitativcriterionof f(x=xxfor)a∗ ethetdixstabilitfference= xy∗ofisbaelessfixedtw∗eenpthanointhet is unitthatfixedthey andpmagnitudeointtheat x =0and x∗ = of f(x)in Figure 6.3 shows that| x =0|is unstable because the slope of f(x)at x =0is greater of the slope at the fixed point must be less than 45◦. That is, if df (x)/dx thanis less thanunitunity. yIn, contrast, the magnitude of the slope of f(x)at x = x is less than unity and the fixed point is stable. In 0App.642857endixforthan6A,runit=0wy. eIn .usecon7?trast,similarThethe magnitudelocalanalyticalslopof theeslopofe argumenothef f(x)acurvt x =tsex∗yistoless=shothanf(wxunit=)xthat∗ydeterminesand the the distance moved ∗ then x is stable; conversely, if df (x)/dx is greater than unit|y, then x | is unstable. Inspection horizontally∗fixedeacpoinht itimes stable.fIn isAppiterated.endix 6A, wex=useAx∗similarslopeanalyticalsteepargumener thants to45sho∗w thatleadsfixedto apoinvaluet isofstable.x furtherIn Appawayendix 6A, we use similar analytical arguments to show that of f(x)in Figure 6.3 shows that| x =0|is unstable because the slope of f◦(x)at x =0is greater x∗ =0is stable for 0x x∗ =0is stable for 0 Numerics: for a given parameter r: - for a given x0, iterate the map and plot the trajectory (n, xn); - verify whether it converges and, in case, to which value(s) - verify numerically if the analytically predicted fixed points x1*, x2* are stable or unstable fixed points Another famous example other equations intrinsically NON LINEAR can show a chaotic behavior for certain values of the parameters. E.g. quadratic recurrence equation Mandelbrot function (in general in the complex field): Z(n+1) = Z(n)2 + C! with C constant (also negative) and n = 0, 1, 2, … Start with an initial value Z(0), then calculate: Z(1) = Z(0)2 + C then: Z(2) = Z(1)2 + C etc etc ... Some examples in the real field n Z1(n) Convergenza 0 0.0000 C = 0.2 and Z(0) = 0 1 0.2000 Convergence to Z*= 0.2764 2 0.2400 3 0.2576 ! Sequenza di valori con 4 0.2664 C = 0.2 e Zini = 0 5 0.2709 6 0.2734 ! Converge al punto 7 0.2748 Z* = 0.2764 8 0.2755 9 0.2759 10 0.2761 11 0.2762 12 0.2763 13 0.2763 14 0.2764 15 0.2764 16 0.2764 Some examples in the real field Previous example: C = 0.2 and Z(0) = 0 => Convergence to Z*= 0.2764 In general: Fixing Z(0) = 0: For 0 For C<~ -0.75 : convergence with damped oscillation For C~-0.76 : bifurcation (two-values attractor) Decreasing C: further bifurcations Further decreasing, at C~-1.42: chaotic behavior (infinite points of attraction; and very small change of Z(0)=> very different behavior of the sequence - “butterfly effect”) Some examples in the real field n Z1(n) Chaotic sequenceI lat c Ca o=s -1.7: 0 0.0000 1 -1.7000 The values of the sequence do not repeat 2 1.1900 However they are within a certain range 3 -0.2839 ! Sequenza caotica 4 -1.6194 (C = -1.7) 5 0.9225 Range including all points of the series: ! I valori si susseguono 6 -0.8491 chaotic attractor or strange attractor senza mai ripetersi 7 -0.9791 ! Tuttavia i valori restano 8 -0.7414 sempre entro certi limiti 9 -1.1503 ! La regione coperta dai 10 -0.3768 punti della successione si 11 -1.5581 definisce attrattore 12 0.7275 caotico oppure strano 13 -1.1707 attrattore 14 -0.3295 15 -1.5914 16 0.8326 Some examples in the complex field - fractal sets Remainder: Z(n+1) = Z(n)2 + C; in general, C and Z(n) are complex numbers. Repeat the iteration either until |z| > 2 or until a maximum number of iterations is reached. For fixed C complex, the set of the obtained Z(n->infinity) not infinite, produces a fractal figure (fill each pixel with black if Z(n->infinity) not infinite, or, better, fill each pixel with a color derived from the number of iterations required at that point) di San Marco, Alcuni esempi: coniglio diAlcDouauni deys, empi:Aldecunindrietes,empi: frattale Alcuni esempi: frattale di San Marco, Alcuni esempi: coniglio di Douady, Alcuni esempi: dendrite, c c== 0.123 +00.7145i23 + 0.745i c = i c = 0.75 − c = i − c = 0.75 − − Lucidi per il corso di Laboratorio – p.3 Lucidi per il corso di Laboratorio – p.4 Lucidi per il corso di Laboratorio – p.7 from: http://www.mun.ca/hpc/hpf_pse/manual/hpf0020.htm#mandel_algorithm_f90 (now obsolete!) Lucidi per il corso di Laboratorio – p.4 – p.7 Lucidi per il corso di Laboratorio Lucidi per il corso di Laboratorio – p.3 Mandelbrot set: the set of those points C in the complex plane for which the value of the function Zn (with