Chaos in a dripping faucet

E u r . J P h v 10 ( I Y X Y ) 99-1115 Prmled In the U K 99 Chaos in a dripping faucet H N Nufiez Yepezts, A L Salas Britots, C A Vargas'' and L A Vicente5 f Departamento de Fisica, Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, Apartado Postal 21-726, Mexico 04000 D F, Mexico $ Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana-Azcapotzalco, Mexico D F, Mexico. Facultad de Quimica, Universidad Nacional Autonoma de Mexico. Mexico D F, Mexico Received 6 April 1988, in final form 10 August 1988 Abstract. Anadvancedundergraduateexperimenton the chaotic behaviour of a dripping faucet is presented. The experiment can be used for the demonstration of typical features of chaotic phenomena and also allows the advanced physics student to learn about the use of microcomputers as data-taking devices. For convenience a brief introduction to the basic concepts of non-linear dynamics and to the period-doubling route to chaos are included. Resumen. Se propone un experimento sobre el comportamiento caotico del goteo en una llave mal cerrada. Este resulta uti1 para la demonstracion de caracteristicas tipicas de 10s fenomenos caoticos, y permite que 10s estudiantes de fisica aprendan a usar una microcomputadora para la toma de datos en un experimento. Hemos creido conveniente incluir una breve introduccion a la dinamica no lineal y. en particular, a la aparicion de caos por sucesivas bifurcaciones subarmonicas. 1. Introduction Chaodynamics is arecentarea of research(Ott suggestion of Rossler(1977), which showsthat 1981, Ford 1983, Bai Lin 1984, Jensen 1987); even drops falling from a leaky faucet behave chaotically its name is recent(Andrey1986).Itconcernsthe underappropriateconditions.Otherexperiments, occurrence of complex and seemingly random pheno-demonstrations or computer simulations have been recently proposed to introduce students to the field mena in non-linear but otherwise deterministic of non-linear phenomena (e.g. Berry 1981, Viet et a/ systems. Common examples of this behaviour include the results of tossing a coin, or the swirling 1983. Salas Brito and Vargas 1986,Briggs 1987), but paths of leaves falling from a tree on a windy day. curiously none of them deals with liquids despite the Similar aperiodic phenomena have been observedin fact that much original work has been done on such systems. In our experiment the students investigate animpressivenumber of experimentalsystems, even in somepreviouslythoughttobevery well the dripping behaviour of a leaky faucet, a system understood, as is the case of the driven pendulum which remains incompletely understood and hence (Koch et a/ 1983).Electrical,optical,mechanical, may still offer some surprises to both teachers and chemical,hydrodynamicaland biological systems students. In this system, the students can measure can all exhibit the kindof dynamical instabilities that the time interval between successive drops, the drip interval-as we, following Martien et a/ (1985), will produce chaotic behaviour (Jensen 1987 and referencestherein).Despitethis,recentdiscoveries in call it-as a function of the flow rate of water. the field of non-lineardynamicsare still not well Thestudentsbecomeacquainted with theconknown to many undergraduate physics students. cepts of non-linear dynamics (as deterministic With the above ideas in mind, we have developed chaos, attractors, subharmonic bifurcations, and the anexperimentthatcanbe useful forintroducing like) by reading the basic literature, paying particusome of the ideas and methods used in the descrip- lar attention to the logistic map (May 1976, tion of non-linear chaotic systems. Our experiment Feigenbaum 1980, Hofstadter 1981, Schuster 1984, follows the work of Martien et a/ (1985). based on a Jensen1987).Then,sincemanyaspects of this 100 H N Nuriez Ybpez et a1 mapping are common to a large class of dynamical systems showing chaotic behaviour, they are encouragedtoexplore it onamicrocomputertoobtain firsthand experience of the behaviour of a chaotic system, before they begin the experiment. In the following we summarise the experimental set-up and show the results obtained so far in our laboratories. Since thedrippingfaucetseemsto follow the period-doubling route to chaos (Martien et a1 1985), after a brief introduction to illustrate the basic concepts of thefield, in Q 2 we examine in some detail the logistic map, a paradigmatic example of a system following such a route to chaos. In Q 3 we describe our experimental device and show the return maps obtained from the data collected. These data confirm the existence of a sequence of period