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Editor'sote: hiss he 02ndnaseriesf eviewnd utorialapersnvariousspectsfacoustics.
Methods of chaosphysics and their application to acoustics
W. Lauterbornand U. Parlitz
Drittes hysikalischesnstitut,Universititbttingen,irgerstrasse2-44,D-3400GiSta'ngen,
Federal epublic fGermany
(Received6 December 987; cceptedorpublicationAugust 988
Thisarticlegives n ntroductiono theresearchreaof chaos hysics.henew anguagend
thebasic ools representednd llustrated y examplesromacoustics:bubblen water
drivenby a soundieldandothernonlinear scillators.he notions f strange ttractors nd
theirbasins, ifurcationsndbifurcation iagrams, oincar6maps, hase iagrams,ractal
dimensions,caling pectra, econstructionf attractorsrom time series,windingnumbers, s
well as Lyapunovexponents, pectra, nd diagrams re addressed.
PACS numbers:43.10.Ln, 05.45. + b, 43.25.Yw, 43.50.Yw
INTRODUCTION
The ast10years aveseen remarkable evelopmentn
physicshat maybe succinctly escribeds he upsurge f
"chaos.t-t6This s,at firstsight,eally uzzling,s heno-
tion of chaos mplies irregularity and unpredictability,
whereashysicssusuallyhoughtobea scienceevotedo
finding he awsof nature,.e., tsorderandharmony. ow,
then,maychaos avebecome subject f seriousnvestiga-
tion in physics--and ot only physics?his is ust the new
insight--that aw and chaos o not exclude achother, hat
even simpledeterministicaws may describe haotic, .e.,
unpredictablend rregular,motion.Thusnot only aw and
order,but also aw andchaos, o ogether nd,evenmoreso,
it seemshat law andchaos re as mportant combination
as aw and order.This statementmay be derived rom the
fact hat chaoticmotion s ntimately elated o nonlinearity
and herealmof nonlinearity y far exceedshatof linearity.
Thisarticle sanattempt o acquainthereaderwith the
ideas and methods that lead to the above statements. The
basicnotions regivenwithout esortingo toomuchmath-
ematics.t is hoped hat this approachwill alsobe honored
by those eaderso whom his snot he irstexposureo the
subject.
I. ATTRACTORS
Theoretical haosphysics tartswith evolutionequa-
tions hat describehe dynamicdevelopment f the stateof a
system a model). Thesemay be continuousmodels
/t=f(x), xR'" m>l, (1)
or discrete ones
x. =gu(x.), x.R'", m>/1, n=0,1 ..... (2)
The stateof thesystems givenby them-dependentariables
x(t) = [xt(t),x2(t) .....x.(t)] or x. = (xl",x " .....
respectively.he ndex/zndicateshat hesystemepends
on a parameter/t (often it will be severalparameters). The
dynamic aws 1 or ( 2 ) determine owa givenstatex (t) or
x.develops. This evolutioncan be viewedwhen the statesof
thesystem redisplayedspointsn a state pace '". n the
continuous ase, he temporal = dynamic) evolution hen
leads o a curve n this space alled rajectory r alsoorbit
(Fig. 1 . In the discrete ase,a sequence f points s ob-
tained,usuallycalledan orbit. The statespace n nonlinear
dynamics s ntroduced bovesa generalizationf the usual
phase pace f HamiltonJan ynamics.Whenp andq are he
generalizedoordinatesndmomenta f a Hamiltonian ys-
tem, thenx = (p, q)R r with m necessarilyven.General
nonlinear ynamical ystemsmay havean odd-dimensional
state space.
An importantquestions how a setof initial conditions
(a volumeof the statespaceR'") evolves stime proceeds.
According o the theoremof Liouville,a volumestays on-
stant n conservativeystems hereast shrinksn dissipa-
tive ones.Here, only dissipative ystemswill be treated. n
this case he question lmostposestself,asto how the vol-
ume shrinksand how the limit set of points n statespace
looks,o which given olume hrinks. hissimple uestion
cannotyet be answeredn generalasobviously n unknown
number f differentimit sets repossible.he imit sets ave
been given the name attractor as trajectoriesout of whole
voluminaof statespacemove towards hesesets, .e., seem
attractedby them. The set of initial conditions points n
statespace)movinguponevolution owardsa givenattrac-
tor is called its basin.
What is alreadyknownaboutattractors nd heir prop-
erties?A certainclassificationan alreadybe given. t often
happenshat all trajectoriesn statespacemove owardsa
FIG. 1. A trajectory n statespace.
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single oint,a ixedpoint Fig. 2(a)]. Thismeanshat the
system oesnot alter with time; t hascome o rest. n the
language f physicists,his is an equilibrium osition.A
standardxamples a pendulumhathascome o restafter
some time of oscillation due to friction.
A morecomplex ossibilitys that the imit setconsists
of a closed rajectory hat is scanned gainand again.An
attractorof thiskind scalleda limitcycle Fig. 2(b) ]. Limit
cyclesegularly ccurwith drivenoscillators.he standard
examples heattractor f thevanderPoloscillator.n phys-
ics,any sinewave or squarewave,etc.) generator isplays
anexample f a limit cycle.The nextkindof attractor illsan
area a two-dimensionalurface) n a, e.g., hree-dimension-
al, statespace. his mayhappenf thesystem scillates ith
two incommensurablerequencies. his attractor consti-
tutes torus Fig. 2 (c) ]. A trajectory n the orus sa quasi-
periodicmotion.Systems ith thisproperty lsoexistexperi-
mentally see,e.g.,Ref. 12). These hree ypesofattractors
have beenknown for a long time.
Quitenew sa furtherkindof attractor, alled trange r
chaotic ttractor Fig. 2(d) ]. In the continuous ase,an at
least hree-dimensionaltatespaces necessaryor a strange
attractor o occur.The properties f strange ttractors re
not yet totallyexplored.An importantproperty s the di-
mension f the strange ttractor,whichusually urnsout to
befractal, i.e., not an integer. n Sec.VIII, we discuss ow
thedimension f general ets anbedefined nddetermined
in practicalsituations.A further property s that strange
attractorsobviouslypossesself-similar tructures;.e., on
magnifyingheattractor,partialstructuresepeat gainand
againon a finerand finerscale. he notionof self-similarity
seemso playan importantpart n chaosphysics sdoes he
notionof fractaldimension.n Sec.X, we discuss ow t may
bebrought boutby thedynamic aw by stretching nd old-
al {b
(c) (d)
FIG. 2. Typesofattractors: a) fixedpoint, (b) limit cycle, c) torus, d)
projection f a strange ttractor.
?nga olume f state pace. uch bjectsbviouslyelongo
thedeepernner tructuref nature.? t maybe nteresting
to note that the discovery f strangeor chaoticattractors
gradually ame hrough heoretical rguing nd that it is
mainly hroughmodels ith chaotic ehaviorhat t hasbe-
come possible o interpret measurementshat were long
known n the language f chaosphysics. coustics assup-
plieda prominent,ndoneof the irst,examplen the ormof
acousticavitationoises-2nd elated xperiments?