Z(0)=0 fixed) does not tend to infinity as the number of iteration n grows In practice, a maximum number of iterations nmax and a maximum value of |Z|=rmax (here: rmax=2) is considered : one-color plots: black pixel: C is in the Mandelbrot set (|Z| remains limited)/ color: C is NOT ; multicolor plots: C points are colored according to the number of iterations n 6.5 Measuring Chaos How do we know if a system is chaotic? The most important characteristic of chaos is sensitivity to initial conditions.In Problem 6.3 for example, we found that the trajectories starting from x0 =0.5 and x0 =0.5001 for r =0.91 become CHAPTERvery differen6. tTHEafterCHAa smallOTIC MOTIONnumber OFof iterations.DYNAMICAL SYSTEMS 171 Because computers only store floatingCHAPTERnu6.mbTHEers CHAto aOTICcertainMOTIONnumOFberDofYNAMICALdigits, theSYSTEMSimplication of 171 this result is that our numerical predictions of the6.5trajectoriesMeasuringare restrictedChaosto small time intervals. CHAPTERThat6. THEis, sensitivitCHAOTICyMOTIONto initialOF conditionsDYNAMICAL6.5 MeasuringimpliesSYSTEMSthat Chaoseven though171the logistic map is deterministic, our ability to make numerical predictions is limited.CHAPTERHow do w6.e knoTHEwCHAif aOTICsystemMOTIONis chaotic?OF DYNAMICALThe mostSYSTEMSimportant characteristic171of chaos is sensitivity CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS How do we know171if atosysteminitialis cchaotic?onditionsThe.Imostn Problemimportan6.3t characteristicfor example,ofwcehaosfoundis sensitivitythat the trajectories starting from 6.5 Measuring Chaos to initial conditions.In Problem 6.3 for example, we found that the trajectories starting from How can we quantify this lack of predictably?Measuring6.5x0 =0In Measuring.5general,chaosand x0 =0 if wChaos.5001e startfor rtw=0o iden.91 bticalecomedynamicalvery different after a small number of iterations. 6.5 Measuring Chaos x0 =0.5 and x0 =0.Because5001 for rcomputers=0.91 becomeonly vstoreery difloatingfferent afternumaberssmalltonaumcertainber of iterations.number of digits, the implication of How do wsystemse know if afromsystemdiisffcerenhaotic?t initialThe mostconditions,important characteristicwe expectof chaosthatis sensitivitythe difference between the trajectories Because computers onlyHothiswstoredresulto we floatingknoiswthatif ansystemourumbnersumericalis ctohaotic?a certainpredictionsThe mostnumimpberofortanofthedigits,t ctraharacteristicjectoriesthe implicationofarechaosrestrictedis ofsensitivityto small time intervals. How do we know if toa systeminitialwillisconditionschaotic?change.IThenmostProblemas aimpfunctionortan6.3 tforcharacteristicexample,of n.Ithiswofe ncfoundresulthaosFigureisthatissensitivitythatthe6.8ourtratowenjectoriesumericalinitialshocwonditionsstartingpredictionsa plot.Ifromn ofProblemof thethetra6.3dijectoriesforfferenceexample,are restrictedw∆e xfoundn vthattoersussmallthe tratimenjectoriesintervstartingals. from xThat=0.is,5 andsensitivitx =0.5001y toforinitialr =0.conditions91 become veryimplies| differen|thatt afterevaensmallthoughnumbertheof iterations.logistic map is deterministic, to initial conditionsx.I0 =0n Problem.5 and x6.30 =0for .example,5001 for rwe=0found.91 bthatecomethevtraeryThatjectoriesdifferenis, startingsensitivitt after afromysmallto initial0 numbconditionser of iterations.0 implies that even though the logistic map is deterministic, for the same conditions as in Problem 6.3a. WeBecauseourseeabilitthatcomputersy toroughlymakonlye storenumericalspfloatingeaking,predictionsnumblners ∆to xaniscertainlimited.is anulinearlymber of digits, the implication of x0 =0.5 and x0 =0Because.5001 forcomputersr =0.91 bonlyecomestoreveryfloatingdifferentnafterumbersa smallto aourncertainumabilitber ofynutiterations.mobmaker ofe digits,numericalthe implicationpredictions ofis limited. | | increasing function of n. This result indicates thatthis resulttheisseparationthat our numericalbepredictionstween theof thetratrajectoriesjectoriesare restrictedgrowsto small time intervals. Because computers thisonlyresultstore isfloatingthat ournumnumericalbers to a predictionscertain numofbertheof tradigits,jectoriesthe implicationare restrictedof to smallHowtimecaninwterve quanals. tify this lack of predictably? In general, if we start two identical dynamical this result is that our numericalexppredictionsonentiallyof the iftrathejectoriessystemare restrictedis chaotic.to