doublings in the system, at least up to period four, before the onset of chaos. Finally, we present our conclusions in D 4. termpredictability in asupposedlydeterministic system. Variousattractors may bepresent in thelongterm behaviour of a dynamical system; in most. its presence or absence is governed by the value of a single control parameter. For example, the magnitude the driving force determines if the pendulum settles to a point or to alimit cycle (or possibly even toamorecomplexattractor(D’Humieres e f a1 1982)). In the case of the dripping faucet, it is the flow rate of water which governs its dynamics: for low values of flow. the dripping is simply periodic and the system is attracted to a stable fixed point: but for much larger valuesof flow, strange attractors canappear.The succession of stationarystates which a system follows prior of the onset of chaos, as the control parameter is varied, determines what is called theroute to chaos followed by the system (Kadanoff 1983). The dripping faucet seems to follow the period-doubling route to chaos (Martien et a1 1985). We will explain this route in some detail below. The evolution of a dynamical system can be described in eithercontinuoustime(a flow) or in discrete time (a mapping). The pendulum is a good example of a system that maybe described by a flow in phase space-although it can also be described by a mapping (Testa et a1 1982). On the other hand. the sequence of drip intervals in a leaky faucet is naturally described by a discrete map.For any given value of the dripping rate. a plot of the next drip interval versus the previous one can give a clear idea of its dripping behaviour and of the possible existence of attractors. This is the representation we use for the data obtained in the experiment (see figure 5); i t is called a return or Poincare map. As anillustration of some of theseideas.and because they offer perhaps the simplest examples of systems undergoing a period-doubling transition to chaos, we shall consider iterative processes of the form 2. Basic concepts and the period-doubling route to chaos Let us first introduce some basic notions and terminology of non-linear dynamics. Consider an harmonically driven pendulum: given the frequency and strength of thedrivingforce,themotion of the system is cpmp!etely determined if the angle 0 and angular speed 0 of the pendulum are known. These variables can be used as coordinates in the phase space of the pendulum; as it swings back and forth, the point representing its state moves along an orbit in phase space. For example, if the strength of the driving force vanishes, due to the effect of friction, no matter how we start its motion the pendulum will come to rest at its point of stable equilibrium after a number of oscillations. From the point of view of phase space, the orbit spirals to fixed the point at the origin. The motion is quite different for non-zero values of the driving force;in this case the pendulum settlestoastationaryoscillation with thesame frequency as the external driving force. These stain a tionary motions in which the system settles after the where f ( x ) is acontinuousfunctiondefined Discussing only transients have died out are examplesof attractors, a suitableone-dimensionalinterval. term which conveys the idea that many nearby orbits one-dimensional mappings as (1)is not as restrictive as it may seem at first. since it can be viewed as a are‘attracted’tothem.Wehavementionedtwo types of attractors, a stable fixed point and a stable discrete time version of a continuous but dissipative dynamical system. The dissipative terms shrink the limit cycle,butthereexistsamorecomplicated attractor, the so-called strange attractors which only volume of phase space occupied by the system until it becomes effectively one-dimensional. In this occur in dissipativenon-linearsystems.Theycapinstance it can be modelled, at least in its universal turethesolution of adeterministicsystemintoa as (1) perfectly defined regionof phase space, butin which qualitativefeatures, by asimplemapping (Collet and Eckmann 1980). In fact, such iterations there is a very complex structure (these objects are been advocated frequently as qualitative usually fractals) and the motion shows every feature have associated with random motion. Such behaviouris a models for many complexphysical systems, from the manifestation of the very sensitive dependence on behaviour of a driven non-linear oscillator (Linsay 1981, Testa et a1 1982) to the onset of turbulence in the initial conditionsdeveloped by thesystem (Gollub and Rayleigh-Benard phenomena (Ruelle 1980). The existenceof strange attractors is the of theresultsheredonot one of the fingerprints of chaos i.e. the loss of long- Benson1980).Most 101 Chaos in a dripping faucet l b) 08 0.8. 