A dynamical ystemmay possesseveral ttractors i-
multaneouslyhat arereached tarting romdifferentnitial
conditionsn statespace. he space f initial conditionss
then divided into different areas, the basins of attraction,
eachof which belongs o its correspondingttractor.One
speaks f coexistingttractors. bviously, ny ypeof attrac-
tor so far known can coexistwith any other type including
the same ype. Thus a systemmay have several ixed points
or severalchaotic attractors and any mixture. An example
for coexistingimit cycless the resonanceurveof a driven
nonlinearoscillatorwhere he maximumof a position oor-
dinate of the limit cycle s plottedversus he frequencyof a
driver. At higher driving, it attains the appearanceof a
breakingwave (Fig. 3). Different oscillatorystatesare ob-
taineddepending n theway thecurve s tracked. his phe-
nomenon s well known as hysteresis. xamples or coexist-
ing chaotic attractorsare, for instance, ound in Ref. 22
where he single-valley uffingoscillator s explored.
Severalquestions oncerning oexisting ttractorscan
immediatelybe posed,suchas, e.g., how many attractorsa
givensystemmay have.This question s usuallynot easily
answered.t may happen hat a systempossessesnfinitely
many coexisting ttractors.For driven nonlinearoscillators
(e.g., hebubble scillator), henumberof coexisting ttrac-
torsgrows apidlywhen he damping s decreased.
The basinsof attraction usually do not have a simple
appearance. ven n the caseof just two coexisting ttrac-
tors, the boundariesof the two basinsmay be incredibly in-
tertwined ndevenbecome fractalset.An example f typi-
cal basinsof attraction s taken from the Duffing equation
+ d -- x + x3 =fcos ot, which s a damped onlinear
oscillatorwith a two-valleypotentialdriven by a harmonic
2.10 -
L93 -
1.76-
Ra-nax.59-
1.42-
1.08 -
40.0 GO.O 280.0
Ps = 20. kPa Rn = 10.
/J [kHz]
FIG. 3. A resonance urveof a bubble n water drivenby a sound ield.For
the model used,seeEq. {9). Radiusof the bubbleat rest R, = l0 urn,
sound-pressuremplitude20 kPa (0.2 bar). in the regionbetween oand
o2, wo coexisting ttractorsare present hat are reached rom different
initial conditions.
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force of amplitudef and frequency o.For a dampingcon-
stant d = 0.2, a forcingamplitude = 1, and a forcing re-
quencyco= 0.85, this oscillatorhas three stableattractors
whosebasins f attractionare shown n Fig. 4 in black, grey,
and white. The coordinatesn the plane are (x,o = Jr) and
are the initial conditions with which the solution of the Duff-
ing equationwas startedat t ---- . A set of 320 by 320 initial
pointshas been used.Each point has beencoloredblack or
grey or left white according o the attractor to which the
solutioncurve ends.The attractorsare two period-2station-
ary solutions nd oneperiod-1 stationarysolution.The black
and white areasare the basins elongingo the period-2at-
tractorsand the grey area belongs o the period-1 attractor.
The fivebig dots epresenthe threeattractors.Thesepoints
are givenby stroboscopicallylluminating the solutioncurve
[x(t),o(t) ] at times ,= n 2rr/co. his eads o onepoint or
the period-I attractor and two pointseach or the two peri-
od-2 attractors.The black basinbelongs o the period-2 at-
tractor represented y the two white dots, he white basin o
the period-2 attractor represented y the two black dots in
the whitearea., nd he greybasin o the period-1 ttractor
represented y the black dot in the grey area. The reader
interested n the questionof basin boundariesmay consult
Reft 23 and from there explore the stateof the art.
II. BIFURCATIONS
When doing experiments, t is found that the system
investigatednormally dependson severalparameters. n a
typical measurement,usually only one of the parameters
(pressure, emperature,voltage,current, etc.) is altered to
learn about the reaction of the system o the alteration. In
theoretical anguage, ne considers one-parameter amily
of systems. he question hen is how an attractor or coexist-
ing attractors alter when a parameter is varied. In chaos
physics, ucha parameter s calleda controlparameter. ys-
tems with 'differentvaluesof the control parameter are dif-
ferent systemsand may have totally dittrent attractors.
Therefore, heremustbe parametervalues t which the type
cl 0.2 f 1.0 co 0.8.5
u
4.0
I0
o.o [
10- i
30
-4.0 [ I I [ I i.[
-3.0 -2.0 1.0 0.0 1.0 2.0 3.0
FIG. 4. Basinsof attraction for the double-valleyDuffing equation
+ d -- x + x3=fcos tot for d = 0.2, f= I, to= 0.85.Thereare three
attractorswith their hreebasins. Courtesyof V. Englisch.)
or grossappearance f an attractor switches o anotherone,
or evenustdisappears,r isgenerated.hischange,nclud-
ing birth anddeath, scalledbifurcation. he setof param-
etervalues t whicha bifurcation ccursscalled ifurcation
set. t is hus subsetfparameterpace, hich,n a general-
izationof the abovenotions,maybe highdimensional.
There re hree asicypes f ocal ifurcation,heHoof
bifurcation,hesaddle-noder tangent ifurcation, nd the
period-doublingr pitchforkbifurcation. hesebifurcations
are called ocalbifurcations, s the phenomena ssociated
with themcanbestudied y inearizinghesystem bout
fixedpointor periodic rbit n the mmediateicinity f the
bifurcationoint of a control arameter). igure5 shows
anexampleor eachof the ypes fbifurcation. he standard
example or a Hopf bifurcation s the onsetof a self-excited
oscillationn the van der Pol oscillator +(x 2-- l)J:
+ co2x 0 at = 0 [Fig.5(a) . In this ase, fixed oint
changeso a limit cycle. ia Hopfbifurcation,limit cycle
may alsochange o a (two-dimensional) orus.
A saddle-nodeifurcation ccurs t the pointsof the
resonanceurveswith thedriving requenciesolandco2n
Fig. 3. At thesepoints, one of the two attractors oses ts
stability nd"jumps," n realityveryslowlymoves,owards
theother ttractor. igure b) showshischangen (pro-
jected) tate paceccordingo the umpatco.A limitcycle
of owamplitudehangesoa imitcycle f arger mplitude.
It is alsopossiblehat a limit cycle s replacedhrough
saddle-node ifurcationby a chaoticattractor. Also, via a
saddle-nodeifurcation,otallynewoscillationrequencies
may be introduced nto a system e.g., subharmonics).
These new" oscillationrequencies,.g.,of period3, are
due to coexistent attractors that take over at the bifurcation
point.
The ast ypeof bifurcation,heperiod-doublingifur-
cation, nlyoperatesn periodic rbits.ts importanceas
become lear only in the last few years seeRefs. 3 and 4)
and hasstressedhe importance f oscillatory ystemsor
our understandingf nature.At a period-doublingifurca-
tionpoint, s hename tates, imitcycle fagiven eriod '
changeso a limit cycle f exactly oubleheperiod, T. This
appears eculiar, ndevenmorepeculiars that his ypeof
bifurcation referentiallyccursn the ormof cascades;.e.,
whena perioddoubling asoccurred,t is very ikely hat,
upon urther lteringhecontrol arameter,furtherperi-
od-doubling ifurcation ccurs ielding T, and soon. In-
deed,via an infinitecascade f perioddoublings, chaotic
attractor anbeobtained. his eads s o themoresophisti-
(
T 2T
(c)
FIG. 5. Examplesor he hree ypes f ocalbifurcations:a) Hopfbifurca-
tion (fixedpoint limit cycle), b) saddle-nodeifurcationlimit cycle
limit cycle), c) period-doublingifurcationlimit cycleof periodT
limit cycleof period2/3.
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catedquestion f possibleequencesf bifurcationshena
parameterf a systems changed.n thecontext f chaos
physics,uch equencesrecalledoutesochaosndcertain
scenarios are observed.