smallHowtimeThiscanintervwedivquanals.ergenceThattify is,thissensitivitoflacktheoyf tpredictably?otrainitialjectoriesconditionsIn general,canimpliesbthateifdescribweveenstartthoughedtwothebidenylogistictheticalmapdynamicalis deterministic, That is, sensitivity tThato initialis, sensitivitconditionsyimpliesto initialthatconditionseven thoughimpliesthe logisticthatsystemsevThemapen differencethoughis deterministic,fromthe dibetweenfflogisticerenoursystems ttwoinitialmapabilit trajectoriesisy conditions,deterministic,fromto mak mayedinffumerical erendivergewetexpinitial predictionsect thatconditions,isthelimited.differencewe expbeecttweenthatthethetradijectoriesfference between the trajectories our ability to make numerical predictions is limited. exponentiallyHow can: we quantify this lack of predictably? In general, if we start two identical dynamical our ability to make numericalLyapunovpredictions is explimited.onent, which is definedwill changeby theas a relation:functionwill cofhangen.InasFigurea function6.8 weofshonw.IanplotFigureof the6.8diwefferenceshow∆axplotn versusof then difference ∆xn versus n systems from different initial conditions, we expect that the difference| betw|een the trajectories | | How can we quantifyHothisw canlackwoef quanpredictably?tify thisInlacgeneral,k of predictably?if we start tInforwogeneral,identheticalsameifdynamicalwconditionse start tforwoasidentheinticalsameProblemdynamicalconditions6.3a. WeasseeinthatProblemroughly6.3spa.eaking,We seelnthat∆x roughlyis a linearlyspeaking, ln ∆xn is a linearly will changeλnas a function of n.In Figure 6.8 we show a plot of the dinfference ∆x versus n | | systems from different initial conditions, we expect that the difference between the trajectories increasing function of n. This result indicates that the| separation| bnetween the trajectories grows systems from different initial conditions, we expectincreasingthat the difunction∆ffxerencen =ofbfore∆ntw.thexeenThis0 samethee resultconditionstra, jectoriesindicatesas in thatProblemthe6.3separationa. We see thatbetwroughlyeen the(6.15)sptraeaking,jectoriesln ∆x|groisws|a linearly will change as a function of n.In Figure 6.8 we show a plot of the difference ∆x| versus| n | | | n| will change as a function of n.In Figure 6.8 we expshowonena| plottiallyn| ofifthethedisystemffincreasingexperenceonenis ∆ctiallyfunctionhaotic.xn versusifofThisthen.nThissystemdivergenceresultisindicateschaotic.of the thattraThisjectoriesthe separationdivergencecan bbeetdescribofweenthetheedtratrabjectoriesjectoriesy the grocanws be described by the for the same conditions as in Problem 6.3a. We see that roughly speaking, ln ∆xn is a linearly | | for the samewhereconditions∆x asisintheProblemdifference6.3a. We bseeeLyapunovtthatweenroughlytheexponensptraeaking,t,jectorieswhicexpLyapunovlnonenh∆itiallysxndefinedatisexpifathetimelinearlyonenbsystemy thet,n.Iwhicisrelation:chaotic.fh theis definedThisLydivapunoergenceby thevofrelation:expthe traonenjectoriest λcan be described by the increasing function of n. This result indicatesn that the separation between the|trajectories| grows increasing function of n. This result indicates that the separation betweenLyapunovthe|trajectoriesexp| onent,growhicwsh is defined by the relation: exponentially if the system isischaotic.positivThise,divthenergencenearbof the ytrajectoriestrajectoriescan be describdivergeed by expthe onentially. Chaotic∆x = b∆ehax eviorλn, is characterizedλnby (6.15) exponentially if the system is chaotic. This divergence of the trajectories can be described by then 0 ∆xnλn = ∆x0 e , (6.15) Lyapunov exponent, which is defined by the relation: | | | | ∆xn = ∆|x0 e |, | | (6.15) Lyapunovexpexponenonent,tialwhicdivh isergencedefined by ofthenearbrelation:y trajectories. | | | | λn where ∆xn is the difference between the trajectories at time n.If the Lyapunov exponent λ ∆xn = ∆x0 e , (6.15) wherewhere∆x∆nxisn theis thedifferencedifferencebetweenbtheetwtraeenjectoriesthe traat timejectoriesn.If atthe timeLyapunon.Iv expf onenthetLyλ apunov exponent λ A naiv| | e |way| of measuring isthepλositivn Lye,apunothen nearbv expisypositivonentrajectoriese, tthenλ nearbisdivtoergey trarunjectoriesexponenthedivtiallysameerge. expChaoticdynamicalonentiallybeha. Chaoticviorsystemisbcehaharacterizedvior is characterizedby by ∆xn = ∆x0 e , is positive, then(6.15)nearby trajectories diverge exponentially. Chaotic behavior is characterized by