0 4 0.4. F 0 08 0 4 0 0 4 0 0 4 08 IC) 0.8. z 0 4- 0 0 4 0.0 Xn 08 X” Figure 1. Return maps, i.e. plots of x,l-, versus x,, for large values of n , obtained from the logistic map for different values of U : ( a ) U = 1.5; ( b )p = 3.3; (c) U = 3.5; ( d ) U = 3.8. This illustrates the dynamics of the map up to the four cycle as well as the chaotic attractor for p > p x . appearance of attractors of period one, two. four and of a one-dimensional chaotic attractor can be appreciated in these plots. Figure 2 illustrates this kind of behaviour in adifferentandmoreglobal way; it shows a plot of the large n behaviour of the f(x) = / d l - x ) ( 2 ) iterates (i.e. the attractors) of the logistic map as a function of thevalue of p . Thisgraph gives a ‘pictorialmeaning’tothe way theonset of chaos where 0+<4 is parameter a measuring the occurs via sequence a of ‘pitchfork’ (periodstrength of thenon-linearity.Withthischoicefor f ( x ) . equation (1) describes a non-linear and non- doubling) bifurcations as the value of p changes. It also shows the critical dependence of the behaviour invertiblemap of theunitintervalonitself.The of p evolution of the sequence of x, generated by this with the value of this parameter. For values between 1 and 3 , andalmost all initialvalues x ( , , simple equation exhibits a transformation from perthere is a single point attractor (figure l(a)). Then. iodic to chaotic behaviour as the control parameter as p is increasedbetween 3 and 4, thedynamics p is increased.Let us seehowthisoccurs.The behaviour of the sequence of iterates is trivial when changes in surprising ways. First, for3 < p S ( l + G) thestationarysolutionbifurcatestoaperiod-two p = 0: for every initial value x,, all the iterates are attractor-theperiod of thesolutionhasdoubled zero.Wecan say thenthatthesolutionquickly x=O; this is reaches an attractor, the single point and its frequencyhalved,hencethenames of called aperiod-onecycle,orbit or attractor.For period-doubling or subharmonic bifurcationgiven to values ofp between 0 and 1, the large n behaviour of the phenomena-as can be seen in the bifurcation the x , is identical;theyapproachthepoint x=0 diagram(figure 2), wherethesolutionhopsback after a certain numberof steps. But for larger values and forth between the upper and lower branches of of p the dynamics is much more interesting as can be the pitchfork, and in figure l ( b ) . As p is increased easily verified using a hand-held calculator. Various further.thesolutionbifurcatesagaintoaperiodtypes of stationary solutions of the logistic map are four attractor, then to a period-eight attractor and exemplified by figures 1 and 2. so on. Thissequence (or cascade) of bifurcations Figure 1 shows return maps (plots of x , , , versus continues indefinitely, but the intervalof values of p x , ) for ,U = 1.5, 3.3.3.5 and 3.8. The successive in which a given periodic orbit acts as an attractor depend on the precise form of the function f ( x ) . as long as it has a single quadratic maximum but to be specific we will analyse the dynamics of the logistic map. This mapping is defined by H N Nlinez Ykpez et a1 102 shrinksveryquicklyatarategoverned universal parameter P E -Pn-l 6 = lim “4.6692.. by the . instabilities in the attractors are not universal-they are specific for the logistic map. (3) 3. The dripping faucet experiment Theapparatusused in theexperiment is rather simple and widely available. We use a Commodore 64microcomputerfordataacquisitionandsubseuntilacriticalvalue = 3.5699... is reached(Feiquentanalysisanddisplay.The inclusion of an genbaum 1978, 1979). This value marks the beginautomatic data-taking procedure is fundamental in ning of the aperiodic regime: the iterates seem to of 2000 wander erratically around a subset of the unit inter- an experiment which requiresthetaking val. If we increase p further, windows of periodic data points every time it is run. In fact. this repreit allows the motion of every integer period reappear. Chaoticor sentsanadditionaladvantage,for periodic motion can be found for suitable values of students to learn simple interfacing techniques and to work with a microcomputer-assisted experiment. ,u>pm.A complete discussion of the properties of The basic apparatus is shownschematically i n the logistic map can be found in the accountgiven by Feigenbaum (1983). For a more complete discussion figure 3. It consists of a large reservoir of water (a