III. ROUTES TO CHAOS
It hasbeen ound that each of the local bifurcationsmay
give ise oa distinctoute ochaos, ndall three asicoutes
havealready eenobserved.: These outes re of impor-
tancebecauset is often difficult to conclude rom ust irreg-
ular measured data whether this is the outcome of intrinsic
chaotic ynamics f the system r simplynoisen the mea-
suringsystem outerdisturbances). hen,uponaltering
the controlparameter, neof the threebasic outes s ob-
served,hen hisstrongly upportshe dea hat hesystems
a chaotic neproducinghe rregular utput hroughtsvery
dynamics tselL n the contextof measurements,n alterna-
tive way to distinguish etween ntrinsicand extrinsicnoise
hasbeendeveloped. his method s discussedn Sec. X.
The scenario asedon a sequence f Hopfbifurcations s
calledquasiperiodicoute o chaos, s a systemwith incom-
mensuraterequencies ndergoes uasiperiodic scillations
(Fig. 6). This route s connected ith the namesof Ruelle,
Takens,and Newhouse.4'2st is a somewhat urprising
routebecause, tarting rom a fixed point, the three-dimen-
sional orus generated fter three Hopf bifurcations s not
stable n the sensehat there existsan arbitrarily small per-
turbation f thesystemalteration f parameters)orwhich
the three-torus ivesway o a chaotic ttractor.This route o
chaoshas been found experimentally n the flow between
rotatingcylinders Taylor-Couette flow) and in Rayleigh-
Bnard convectionwhere a liquid layer is heated rom be-
low. TM
The route o chaosmediatedby saddle-node r tangent
bifurcations omesn different ypes,but all with the appear-
anceof a direct transition rom regular o chaoticmotion.
The mostprominent ype s called he intermittency oute o
chaos. This route is connected with the names of Pomeau
andManneville?:? t onlyneeds single addle-nodeifur-
cationand s not easilyvisualized n its propertiesn a single
diagram Fig. 7). It is n a senseeallya route o chaos and
not ust a ump), as n the immediatevicinity after the bifur-
cationpoint tc, thetrajectory ontainsong ime ntervals f
(almost) regularoscillation so-calledaminarphases)with
only short bursts nto irregular motion. The period of the
oscillations qualsapproximately hat beforechaoshas set
in. With increasing istance it --/% [ from the bifurcation
point tc into the chaoticregion, hese aminar phases e-
FIG. 6. Quasiperiodicoute o chaos ia a sequencef Hopfbifurcations.
IJ.,I=
FIG. 7. Intermittency oute o chaos ia a saddle-nodeifurcation.
comeshorterand shorter,and the intervalsof visiblychaotic
oscillations arger and larger, until the regular oscillation
intervalsdisappear. haos s reallydeveloped nly at It val-
uesat somedistance rom tc. This route has, or instance,
beenobservedn Rayleigh-Bnardexperiments. esideshe
intermittencyroute, there is a different ype of transition o
chaos connected with saddle-node bifurcations. It consists of
a direct ransition rom a regularattractor (fixedpoint, imit
cycle) to a coexisting haoticone without the phenomenon
of intermittencydescribed bove.This type is usuallyen-
countered n systemswith many coexisting ttractors,as n
the caseof bubblesn a liquid drivenby a sound ield.
The route to chaosencounteredwith the period-dou-
blingbifurcation s called heperiod-doublingoute o chaos
(Fig. 8). This route s connectedwith many names,with
Sharkovskii, Grossmann, Thomae, Coullet, Tresser, and
Feigenbaumeing hemostprominent nes.- Aperiodi-
city is introduced here in steps,as every period doubling
transformsa limit cycle at first only into a limit cycle of
doubledperiod.But when he sequencef successiveeriod
doublingsconsists f infinitely many doublings, he limit
will be a periodof infinity, .e., an aperiodicmotion.This, of
course,can only happenat a finite value of the control pa-
rameter t, when the intervals n/z betweensuccessiveou-
blingsget smallerat a sufficientlyapid rate. This is indeed
the case.Perioddoubling s governed y a universal aw that
holds n the vicinity of the bifurcationpoint to chaos tc.
Actually, there are several aws. One of these states that
when the ratio 5,of successiventervalsof/, in each of
which there is a constantperiod of oscillation, s taken,
5, = (it,, -it,_ )/(It,,+ --It, ), (3)
whereIt,, s hebifurcation oint or theperiod rom2" T to
2" * ' T, then in the limit n- o a universalconstant s ob-
tained,' which or usualphysical ystems as he value
lim 5,,= = 4.6692-" . (4)
This number s called the Feigenbaumnumber5, because
FIG. 8. Period-doublingoute o chaosvia an infinitecascade f period-
doublingbifurcations.
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Feigenbaum iscoveredts universality. he numberhad
been oundpreviouslyn the nverseascadeonsistingf
bandsof periodic haos onvergingowards he accumula-
tion point/t from the opposite ide? It is not known
whetherhisnumber anbeexpressedyothernumbersike
r or e, or isa totallynewnumberof a similarkind.At pres-
ent, t canonlybedetermined umericallyo some ccuracy
(like rande). Thepe_riod-doublingoute ochaos asbeen
found n manyexperimentsy now,but among he irstwas
a purelyacousticalxperiment,'8-2he acoustic avitation
noise,whichwill bediscussedn greater etail n a forthcom-
ing article.
Perioddoubling asbeen oundexperimentallyn sig-
nificantly ifferent ystems-u and n areas s different s
physics hydrodynamics, coustics, ptics), electronics,
chemistry, iology, ndphysiology. large lass f physical
systemshat sconsideredf specialmportances thedriven
nonlinearscillators(bubble scillator,2-36 uffing scil-
lator, vanderPoloscillator,?Todaoscillator38).heyall
showperiod-doublingndsaddle-nodeifurcationso chaos
(the vanderPol oscillator, lsoHopf bifurcations) nddis-
playcommoneatures onnected ith theirresonancerop-
erties. The bubble oscillator will be described in more detail
in a separate rticle.
IV. BIFURCATION DIAGRAMS
Severalmethodsare available o handleexperimental
data in the attempt to determinewhich route to chaosmay
apply.They are not all similarlywell suited o each oute so
that usuallyoneshould ry all of them. n the intermittency
route o chaos,hedirectlymeasuredimedependencef the
variable considered is taken to observe the continuous
shortening f the aminarphases ith change f the control
parameter.n the quasiperiodicoute, he time dependence
of a variableusuallyhasno specificeatureshat wouldeasi-
ly be detectable.n thiscase he Fourierspectrum houldbe
calculated,which immediatelydetects he new frequencies
appearing ponalterationof the controlparameter. he pe-
riod-doubling oute to chaos, oo, is best observed n the
spectrum f the data,as he successiveppearance f linesat
half the owest ine (and their harmonics) s verycharacter-
istic.But in thiscase lsoa plot of the imedependenceften
yieldsgoodhints.
The methods recommended so far stem from usual data
analysis. heyare not quitesatisfactoryor copingwith the
problemof chaoticmotion.Therefore,chaos esearch as
invented nd ntroducedts ownspecificmethodso display
its results.
Among hesemethods re the bifurcation iagrams hat
havebecome powerfuland standard ool in visualizing he
properties f a system. n a bifurcationdiagram, he attrac-
tors of a system re plottedversus he controlparameters.