where ∆xn is the differencetwicebetweenwiththe traslighjectoriestlyatditimeff|erenn.I|tf initial|the expLy| apunoonenconditionsvtialexpdivonenergencet λandexpexpofonenonenmeasurenearbtialtialydivtraergencedivjectories.theergenceofdinearbfferenceofy nearbtrajectories.yoftrathejectories.trajectories as a is positive, then nearby trajectories diverge exponentially. Chaotic behavior is characterized by where ∆xn is the difference between the trajectories atA timenaiven.Iwayf theof measuringLyapunoA naivve theexpwayonenLyof apunomeasuringt λ v exptheonenLyapunot λ isv expto onenrunttheλ issameto rundynamicalthe same dynamicalsystem system exponential divergence of nearbfunctiony trajectories.of n.We used this method to generate FigureA naiv6.8e .wBecauseay of measuringthe ratethe ofLyseparationapunov exponenof thet λ is to run the same dynamical system is positive, then nearby trajectories diverge exponentwicetiallywith. Chaoticslightlybehadiviortffwiceereniswithtcinitialharacterizedslightlyconditionsdifferenby t initialand measureconditionstheanddimeasurefferencetheofdithefferencetrajectoriesof the trajectoriesas a as a trajectories might depend on the choice of x ,afunctiontwicebetterwithof nmetho.Wslighe usedtlydthisdiwffouldmethoerentdbinitialteo generateto computeconditionsFigure 6.8.theandBecausemeasureratetheofratetheof separationdifferenceof theof the trajectories as a A naive way ofexpmeasuringonential thedivergenceLyapunoofv expnearboneny trat λjectories.is to run the samefunctiondynamicalof n.Wsysteme0used this method to generate Figure 6.8. Because the rate of separation of the twice with slightly different initial conditions and measure the difference of the trajectories as a trajectories might depend on the choice of x0,abetter method would be to compute the rate of separation for many values of x0. This method wfunctionould beoftedious,n.We usedbecausethis methowe wd ouldto generatehave toFigurefit the6.8. Because the rate of separation of the A naive way of measuring the Lyapunov exponentrajectoriest λ is to mighrun thet depsameseparationend dynamicalon thefor manchoicesystemy valuesof xof0,ax0.bThisettermethomethod wdouldwouldbe tedious,be to bcomputeecause we thewouldratehaveofto fit the function of n.We used this method to generate Figure 6.8. Because the rate of separation of the trajectories might depend on the choice of x ,abetter method would be to compute the rate of separation for many vseparationalues of xto.(6.15This) formethoeachdvaluewouldof xb0eandtedious,then determinebecause0 anweawverageouldvhaalueveoftoλ.fit the trajectories might deptwiceendwithonseparationtheslighchoicetly diofffxerento0,at(binitial6.15etter )methoconditionsfordeacwouldhandvbalueemeasureto computeof xthe0theanddiffrateerencethenof of thedeterminetrajectories0 anas aaverage value of λ. separation for many values of x0. This method would be tedious, because we would have to fit the separation for manyfunctionvalues ofofx0n. .WThise methoused thisd wouldmethobe dtedious,to generatebecauseFigurewseparatione would6.8. haBecausevetoto(6.15fitthethe) forrateeacofA hmoreseparationvalueimpofortanxof0ttheandlimitationthen determineof the naive anmethoaveraged is thatvaluebecauseof λthe. trajectory is restricted A more important limitation of the naive methotoseparationthe unitd isintervtothat(al,6.15theb)ecauseseparationfor eachthev∆aluextraceasesofjectoryx toandincreaseisthenrestrictedwhendeterminen becomesansuaffiverageciently vlarge.alue of λ. separation to (6.15)traforjectorieseach valuemighof xt 0depandendthenondeterminethe choiceanofavxerage,abvalueetterofmethoAλ.mored wimpouldortanbe tot limitationcompute theof theratenaivof e method isn that b0ecause the trajectory is restricted 0 However, to make the computation of| λ as| accurate as possible, we would like to average over as A more important limitationto theof theunitnaive inmethotervd al,is thatthebecauseseparationthe trajectory∆isxnrestrictedceases toA increasemore importanwhent limitationn becomesof thesunaivfficiene methotly large.d is that because the trajectory is restricted separation for many values of x0. This method wouldtobthee tedious,unit inbtervecauseal, wmanthee wyouldseparationiterationshave astop∆fitossible.xthen ceasesFortunatelyto increase, there is whena betternprobecomescedure. Tsuo understandfficiently large.the procedure, to the unit interval, the separation ∆x ceases to increase when n becomes