largeMariottebottle)keptataconstantpressure of theperioddoublingas well asotherpossible with the help of a float valve. The water can flow routes to chaos in a dynamical system see Kadanoff through a valve to a plastic tube with a nozzle at the (1983). well as the float valve,were As with manyotherpropertiesdiscovered in end.Thisvalve,as obtained from a used automobile carburetor. With systemsmaking a period-doublingtransitionto which is the chaos, the constant d is universal in the sense that it its help we can control the dripping rate, is found to be valid for a large number of systems control parameter in our experiment. Drops falling the nozzle pass through an optocoupler and not only for the logistic map. For example, if the from (GeneralElectric H23L1, with aSchmidttrigger dripping faucet effectively follows the periodincluded at the output) which produces a TTL pulse doubling route to chaos and we were able to calculate 8, we should find a numerical value very close to for each drop. The pulses are sent,via a very simple interface (figure 4), to the user port of the that given in (3). Now, obviously, not every feature Commodore64microcomputer.Thecomputer is of the logistic map is shared by other systems, for of used to store the data, to compute the drip interval example, the values quoted above for the onset n-= ,P,+, -P,, Figure 2. A section of the bifurcation diagram of the logistic map. The graph shows the asymptotic behaviour of x,, for values of ,U between 2.94 and 4. 0 75 2 c v 2 050 + c 4 0.25 3.0 3 4 P 38 103 Chaos in a dripping faucet WoterInlet ~- " - M o d i f i e d carburetor valve Interface Emltter-detector par Mlcrocomputer Figure 3. Schematic diagram of the experimental set-up. We use a float valve (marked 'level control' in the diagram) to maintain the water level in an upper reservoir (not shown). and to display the return maps obtained. With this arrangementstudentsareabletotake,storeand analyse up to3072 drips (using 6 Kbyte of memory). The machine-language subroutine used for acquiring the data and measuring the drip interval T, is capable of taking data up to a rate of 1.2 kHz, far above the dripping rates occurringin the experiment, and has an estimated resolutionof 50 p . This estimation has been tested with good results with the help of a signal generator (Wavetek 181) used as the input of our data-taking device. The flow rate is controlled by means of the carburetor valve, but we do not measure it directly. preferringinsteadtousethe valve settingasan indicator. The program we use to analyse the data computes a mean dripping rate. The mean dripping rates students are able to investigate under experimental conditions vary from 0.1 to 40 drips/s, a rate at which the drops become a continuous stream of water.Inthisinterval,thesystemmovesfroma stableperiod-oneattractorandundergoesperiod doublingsuntilstrangeattractorsappearfordrippingratesgreaterthan7drips/s.Atsuchlarge dripping rates the behaviouris irregular and, surely, is very complex (figures 5 and 6). In fact, much to our surprise the dynamics of the system is very rich Figure 4. The interface is a single 74LSOO chip. The connections to the microcomputer user's port are shown Motched emltter-detector par H23 L1 " " " " " I I I I I I I I I I l I User's p o r t H N Nunez Ykpez et a1 104 -.,. l i 412 4 L , l l , 384- 384 l I 412 440 1 1 105 i 105 I 0 64 128 173 188 l " " 123 0 60 , v l 114 30 0 33 l i 158 , - 66 158 Figures. Example of the experimental results shown as T,,,, (vertical axes) versus T,, (horizontal axes) graphs redrawn from the printout of our data. Periodic behaviour, ( o ) - ( c ) : complex chaotic behaviour. ( d ) - ( f ) , AI1 values of time are in milliseconds. and shows patterns not discussed in Martien et al. All of this hasgeneratedagreatdeal of interest among our students. T,,,, Typical experimentalresultsareshownas versus T,, plots in figures 5 and 6 (notice the qualitative similarity of figures 5(a)-(c) with figures l ( a ) (c). These are plots of the 2000 typical points taken each time the experiment is run. The beginning of a 114 ; 114 I I 122 130 I I Figure 6 . Another example of an attractor in the chaotic region. Note the folding and separation developed as i t becomes a more complex attractor. Axes and units as for figure 5 . period-doublingsequencecan be appreciated;the dripping behaviour shows attractors of period one, two and four prior to the chaotic regime. With the current experimental arrangement it is not possible to ascertain precisely the ranges of stability of the attractors but. roughly, the students have found the periodic attractors to be present up