This is the idea that, however, usually cannot be fully real-
ized due to the dimension of the attractors and also of the
parameter pace.As for visualization urposes, ormally
only theplaneof the paper savailable;ust onecoordinate f
the attractor (a projection) s plotted versus singlecontrol
parameter.This works or discretesystems 2). In the case
of continuous ystems, discretization seeSees.V and VI
below) is necessary.
The standard xample f a bifurcation iagrams here-
fore given by one-dimensionalterated maps x,+,
=f,(x,), x,[a,b]; x,,a,bR, dependingna single on-
trol parameter/t,/t, as, n this case, he attractors re at
mostone dimensional. his map showsup, for instance, n
(strongly) dampedoscillatorysystemswhen Poincar6sec-
tionsare taken (seeSec.V). Figure 9 showsa bifurcation
diagramof the so-calledogistic arabolan the formx, +
= 4/tx, ( 1 -- x, ). For small/t, the attractor sa fixedpoint.
It splits nto a periodic ttractorof period2, thenperiod4,
etc., until at the accumulation oint/t the period2% i.e.,
aperiodicity,sobtained. fterwards,hese eriods repres-
ent in the form of bands hat combine, n a reverseway to
perioddoubling,o a single haotic and. 9 n the chaotic
region,parameter ntervalswith periodicattractorsappear,
e.g.,of period3. Most of the knownproperties f thissystem
are collected n Refs. 11 and 39; seealso Refs. 3 and 4.
A variant of this kind of bifurcationdiagramhasbeen
introducedyus n thecontext facoustichaos'Sndcalled
spectralifurcationiagram?2Of course,lso n thiscase
only a singlecoordinateof an attractor can be handled,
whosepowerspectrums plottedversus he controlparam-
eter. Sincea powerspectrum tselfneeds wo dimensionso
be plotted,additionaldifficulties ppearwhichmay be over-
comeby usinggrey scales r, evenbetter,color graphics see
Ref. 1, color plate VI or Ref. 20, plate I). Spectralbifurca-
tion diagrams re especiallywell suited o experitnental ys-
tems, where external noise usually is hard to avoid. This
noise s distributed vera wide spectral ange,whereas he
energyn the systemsconcentratedn a few ineswhen,e.g.,
perioddoublingaddsnew ines.These hen stick out from
the noisebackground nd are easilydetected.n acoustics,
spectralbifurcationdiagramsare well known as "visible
speech"when ime is consideredsa "controlparameter."
V. POINCAR SECTIONSAND POINCARiMAPS
When trying to displaybifurcationdiagrams, s men-
tionedabove,difficulties risecoming rom the dimension f
the space eededor thispurpose. his problemhad already
beenencountered y Poincar6n the contextof copingwith
the problemof the stabilityof the solarsystem.He invented
what today s called a Poincar$section,wherebyone dimen-
1.0
X
0.5-
FIG. 9. The bifurcation diagram of the logistic parabola n the form
x.+ = 4/L. (1 --X.).
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sioncan be gained,and a continuous ystem f the kind in
Eq. ( 1 is transferred o a discrete ystem f the kind in Eq.
(2). When nvestigating igh-dimensionalystems,his s of
little help, but when workingwith low-dimensional, spe-
cially three-dimensional, ystems, he limit of visualization
of the properties f a systems shifted or quitea largeclass,
among hem the driven one-dimensional scillators.When
havinga three-dimensionaltatespace, Poincarsections
simply a planeS (a hyperplane n higher dimensional ys-
tems) in the statespace hat is intersected y all trajectories
transversallyFig. 10). Usuallysucha planemay only exist
locally,but globalPoincarsections re alsoencounteredn
certainspecial ystems, .g., n our drivenbubbleoscillators.
We thereforeconsideronly globalPoincar6sections ere.
Sectionplanesare well suited o investigate he stabilityof
periodicorbits. n chaos esearch hey are now frequently
used o displaystrangeattractors,wherebyonly the section
pointsof the trajectoryof the attractor n the planeS are
plotted.n Fig. 11,a strangeubblettractor,.e.,anattrac-
tor of a drivenbubblen a liquid,as t appearsn a Poincar
section, s given.The sectionpointsarrange hemselves n
lines oldedover and over again.They hop aroundon this
structure n a hardly describablemanner.To show ts fractal
natureand self-similarity, spointedout in Sec. , a blowup
is given n Fig. 11 b) that reveals he occurrence f the same
structure on a finer and finer scale.
It is possibleo considerhe dynamics f a givencontin-
uous ystemn thesection lane$ only.Whena pointQ1 _S
is taken, t is imagedvia the dynamics f the system o the
point Q2 = P(Q, )_S. n this way, a continuous ynamical
systems transferredo a discrete ynamical ystem, iven
by a map romS to S. Thismap scalled PoincarmapPer
also irst returnmap. t is immediately een hat a periodic
orbit of the continuous ystem ecomes fixedpoint of a
correspondingiterated) Poincar6map.One dimension as
beensaved n this way. A quasiperiodic rbit consisting f
two incommensuraterequencieshen ooks ike a limit cycle
in the Poincarsection a cut transversehrougha two-di-
mensionalorus).This"limit cycle,"however,smadeup of
pointshopping round,not by a smooth eriodic rajectory
in the section lane.
periodicrbit
FIG. 10. Poincarsection laneS and the Poincar6map P. The section
pointQ of a periodic rbit?, sa fixedpointof thePoincarb apP.
(a) Rn= 10. am
Ps = 90. kPa u
2.60
1.32 -
Up o.o4-
[m/.l_ 1,24
-2.52
-3.80
0.50 1.15 1.40 1.65
(b) Rp/Rn
1.30 -
1.90
0.g8 -
Up 0.66-
[m/,] 0.34
0.02
-0.30
.42
1.44 I J46 lJ46 1.50
Rp/Rn
FIG. 11.A strange ubble ttractor n a Poincarsection lane. a) Total
view, (b) exploded iew that indicates he self-similarity f the bandstruc-
ture.
With the helpof the Poincarsectionmethod, he peri-
od-doubling oute to chaoscan adequately e displayed
without esortingo spectralepresentations.pon he irst
period-doublingifureation,he ixedpointsplitsntoa peri-
odicorbitconsistingf twopoints hatare maged ackand
forth. At eachsuccessiveerioddoubling, achof the pre-
viouspointssplits nto two new pointsuntil in the limit at
/x =/z oo=/% the point set of an aperiodicattractor s ob-
tained.For a bifurcationdiagram,usually he pointsof a
Poincarsection re used,wherebyone again s forced o
takeonlyonecoordinate f a point n thesectionor simple
visualization.
The Poincar6mapP defined n a surface determines
where he pointsors are magedunderP. ThusP considers
wholeset of initial conditions imultaneously.n physics,
usuallyonly single rajectories anbe followed e.g., when
measuring pressuren dependencen ime). The trajectory
may thenbe describedn discreteorm through he series
P(xo ), P (P(xo) },..., xo being he nitial condition.This leads
to iteratedmaps compare q. (2) ]:
x,+ =P(x,), n=0,1,2 ..... (5)
The simplestmap P with nontrivialdynamicshat may be
encountered ill be a one-dimensional apwith some unc-
tion .'
x+=f(x,, xR, n=0,1 ..... (6)
Whena controlparameter/zs introducedor comparison
withexperiments,nes ed oa family fmaps,:
x,+,=f,(x,), xR, /zR, n=0,1 ..... (7)
Thus hePoincar6mapconnectsontinuousynamical ys-
tems (differential quations)with iteratedmaps. terated
mapshave ongbeen nvestigatedy pure mathematicians
who collected a wealth of beautiful results that now find
1980 d. Acoust. Sec. Am., Vol. 84, No. 6, December 1988 W. Lauterborn and U. Parlitz: Chaos acoustics 1980
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applicationsn physics.With respecto the corresponding
differential quations,teratedmapsare muchsimplerand
canbe investigated uchmoreeasilyboth analytically nd
numerically.Yet the samerichness n behaviorcan be ex-
pected s n the originaldifferential quation.How involved
a behaviormust be envisageds strikingly demonstrated y
the "simple" exampleof the quadraticmap or logisticpa-
rabola
x,+, =4/x,(1--x,), x,[0,1], /[0,1], (8)
the bifurcation iagramof which, n a sense, ancompletely
be givenand s plotted n Fig. 9.