su|fficiently| large. | | separationHotow(e6.15v| er,) nfor| toeacmakh valuee theof x0computationand then determineHowevofer,anλtoaasmakverageaccuratee thevaluewetocomputationtaktheofeasλthe.unitpnaturalossible,inoftervlogarithmλ al,as waccuratethee ofwseparationbouldothassidesplikossible,ofe (6.15t∆oxw)naeandveragewceasesouldwrite likλtooaseveincreasetor aavseragewhenover ans becomes sufficiently large. However, to make the computation of λ as accurate as possible, we wouldmanlike ytoiterationsaverage ovaser apsossible.HowevFer,ortunatelyto mak,etherethe computationis a better procedure.of| λ as| Taccurateo understandas possible,the procedure,we would like to average over as many iterations as possible.A moreFmanortunatelyimpy ortaniterations, theret limitationis a betteras pofproossible.thecedure.naiveTFmethoo ortunatelyunderstandd is thatthe,bproecausetherecedure,theis atrabjectoryetter isprorestrictedcedure. To understand1 ∆xthen procedure, we take the natural logarithmmany iterationsof both assidespossible.of (6.15)Fortunatelyandλ =writeln λ, thereas . is a better procedure. To understand(6.16) the procedure, we take the naturaltologarithmthe unitofinbtervoth al,sidestheof (separation6.15) and write∆xnλ asceases to increase when n becomes sufficiently large. n ∆x we take the natural logarithm of both sides of (6.15) and write λ as ! 0 ! However, to make the computation of| λ as| accurate as possible, we wouldweliketaktoeavtheeragenaturalover aslogarithm of both sides! !of (6.15) and write λ as Because we want to use the data1 from∆xnthe entire tra! jectory! after the transient behavior has ended, 1 ∆xn λ = ln . ! ! (6.16) many iterations as pλossible.= lnFortunately. , there is a better procedure.(6.16)To understandwe use thethefactprothatcedure, n ∆x 1 ∆xn n ∆x0 1 ∆x we take the natural logarithm! of0b!oth sides of (6.15) and write λλas= ln . ! ! λ = ln (6.16)n . (6.16) ! ! ! ∆xn ! ∆x1 ∆x2 ∆xn Because we want to use the data from the entire tra! jectory! after the transienBecauset behaviorwehaswaended,nt ton use the∆xdata0 from the entire tra! jectory=! after the transienn . t∆bxeha0 vior has ended, (6.17) ! ! ! ! ! ∆x0 ! ∆x0 ∆x1 ···∆xn 1 ! ! we use the fact that 1 ∆wexnuse the fact that ! ! − ! ! λ = ln . Hence,Becausewe canwe expresswan(6.16)t toλ asuse the data from the entire tra! jectory! after the transient behavior has ended, Because∆x w∆exw∆axnt to∆usex thendata∆xfrom0 the entire tra! jectory! after the transient behavior has! ended,! n 1 2 n ! ! ! ! ∆xn ∆x1 ∆x2 ∆xn = . (6.17) we use the fact that n 1 we use∆x0 the∆xfact0 ∆x1that···∆xn 1 ! ! = 1 − . ∆x (6.17) Because we want to use the data from −the entire tra! jectory! after the transient behavior has∆xended,0 ∆x0 ∆x1 ···λ =∆xn 1 ln i+1 . (6.18) ! ! n∆−xn ∆∆x x1 ∆x2 ∆xn Hence, we can expresswe λuseasthe fact that i=0 != i ! . (6.17) Hence,∆xwne can∆expressx1 ∆xλ 2as ∆xn " ! ! ∆x0 ! ∆x0! ∆x1 ···∆xn 1 n 1 = . (6.17) − 1 − ∆x ∆xn ∆x1 ∆x2 ∆xn ! ! i+1 = ∆x0 . ∆x0 ∆x1 ···∆xn (6.17)1 n 1 λ = ln . (6.18) Hence, we can express1 −λ as ∆x n ∆xi∆x0 ∆x0 ∆x1 ···∆xn 1 − i+1 i=0 ! ! − λ = ln . (6.18) " ! ! n ∆x Hence, wHence,e can expresswe λcanas express! ! λ as i=0 ! i ! n 1 ! ! " ! ! 1 − ∆xi+1 ! ! λ = ln . (6.18) n 1 n 1 ! ! n ∆xi 1 − ∆xi+1 − i=0 ! ! λ = ln . 1 ∆xi+1 (6.18) " ! ! n ∆x λ = ln . ! (6.18)! i=0 ! i ! n ∆x ! ! " ! ! i=0 ! i ! ! ! " ! ! ! ! ! ! ! ! CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 173 CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMSCHAPTER 6. THE CHAOTIC173MOTION OF DYNAMICAL SYSTEMS 173 1.0 1.0 1.0 ! ! ! CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS0.0 0.0 171 0.0 6.5 Measuring Chaos CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 171 How do we know if a system is chaotic? The most important characteristic of chaos is sensitivity -1.0 -1.0 to initial conditions-1.0 .In Problem 6.3 for6.5example,Measuringwe found thatChaosthe trajectories starting from x0 =0.5 and x0 =0.5001 for r =0.91 become very different after a small number of iterations. Because computers only store floating nHoumwbersdo wtoe aknocertainw if a nsystemumber isofchaotic?digits, theTheimplicationmost importanof t characteristic of chaos is sensitivity this result is that our numerical predictionsto initialof the traconditionsjectories.Iaren restrictedProblem 6.3to smallfor example,time intervweals.found that the trajectories starting from -2.0 That is, sensitivity to initial conditions