to 7 dripsis. For greater dripping rateswe observe chaotic behaviour, signalled by whatseem to bestrangeattractors: typical examples are shownin figures 5(d)-V) and in figure 6. This last attractorhasbeen singled out because it illustratesthefolding,stretchingand fractioning that occur in the attractors in the process of becoming more complex, as a result of increasing thedrippingrate.Wehavenot been abletosee periodic attractors of period larger than four. due perhaps to the inherent noise in the system or to the somewhatpoorcontrolofdrippingratesallowed by the carburetor valve. But, occasionally, students were able to observe cycles of period three As theseobserimmersed in thechaoticregime. vations are very sensitive to the valve setting and to vibrationsproducedneartheapparatus. we have beenunabletoreproducethemat will with the current experimental arrangement. The result of the experiment has been taken as an indication of a period-doubling route to chaos in the system, but to be conclusivefurtherevaluation is needed. For example, it may requirethecomputation of universal parameters like d. But before we Chaos in a dripping faucet 105 levelandinterest.Anotherusefulfeature of the can determine such parameters we must be able to measure with greater confidence the stability interexperiment is that itallows advanced physics stuvals of the attractors and to discern at least a period-dents to learn about simple interfacing techniques eight attractor. and the use of microcomputers as data-taking Asfigures 5 and 6 show,theaperiodicregime devices in physics experiments. exhibits patterns of behaviour which seem to have Finally, we must say that a similar experiment is an underlying one-dimensional structure somewhat being developed at Universidad Simon Bolivar blurred by the noise in the system. This quasi-one(Venezuela) by Professor C L Ladera at the suggesdimensional appearance of the attractors is an indi- tion of one of us (ALSB). cator of chaotic behaviour as a characteristic of the system and not a result of external noise generated, for example. by the carburetor valve or produced Acknowledgments by smallaircurrents.Ontheotherhand,these results show that a qualitative model in terms of a We wish to thank C Carbajal and J Sandria for developing the machine-language subroutine used by the data-taking one-dimensionalmappingmaybeappropriate. In procedure and for their help in setting up the experiment. fact.toanalysetheresults of theirexperiment We also wish to thank F D Micha for her help in revising Martien et a/ proposed very a simple onethe manuscript. dimensional analogue model. It is worth mentioning herethatthesystemexhibitshysteresisandthe bifurcationpointsmaydifferforincreasingand References decreasing dripping rates. Despite the fact that the system is expectedtoshowhysteresis, we believe Andrey L 1986 Prog. Theor. Phys. 75 1258 our observations to be due mainly to thevalve used Bai Lin H 1984 Chaos (Singapore: World Scientific) to control the flow of water. We are now trying to Berry M V 1981 Eur. J . Phys. 2 91 Briggs K 1987 A m . J . Phys. 55 1083 improve the arrangement and to use a good needle Collet P and Eckmann J P 1980 Iterated Maps of [he valve in order to determine this. Interval as Dynamical Systems (Boston: Birkhauser) 4. Conclusions In summary, we have presented an experiment in which students can investigate the non-linear behaviourandtheroutetochaos in adrippingfaucet. Students are able to observe a sequence of period doublingsprecedingchaosandtheexistence of a chaotic regime with various types of strange attractors. They can also convince themselves that despite the large number of variables involved in the phenomenon i t can be qualitativelymodelled by a onedimensional mapping (although we may expect better agreement with a mapping of greater dimensionality). In view of the above results, and to the relative simplicity of theexperimentalarrangement, we think that this system is very suitable for introducing theconcept of non-lineardynamicsandthetechniques for its experimental study. The experimentis of thetype of behaviour a very goodexample possibleinclassical dynamicsystems. Our experimental set-up can also be useful as an exhibit or to inform conferences addressing wider audiences. On the other hand, when usedin an open-ended investigation. it has allowed our students to explore the many features of the transition to chaos at their own D'Humieres D, Beasley M R. Huberman B A and Libchaber A 1982 Phys. Rev. A 26 3483 Feigenbaum M J 1978 J . Stat. Phys. 19 25 "-1979 J . Slat. 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