One-dimensional aps ike thoseof Eq. (7) canbecon-
structed rom the Poincar6mapP only n those aseswhere
the (strange)attractor n thesection lane esembleslmost
a (thin) curve. Then either a projection n some suitable
directionwill do, or somecoordinates long he curvemust
be ntroduced ith respecto whicha one-dimensionalap
may be formulated.Such maps are often called reduced
(Poincarb) aps r, assuggestedy us,attractormaps?
Figure12 a) shows Poincar6 ection lanewherea strange
attractor can be seen or a bubblewith radiusat rest of R,
= 10/m, drivenat a sound-pressuremplitudeof 276 kPa
and a frequency f 530 kHz. It is noticed hat the section
pointsmake up eight short line segments, nd thus a one-
dimensionalmap may be constructedor eachof them. This
is indeedpossible, s he dynamics n the attractorare both
regular nd chaotic. he regularity onsistsn the fact hat,
from one sectionof the chaotic rajectory o the next, the
section oints oaround romonesegmento thenext n the
manner ndicateduntil all eight segments avebeenvisited.
Then the sequencetartsagain.The chaos n the dynamics
comes rom the fact that on a segmenthe pointscomeback
Up
Rn = 10. /.zm
Pa = 276. kPa
18.0 -
14.4
10.8
7.2-
3.6-
0.0
v = 530.0 kl/z
1.2 1.5 1.8 .1
Rp/Rn
1.90
1.86 -
R a 1.82-
Rn 1.78-
1.74-
1.70
1.70
.90
FIG. 12. Period-8 chaotic attractor of a bubble oscillator in a Poincar6 sec-
tion plane a) and a subharmonicttractormap of ordereight (b). The
encirclednumbersn (a) indicate he successionf the sectionpoints.
in an rregularway or whichno ong-termprediction anbe
made. For this type of behavior,known from the inverse
cascadesf the ogistic arabola, he termperiodic haos as
been oined.9Figure12(a) thusshows period-8 haotic
attractor. n thiscase, trong ines n the Fourierspectrum f
the motionat - he driving requencynd heirharmonics
appear seeSec.VI). If onlyeveryeighthsection ointwere
plotted,onlyone ine segment ouldshowup. Sucha plot s
calleda subharmonic oincark ection lot of ordereight.
From one line segment,which, to be sure, s only ap-
proximately line,a reduced oincar6mapor attractormap
maybe constructedn the followingway. Everyeighthsec-
tionpoint s akenand hepoints f thissequencereplotted
versus he previous ne. This means hat x + 8 is plotted
versus ,, x being he original terationpointswith n = 1,
9,17,... When one coordinate, n this case he radius of the
bubble, s taken,a map ike that shown n Fig. 12(b) is ob-
tained or a certainsegment. here will be eightdifferent
attractormaps f thiskind,dependingn thestarting oint,
i.e., hesegmenthosen. he typeof mapgiven n Fig. 12(b)
is called a subharmonic ttractor map of order eight. It
strongly esembleshe ogistic arabola 8). Whena param-
eter in the originaldifferential quation s altered, t may
happenhat hecorrespondingttractorn thePoincar6 ec-
tion planealters n sucha way that the correspondingsub-
harmonic)attractormapalters ike a logistic arabolawhen
producing erioddoubling. hen perioddoubling n the
continuous ystemmay be said o occuras in the logistic
parabola. his ollowsrom he universalityf thescaling
lawsgoverningeriod oubling)
Whenever he attractor s more complicated,wo-di-
mensional apsmust econstructednd nvestigated.ince
the time that the connection etweenteratedmapsand con-
tinuous ynamicalystemsasused y physicists)asbeen
clearly oticed, otonlymathematiciansutalsophysicists
(and otherscientists) ork ntensely n the propertiesf
iteratedmaps,with muchcomputerworkgoingon. n any
case,he occupation ith iteratedmapss stronglyecom-
mendedor thosewhowish o geta deeper nderstandingf
chaos. he newcomermaystartwith the articleof May in
Nature. t
Vl. DEMONSTRATION OF SOME METHODS OF CHAOS
PHYSICS CHOOSING A BUBBLE OSCILLATOR
In this section,we pause o demonstrate omeof the
methodsoherentlyna bubble scillator.hebubble scil-
lator used s a nonautonomousifferential quationof sec-
ond order of the form 35
pc dt
with
P(R,,t) = P R) -- 2a/R -- 4/(//R) -- Pta
and
+ PO - P sin(2rrvt)
P,(R) = (P,t -P + 2rr/R,, (R,,/R) 3':,
whereR = R (t) is the radiusof the bubbleat time t, R,, is the
1981 J. Acoust. oc.Am.,Vol.84, No. 6, December 988 W. Lautorbornnd U. Parlitz:Chaosacoustics 1981
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Rn = 10. /rn Ps = 90. kPa p = 207. kl'iz
1.84
1
R 1.o6-
Rn
0.67 -
0.28
1,449 1,490 1.511 1.542 1.573
t
1.804
Rn = I0. p.m Ps = 90. kPa p = 197. kHz
1.84
1.45 -
R 1.o6-
Rn
0.67 -
0.28
1.5220 I. 8 1.5878 1.6204 1,6532
t
Rn = 10. /m Ps = 90. kPa = 193. kHz
1.8880
1.45 -
l.O6-
Rn
0.67 -
0.28
1.5540
1.6206 1.6542
t [ms]
1.5874 I.876
1.7210
Rn = 10. /m Ps= 90. kPa p = 192.5 klz
1.64
1.45 -
.o6-
0.67 -
0.28
1.5580 1.5914
1.6246 1.6582
t [m
1.6916
1.45-
R 1.00
0.67 -
0.26 -
1.578
Rn= 10. /m P
= 90. kPa p = 190.0 kl'Jz
1.7250
Rn = iO. itrn
Ps = 90. kPa
56.0 -,
t, = :207.0 kl'lz
32.8 -
0.6-
-13.6'
-36.8 '
-60.0
0.28
0168, 'JOB 1J48 1.88
R/Rn
P.= 90. kPa u= 197.0 kJz
56.0-
32.8
9.6-
13.6-
-36.8 -
-60.0
0.28 olo8 ,JoB
R/R
P,= 90. kPa u= 193,0 kHz
56.0
1.88
32.8
0.6
-13.6
-36.6
-60.0
0.:8 0168 1J08 11,,8
R/Rn
1.88
Ps = 90. kPa u= 192.5 kHz
56.0
32.8
0.6-
-13.6-
-36.6 -
-60.0
0.28 0J68
I JOB I J46
R/Rn
1.88
P = 90. kPa t,= 190.0 kJz
,56.0
32.8
u 9.8-
["'/']-1a.8
-36.5
-60.0
0.28 oj68
.612 1.646 1.660 1.714 1,748 I.'08 IJ48 1.88
t [ms]
FIG. 13.Period-doublingouteochaosemonstratedy he ttractorsora bubblescillator.eft olumn:adius-timeolutionurves;iddleeft olumn:
trajectoriesnstatepace;iddleight olumn:oincar6ectionlots;ight olumn:owerpectra.adiusfbubbletrest , = 10pro,ound-pressure
amplitude0kPa 0.9bar),drivingrequencyst ow: 07kHz;2rid ow:197 Hz,3rd ow:193 Hz,4th ow:192.5 Hz,5th ow:190 Hz.