impliesx0 =0.that5 andevxen0 =0though.5001the0.7forlogisticr =0.map91 becomeis0.8deterministic,very differen0.9 t after a small1.0number of iterations. -2.0 CHAPTER 6. THE CHAOTIC MOTIONMeasuringOF DYNAMICAL SYSTEMS chaos173 -2.0 r our ability to mak0.7e numerical predictions0.8 Becauseis limited.computers0.9 only store1.0 floating0.7numbers to a certain0.8 number of digits,0.9 the implication1.0 of 1.0 r r How can we quantify this lack of predictably?this result isInthatFiguregeneral,our6.9:numericalifTheweLystartapunopredictionsvtexpwoonenident calculatedticalof thedynamicaltrausingjectoriesthe methoared inrestricted(6.20)asafunctionto smallof timethe intervals. ! Figure 6.9: The Lyapunov exponent calculatedThatusingis,thesensitivitmethocond itrolny(6.20tparametero initial)aFiguresafunctionrconditions. 6.9:CompareTheof thetheLyimpliesapunobehaviorv expthatof λonentoevthetencalculatedbifurcationthoughusingdiagramthe logisticthein methoFiguremapd6.2in.isNote(6.20deterministic,)asafunction of the systems from different initial conditions, we expect thatthat λthe< 0 diforffr A PROBLEM in a numerical approach: ROUNDOFF: small initial errors are exponentially amplified in time; after some (?) iterations the trajectories can diverge! How to calculate λ ? FIT over several trajectories Chaos in classical billiards Billiards MODEL BILLIARDS (conservation of energy law, reflection law of geometric optics) calculate trajectories (which depend on: shape of the billiard; initial position and velocity) Billiards Circular billiards support regular (periodic or non - periodic) trajectories, but in any case non - ergodic. (note also: conservation of angular momentum, incidence angle constant) In phase space (q(t),p(t)): limited region (a line: q(t) varies, p(t) constant) Billiards Also elliptical billiards support regular trajectories: The convolution of a trajectory can be: ellipse, hyperbole, regular polygon Billiards Rectangular billiards also support regular (periodic or non - periodic) trajectories, which in this case can be also ergodic Electronic and Optical Billiards - Taylor Lab http://materialscience.uoregon.edu/taylor/science/taylor_lab.html between the layers of GaAs and AlGaAs. By shaping the gate patterns to form an enclosed region, the device becomes analogous to a billiard table (Fig. 3). Billiards In a rectangular/square/elliptic billiards the trajectories are regular but also stable, i.e. changing the initial conditions, they remain close each other By insertingFigure 3 axxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx circle in a rectangular or square billiard, Figure chaotic 4 The currenttrajectories, project stronglyinvestigates dependent billiard shapes on thedesigned initial to conditions, induce chaos are in the classicalgenerated electron trajectories. The Sinai billiard, shown in Fig. 4, is of particular interest.(“Sinai The 'empty'billiard”, square from has the been name predicted of the Russian to support chaologist, stable 1972) (i.e., non-chaotic) trajectories. By inserting a circle at the centre of the square, the billiard is transformed into the 'Sinai' geometry, named after the Russian chaologist who, back in 1972, predicted that this billiard would generate chaotic trajectories. Figure 5 shows the state-of-the-art multilevel gate architecture we use to investigate Sinai's proposal - the fundamental transition to chaotic behaviour in a controllable, physical environment. Whereas transitions to chaos have previously been observed in systems such as a pendulum and a dripping tap, here we induce the transition in the flow of fundamental particles - electrons. In addition to addressing fundamental aspects of chaology, the results are of interest to the electronics industry, where the ability to exploit the extreme sensitivity of chaotic behaviour is important. This work serves as a demonstration of the precision with which semiconductor technology can tune electronic properties of small devices. 2 of 8 5-12-2005 0:59 Electronic and Optical Billiards - Taylor Lab http://materialscience.uoregon.edu/taylor/science/taylor_lab.html between the layers of GaAs and AlGaAs. By shaping the gate patterns to form an enclosed region, the device becomes analogous to a billiard table (Fig. 3). Billiards In a rectangular/square/elliptic billiards the trajectories are regular but also stable, i.e. changing the initial conditions, they remain close each other By insertingFigure 3 axxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx circle in a rectangular or square billiard, Figure chaotic 4 The currenttrajectories, project stronglyinvestigates dependent billiard shapes on thedesigned initial to conditions, induce chaos are in the classicalgenerated electron trajectories. The Sinai billiard, shown in Fig. 4, is of particular interest.