1982 J. Acoust.Soc. Am., Vol. 84, No. 6, December 1988 W. Lauterbom and U. Parlitz: Chaos acoustics 1982
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Rn = 10. Hm
P = 90. kl'a , = 07. kflz Rn = 10. /_Lm Ps = 90. kPa = 207.0 kliz
2.60
1.37
0.04
O.gO
1115 I J40 IJ65
Rp/Rn
2.60 '
1.32-
0.04 -
up
[,/sl -I.24
-2.52
-3.80
Ps= 90. kPa u= 19%klz
L90 1115 I.'40 IJ65
Rp/Rn
1.90
Ps= g0. kPa u= 193. kliz
2.60j
.32
Up .04
[=/.1-1.24
0.90
Rp/Rn
l.g0
Ps = 90. kPa u = 192.5 kl'lz
2.60-
1.32 -
Up 0.04-
[m/s] -i.24 -
-2.52-
-3.80
0.90
ee
l.g0
IJ15 I.'40 .'6 1.90
Rp/Rn
Ps = 90. kPa u= 190.
Up0.04
[m/s]-1.24
0.90 1.15 1.40 1.65 1.90
Rp/Rn
1o
lO
10 "
1
0.0 I 03.5 207.0 31'0.5 414.0 51'7.5
f [kHz]
Rn = 10. /am Ps= 90. kPa u = 197.0
1o'
td'
::
lO-:'
0.0
1
98.5 lirLO 295.5
f [kHz]
Rn = 10. /m Ps -- 90. kPa
621.0
394.0 492.5 591.0
s
lO'
lO
lO-'
0.0
193.0 kliz
96.5 193.0 289.5 386.0 482.5
f [kHz]
lO'
lff
1o-
1o-:
Rn = 10./zm Ps = 90. kPa v= 192.5 kl{z
96.25
192.50 288.75
f [kHz]
I
0.00
,,
85.00 481.25
v = 190.0 kl'lz
S
ld
10-'
10-:
0.0
Rn = 10. /xm Ps = 90. kPa
285.0
f [kHz]
579.0
95.0 190.0
5'77.50
380.0 475.0 fi70.0
FIG. 13. (Continued.)
1983 d. Acoust.Soc. Am., Vol. 84, No. 6, Docombor1988
W. Lauterborn and U. Parlitz: Chaos acoustics
1983
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radiusof the bubbleat rest,v is the frequency f the driving
sound ield, Ps is the amplitudeof the driving sound ield,
stat 100 kPa is the staticpressure, v = 2.33 kPa is the
vapor pressure, = 0.0725 N/m is the surface ension,
p = 998kg/m s hedensityf he iquid,/.t 0.001N s/m3
is the viscosity,= 1$00 m/s is the sound velocity, and
c= 4/3 is the polytropicexponentof the gas n the bubble.
Here, denotesifferentiationf he adius ith especto
time t. The model 9) describes spherical asbubbleof
radius R(t) in water, set into motion by a sound ield of
sinusoidalimedependenceonstant t anyone ime all over
the bubble urface. or this article, t may sufficeo ust give
the bubblemodel without discussingts derivationor rela-
tion to other bubblemodels.A more thoroughdescription
will be givenelsewhere,ogetherwith a detaileddiscussion
of the chaoticproperties f someof its solutions.
When the sound-pressuremplitude (control param-
et. r) of the drivingsound ield s ncreased t constantre-
quency, r when he frequency anothercontrolparameter)
of the drivingsound ield s alteredat constant ound-pres-
sureamplitude,peculiar hingsmay happenwith the radial
( ) oscillationof the bubble (Ref. 35; see,also,Refs. 32 and
34). Despiteperiodicexcitation,chaoticoscillations re en-
countered or someparametervalues.An examplewhere
thishappens ia a period-doublingoute s given n Fig. 13,
which displays20 diagrams n four columns.A bubbleof
radiusat restR = 10m, drivenat a sound-pressurempli-
tude of 90 kPa (0.9 bar), hasbeenchosen. he frequency f
the driving sound field is lowered from v = 207 kHz to
v = 190kHz. In fivesteps,he stationary olutions nd heir
spectra (after transientshave decayed) are plotted for
v = 207, 197, 193, 192.5, and 190 kHz, the same or all dia-
grams n a row. The dots n the diagrams f the radius-time
curves left column) correspondo a certainphaseof the
driving sound ield. Their interval thus correspondso the
period T of the driving. The rows contain to the left the
radius imecurves?ollowed y phase pace velocity ersus
radius) curves (trajectories), Poincar6 sectionplots, and
powerspectra f the radius-timecurves. n the first row, the
bubbleoscillatesn a stationarystatewith the periodof the
driving. In the languageof chaos heory, we have a limit
cycleasan attractorwhichhas heperiodTofthe driving. n
the radius-time plot, the thick dots therefore all lie at the
same evel R/R,. The limit cycle rajectory n the second
columnalso s markedby just one dot, as t exactlyrepeats
after one period.With periodicallydrivenoscillators, glo-
bal Poincar6 laneof section anbe defined y a fixedphase
of the driving.The plane hen s madeup of the radiusR e of
the bubble (given here normalized with R, as Re/R, ) and
its velocity U,, where the index P stands or Poincar6 o
notify that it is a sectionplane, in contrast o the second
columnwhere he velocityall along he trajectory s plotted.
Thus the firstdiagram n the third columnsimplycontains
singlepoint, the one sectionpoint of the limit cycle. The
appearanceof only one point in the sectionplane indicates
that the limit cyclehas he period Tofthe driving, as s also
learned rom the otherdiagrams.The first picture n the last
columngives he corresponding owerspectrum.As expect-
ed, the lowest frequency n the spectrum s v = I/T, but
higherharmonics represent ue o the nonlinear atureof
the oscillation. In the second ow, at v --- 197 kHz, we see n
the radius-timecurve o the left that the two pointsnow lie
on two horizontal ines, ndicating hat the oscillation nly
repeats fter two periods f the driving.The corresponding
trajectory s againa limit cycle,but of more complex orm.
The two thick points ndicateperiod2 T for the imit cycle,as
do just the two points n the Poincarsectionplane.The
powerspectrumof the radius-timecurve o the right now
hasa lowest requency f v --- 1/(2 T). Again,as heoscilla-
tion s nonlinear, he harmonics fv = 1/(2T) showup; .e.,
lines at v-- 3/(2T), 5/(2T) .... are newly introduced nto
the spectrumwith respecto the first row. Clearly, his sec-
ond row indicates hat perioddoublinghas aken place. n
the third and fourth rows, t is demonstratedow period
doublingproceeds,eading o rapidlymorecomplex rajec-
toriesmakingup the limit cycle.The spectrum s filled up
with new inesexactlybetweenwo old inesof the spectrum
at eachperioddoubling. n this way, the irregular,chaotic
state n the ast ow sreached. s the rajectorysaperiodic,
the radius-time urvenever epeats, nd only gives short
segment f the actualpossible scillation equencef larger
and smalleroscillations. he trajectorynow formspart of a
strangeattractor createdout of a limit cycle. Only a few
revolutionsare plotted, as otherwisea whole area would
have urnedblack, eavingnodiscernible tructure.Also, the
points ndicating he elapse f oneperiodof the drivinghave
beenomittedso as not to disturb he picture.The spectrum
now contains some amount of noise which is intrinsic, i.e.,
coming rom hedeterministic ynamicstself.The Poincar
section lot containsmanymoresection ointsof the attrac-
tor thancorrespondo the turnsof the trajectory lotted n
the last row (second olumn) and thus bestdisplays he
involvednatureof the chaoticattractor.However, he way
pointsare maged rom onepart of the attractor o another
cannotbe givenby this type of staticpicture.