(“dynamical The 'empty' billiard” square has or been“Sinai predicted billiard”, to support 1963) stable (i.e., non-chaotic) trajectories. By inserting a circle at the centre of the square, the billiard is transformed into the 'Sinai' geometry, named after the Russian chaologist who, back in 1972, predicted that this billiard would generate chaotic trajectories. Figure 5 shows the state-of-the-art multilevel gate architecture we use to investigate Sinai's proposal - the fundamental transition to chaotic behaviour in a controllable, physical environment. Whereas transitions to chaos have previously been observed in systems such as a pendulum and a dripping tap, here we induce the transition in the flow of fundamental particles - electrons. In addition to addressing fundamental aspects of chaology, the results are of interest to the electronics industry, where the ability to exploit the extreme sensitivity of chaotic behaviour is important. This work serves as a demonstration of the precision with which semiconductor technology can tune electronic properties of small devices. 2 of 8 5-12-2005 0:59 2014 Abel Prize from the Norwegian Academy of Science and Letters awarded to the mathematical physicist Yakov Sinai (now Princeton University, New Jersey). for his “fundamental contributions to dynamical systems, ergodic theory, and mathematical physics” http://www.abelprize.no/ CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 198 where φ is the angle of the needle with respect to a fixed axis along the field, µ is the magnetic moment of the needle, I its moment of inertia, and B0 and ω are the amplitude and the angular frequency of the magnetic field. Choose an Billiardsappropriate numerical method for solving (6.53), and plot the Poincar´e map at time t =2πn/ω.Verify that if the parameter λ = 2B0µ/I/ω > 1, then the motion of the needle exhibits chaotic motion. Briggs (see references) discusses how to Stadium (Bunimovich) billiard has a geometry !simpler construct the corresponding laboratory system and other nonlinear physical systems. than Sinai billiard, also resulting in chaotic trajectories L L r r (a) (b) Figure 6.14: (a) Geometry of the stadium billiard model. (b) Geometry of the Sinai billiard model. Project 6.24. Billiard models Consider a two-dimensional planar geometry in which a particle moves with constant velocity along straight line orbits until it elastically reflects off the boundary. This straight line motion occurs in various “billiard” systems. A simple example of such a system is a particle moving with fixed speed within a circle. For this geometry the angle between the particle’s momentum and the tangent to the boundary at a reflection is the same for all points. Suppose that we divide the circle into two equal parts and connect them by straight lines of length L as shown in Figure 6.14a. This geometry is called a stadium billiard. How does the motion of a particle in the stadium compare to the motion in the circle? In both cases we can find the trajectory of the particle by geometrical considerations. The stadium billiard model and a similar geometry known as the Sinai billiard model (see Figure 6.14b) have been used as model systems for exploring the foundations of statistical mechanics. There also is much interest in relating the behavior of a classical particle in various billiard models to the solution of Schr¨odinger’s equation for the same geometries. 