The readerwill surelyagree hat by simply ookingat a
radius-time curve (a solution curve of a differential equa-
tion) in the chaotic egion, he statement hat intrinsicaper-
iodicity s present annotbemade.However, hat no period-
ic solutions houldbe present n certain parameter egions
hasbeenarguedneverthelessrom repeated nd strong ri-
als, 2 inceheseegionsannot eoverlooked.ven searly
as 1969 t had beenobserved hat these egionsoccur near
regionswheresubharmonicsre present, nd a connection
between ubharmonicsndnoise asbeen onjectured.
The regionsof chaosare bestvisualized n bifurcation
diagrams,as then the route to chaoscan automaticallybe
discerned.Figure 14 ust givesone exampleof a bifurcation
diagramwhere he normalized adiusof the bubbleat a cer-
tain phaseof the driving sound ield (when transientshave
decayed) s plottedversus he driving requency s control
parameter.The plot hasbeenobtained n the followingway.
First, a maximum of 100 oscillations s calculated to let tran-
sientsdie out. Then 100pointsof the radiuscoordinateof the
Poincar6section lane (givenby constantphaseof the driv-
ing) areplotted.When here s a limit cycleof periodT, these
100 pointswill neatly fall one upon the other and ust one
point will showup in the diagram.Then, starting rom this
1984 J. Acoust.Soc. Am., Vol. 84, No. 6, December 1988 W. Lauterborn nd U. Parlitz:Chaos acoustics 1984
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7/24/2019 1988 Methods of Chaos Physics and Their Application to Acoustics
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Ps = 275. kPa Rn = 10. /m
2.2
1.58
1.27
0.961
51'0.0 I t
60.0 435.0 585.0 660.0 735.0 110.0
FIG. 14. Bifurcation iagramof a bubbleoscillator.
value as the initial condition, the next seriesof oscillations s
calculatedora slightlyncreasedor decreased)requency
yieldinghenextattractor.f it is of period T, twopoints
will showup in the diagram. n the caseof a chaoticattrac-
tor, 100pointswill beplotted, catteredlong verticaline
at the given requency.n this way, the properties f the
system avebeenmadevisible or 1350 requency oints.
Figure14 shows n interestingypicalpicturewith period
doublingo and romchaosmakingup a complicatedbub-
ble" structure.
Again, he eaderspointedo a forthcomingrticle or
moredetails.There, the growthof thesebubbles nd their
distributionn parameterpace long esonanceorns ield-
inga superstructure2of bifurcationsill bediscussed.his
superstructures conjectured o be universal n somesense
and for a certain class of driven nonlinear oscillators.
VII. PHASE DIAGRAMS (PARAMETER SPACE
DIAGRAMS)
When there s more than one controlparametern a
system,tspropertiesanonlybegiven n a series f bifurca-
tiondiagrams, here neparameterschosenscontrol a-
rameter, he otheronesheld fixedand only changed rom
onediagram o thenext.Experiencehowshat t is noteasy
to grasp the grosspropertiesof a system rom such se-
quences. morecondensediewcombining uchSequences
wouldbedesirable.hiscanbedonewith thehelpofphase
diagrams r, equivalently, arameter pace iagrams. he
name"phase iagram"comesrom thermodynamics, here
in apY diagramheareas remarkedwhere, .g.,a liquidor
gaseous hase s present, nd curvesare plotted o denote
theirboundaries. t the pointsof the curves, phase ransi-
tion takesplaceand alsocoexisting hases re known. n
exactly he same ense, hase iagrams f (nonlinear)dy-
namical systems re to be understood. heoretically, hey
are heplotof thebifurcation et n parameter paceogether
with the ndication f thekindof attractor "phase") n the
areascreatedby the curvesor planesof the bifurcation et.
To calculate vena fairly complete hase iagramof a dy-
namicalsystems a time consumingaskand usuallyneeds
hoursof computerime. An earlyexample f a phase ia-
gram or a bubble scillators shown n Fig. 15, aken rom
6
bor
5
2
1
o
1.6
1
T
I
1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.
Vo
FIG. 15. Phasediagramof a bubbleoscillator just a few bifurcation ines
from the actually nfinitelymany).
Ref. 32. In the parameterplane spanned y the (normal-
ized) driving requency /vo and hesound-pressurempli-
tude Pa, one curvebelongingo the bifurcation etof the
bubbleoscillator see Ref. 32 for the equationused) is
drawn, separatingegions f period-l, period-2,period-3,
and period-5oscillation. elow he curve,period-1oscilla-
tions, .e., oscillations ith the sameperiodas the driving
(period-I limit cycles),occur;above his curve,but only
very near to it, period-2, 3, and -5 limit cycles re present.
The region bovehecurvewill be urtherdivided ya com-
plicated nfiniteset of bifurcationines,sinceperioddou-
bling sets n and further saddle-node ifurcations ccur.One
more phasediagramusing he samebubblemodelas n Ref.
32 sgivenn Ref.34.Verydetailedhaseiagramsf the
Dufiing scillator + d + x + x3 =fcos cot nd heToda
oscillator d- d d- e -- I =fcos cot an be found n Refs.
21 and 38, respectively. hasediagrams an alsobe mea-
sured.An example f a measured hase iagram f a simple
electronic scillators given n Ref. 41.
The determination f phase iagrams f dynamical ys-
tems s one of the main tasksof chaos esearch s, n a sense,
theygivea complete ualitative ndevenpartlyquantitative
overviewof possible ehaviorof a nonlineardynamical ys-
tem.Specialmethods represently eingdevelopedo locate
and ollowbifurcation urvesn parameter pace,o speed
up the calculations. ue to increasen computer peed nd
availabilityof computer ime, the near future will seea
quicklygrowing etof phase iagramsor various ystems.
VIII. FRACTAL DIMENSIONS
The notionof the dimension f an object physicalor
mathematical)had ongoccupied hysicistsnd mathema-
ticiansuntil a solution ame nto sight.Cantorand Poincar6
both put great effort into this questionbut failed. It was not
until Hausdorff's article, "Dimension und/iusseres Mass"
("Dimension ndexternalmeasure"),hata satisfactorye-
finition ouldbeput orward? Thisdefinition f thedimen-
sion fa setof points aturallyeadsofractal imensions,
i.e.,dimensionshatarenot ustnaturalnumbers. or a long
time,setswith these ropertiesfractals)were hought o be
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purelymathematicalbjects ntil theirobviouslybundant
occurrencen naturewasdemonstratedy Mandelbrot?