1. Write a program to simulate the stadium billiard model. Use the radius r of the semicircles as the unit of length. The algorithm for determining the path of the particle is as follows: (a) Begin with an initial position (x0,y0) and momentum (px0,py0)of the particle such that p =1. | 0| (b) Determine which of the four sides the particle will hit. The possibilities are the top and bottom line segments and the right and left semicircles. Billiards NON Ergodicity Ergodicity of circular of chaotic billiards billiards Conservation of the energy, but in one case (stable trajectories): - another physical constant (e.g. angular momentum in case of circular billiards) - no other physical constant for stadium billiards our model point-like spheres no friction: forces normal to the boundaries => v’// = v // => v’ = - v perfectly elastic collisions: energy conservation: |v’| = |v| the algorithm given x,y,vx,vy at time t calculate : time to the next collision the position of collision velocity after the collision (reflection) Iterate N times (N collisions) CHAPTER 6. THE CHAOTIC MOTION OF DYNAMICAL SYSTEMS 198 where φ is the angle of the needle with respect to a fixed axis along the field, µ is the magnetic moment of the needle, I its moment of inertia, and B0 and ω are the amplitude and the angular collision timefrequency of the magnetic field. Choose an appropriate numerical method for solving (6.53), and plot the Poincar´e map at time t =2πn/ω.Verify that if the parameter λ = 2B0µ/I/ω > 1, then the motion of the needle exhibits chaotic motion. Briggs (see references) discusses how to ! construct the corresponding laboratory system and other nonlinear physical systems. Calculation of time to the next collision: L L r x(t) = x0 + vx t r y(t) = y0 + vy t (a) (b) boundaries: f(x,y)=0 : (e.g.Figure 6.14: (a): Geometry y0 +of thevstadiumy tc billiard=0)model. (b) Geometry of the Sinai billiard model. Project 6.24. Billiard models Consider a two-dimensional planar geometry in which a particle moves with constant velocity along straight line orbits until it elastically reflects off the boundary. at the collision time tc: This straight line motion occurs in various “billiard” systems. A simple example of such a system is a particle moving with fixed speed within a circle. For this geometry the angle between the particle’s momentum and the tangent to the boundary at a reflection is the same for all points. Suppose that we divide the circle into two equal parts and connect them by straight lines of length L as shown in Figure 6.14a. This geometry is called a stadium billiard. How does the motion f(x(tc),y(tc))=f(x0 + vx tc, y0 + vy oftca)=0particle in the stadium compare to the motion in the circle? In both cases we can find the trajectory of the particle by geometrical considerations. The stadium billiard model and a similar geometry known as the Sinai billiard model (see Figure 6.14b) have been used as model systems for exploring the foundations of statistical mechanics. There also is much interest in relating the behavior of a classical particle in various billiard models to the solution of Schr¨odinger’s equation for the same geometries. 1. Write a program to simulate the stadium billiard model. Use the radius r of the semicircles as the unit of length. The algorithm for determining the path of the particle is as follows: (a) Begin with an initial position (x0,y0) and momentum (px0,py0)of the particle such that p =1. | 0| (b) Determine which of the four sides the particle will hit. The possibilities are the top and bottom line segments and the right and left semicircles. collision point (half) circular boundary: equation: 2 2 [x( tc) - xc] + [y ( tc) - yc] = 1 i.e.: 2 2 (x0 + vx tc - xc) + (y0 + vy tc - yc) = 1 => 0, 1 o 2 solutions: (0 sol.) no collision (1 sol.) collision (tangent line) (2 sol.) collision (consider only the larger tc) velocity after collision For reflection off of a circular boundary: 2 2 (x - xc ) + y = 1 2 2 v’x = (y - (x-xc) ) vx - 2 (x - xc)y vy 2 2 v’y = - 2 (x-xc) y vx + ((x - xc) - y ) vy 2 2 (valid if vx + vy = 1 ) Lyapunov exponent Dynamics is chaotic: start with two particles with almost identical positions and/or momenta (varying by say 10−5); compute the difference ∆s of the two phase space trajectories as a function of the number of reflections n, where: s 2 2 ∆ n = !|r1,n − r2,n| + |p1,n − p2,n| Lyapunov exponent can be calculated by a semilog plot of ∆s versus n - L dependence? - role of single/double precision? Some programs: on $/home/peressi/comp-phys/XIII-chaos/ map.f90 billiard.f90 Chaos and billiards also available in Java: (Lab experiences with High School students) http://www.laureescientifiche.units.it/ (then follow the link on the banner “Fisica”=>”Laboratori”=>”Fare scienza con il computer”=>”materiale didattico”) or directly: http://www.democritos.it/edu/index.php/Main/Biliardi Billiards possibility of observing "Quantum Chaos": delocalization of the wavefunction in chaotic billiards Billiards possibility of observing "Quantum Chaos" Iron on Copper (111)