Chaos hysics asstrongly xpandedhe mportance f frac-
tals and their dimensions. Chaotic attractors are known for
their usuallypeculiar hapes, hichpoint o fractaldimen-
sions.Th raises he question f how to determine ractal
dimensions,specially hen heyare encounteredn experi-
ments. ndeed, oncepts avebeendeveloped hereby rac-
tal dimensions anbe determined oth rom numericallycal-
culatedtrangettractorsnd rommeasuredata. 3-s3
An infinityof differentdefinitions f a dimension as
been ntroducedhaving their value n describinghe in-
homogeneityf theattractor.Onlya simplified otionof the
Hausdorffdimension,alledcapacity o, and the mostoften
usedcorrelation imension 2 will be discussedn morede-
tail here.
First, thedefinition f the dimension oof a setof points
A CR ' is given:
do= lim log M(r)/log (l/r), (10)
wherer is the edge engthof an m-dimensionalubeand
M(r) is the lowestnumberof m-dimensional ubesof edge
length to cover hegiven etA (a chaotic ttractor).When
this definition s applied to a point, a line, an area, and a
volume, he dimensions o = 0, 1, 2, and 3 are obtainedas
they shouldbe. But the definition s muchmore powerful.
Also, Cantorsetsnow geta dimension, sually ractalas t
turnsout.Figure16giveshestandard xample f a Cantor
setwhereby, tartingwith theunit nterval 0,1 , themiddle
third without heendpoints s successivelyakenout of the
remainingntervals.n the igure, o the eft, theedge ength
r is given hat is convenientlyaken o cover he set,and to
theright henumberMofeubes intervals) hat sneededo
cover he set.According o the definition f do, one then
easilygets,by simply nsertinghe sequences given n the
figure nto Eq. (10),
do imog '_ log= 0.6309-'-, (11
-1og 3 log3
a noninteger umber.This is the fraetal dimension f the
Cantor set.
It tums out that for systemswith a high-dimensional
statespacehedirectuseof the definition f do (box count-
ing) is not practical n riumericalapplications. rom the
nextmembers f dimensions,he nformationdimension ,
FIG. 16.Cantorset onstructionysuccessivelyakingout hemiddle hird
without ts end points.
the correlationdimension 2, and the higher-order imen-
sionsd3,d4 .... the correlationdimension 2 is the mostat-
tractiverom heexperimentalist'sointof view. 3'44or the
sequence of 'dimensions, the relations
d,>d,_ >'">d2ddo hold, and often the d, are
nearly all the same.Thus da, which is very convenient o
determinenumerically, s a good estimateof "the" fractal
dimension f a strange ttractor.
The correlation dimension is defined as
d = lim log C, (r)/log r, (12)
where agains theedgeength f an m-dimensionalube
and C, (r) is the so-called orrelationsum
1 N
C,,(r) lira
o -k, I
(13)
In this expression, is the numberof points n R of the
(strange)attractoravailable romsome alculations r mea-
surements, is the Heaviside tep unction H(x) = 0 for
x 0 and, usedhere, H(0) = 0], p are
thepoints f theattractor, nd [']] s a suitable orm,e.g.,
the Euclidean orm.The dimension oesnot depend n the
norm; therefore,any norm may be chosen hat is most con-
venient or numerical omputation.n our determination f
cavitationnoiseattractors,we usedd: togetherwith the
maximum orm. 7Equation12) immediatelyhowshat
d: can be determinedrom the slopeof the curveobtained
whenC,, (r) is plottedversus, eachon a logarithmic cale.
An example or the determination ld: is given n Fig.
17 for the strange ubble ttractorof Fig. 11. n Fig. 17(a),
the og C(r) vs og r curve s plottedwith 100000 pointsof
the attractor in the Poincar sectionplane. Figure 17(b)
shows he local slopeof the curve n Fig. 17(a) whereby
"local"means fit of theslope vera region f a quarterOfa
decade.t is seen hat a plateau n the ocalslope nlyoccurs
for values f logr between - 1.5and - 0.5 givinga fractal
dimensionf de = 1.3+ 0.1. For larger 's thegross truc-
ture of the attractor naturally leads o a decreasingocal
slopeuntil at r's above he sizeof the attractor he slope s
zero,because ll pointsof the attractor it into the cubeof
edge ength . At low r's the followinghappens. ecause f
the finite precisionn the calculations,he exact ocationof
the attractorpoints n the Poincar sectionplane s only
known o a limited numberof digits.Therefore, he attractor
looks more and more noisy on smallerand smaller scales.
For these easonshe localslope or small 's tends o 2,
which is the dimensionof the Poincar sectionplane. Thus
onlya region etween malland argevalues fr is available
for the determination of the fractal dimension. The fractal
dimensionld2 = 1.3 s valid or the attractor n the Poin-
car( ection lane.To get he ractaldimension f the whole
attractor, one dimensionhas to be added giving de ---- .3.
Some emarksmay be n order to warn the readerwho wants
to apply his ormalism or fractaldimension etermination.
The methodmustbe usedvery carefully,especiallywith ex-
perimental atawhenused n conjunction ith reconstruct-
edattractors seeSec. X). The errorsareusually ery arge.
1986 J. Acoust. oc.Am.,Vol.84, No.6, December988 W. LauterbornndU. Parlitz: haos coustics 1986
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Pa = 90. kPa v = 190.kffz Pat -- 10./m
(.) J
ogO-)
FIG. 17.Determinationf thecorrelation imension for hestrange ub-
bleattractorgiven n Fig. I I. (a) CorrelationsumC(r) versushecubeedge
length on a doubly ogstithmicscale, b) localslope fit to thecurve n (a }
overa rangeera quarterera decade]versus . The fit region s from -- 1.7
to -- 0.5. In this egion he ocalslope asa plateau ivingd = 1.3 0.1.
The stateof the art of the techniquesvailables discussedn
Ref. 46.
Thus ar wehavediscussedwo typesof dimensions,o
and d2, out of the series ,, n = 0,1,2 .... The definitionof
dimensionaneven eextendedo dq,where is any eal
number.sTo an experimentalist,hisgeneralizationay
seem ather sophisticatednd beyond nything hat can be
measured.heopposites true? s4Only he ull set dq,
qR}, or equivalentlyhe smooth) calingpectrumfa)
of (local)scalingndices on heattractor,9describeshe
globalstructureof the attractorsatisfactorily nd simulta-
neouslyn a measurable ay.Thescaling pectrumfa) has
a quitesimplemeaning. ake a pointof a (strange)set (at-
tractor) and a smallsphere round t. Then the numberof
pointsof theset nside hespherewill scalewith some xpo-
nent (index) a when the radiusof the spheregoes o zero.
Differentpointsof the setmay havedifferent ndices . The
spectrumfa) characterizeshestrength ra scalingndex
or moreprecisely,fa) is the global Hausdorff)dimension
of the subset f pointsof the set (attractor) with scaling
indexa. The relationf(a)
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found o thisproblem,5-57 hichmaybe ormulated s he
problem f reconstructingn attractorof a dynamical ys-
tem from a time seriesof one (measured) variable only. The-
ory states:For almosteverystatevariablep(t) and for al-
mostevery ime intervalT, a trajectory n an n-dimensional
statespace an be constructedrom the (measured)values
[p(kt), k = 1,2 ....N] bygrouping valueso n-dimension-
al vectors:
p(") [p(kts),p(kt + r),...,p(kts (n- l)r)],
(14)
where t is the sampling interval at which samplesof the
variablepare taken. t is advisableo choosehe delay ime T
as a multiple of t,, T = lt, IN, to avoid nterpolation.The
aboveconstruction ieldsa point set A t" = (p}", k = 1,
2,...,N -- n ) in the embedding pace", whichrepresentshe
attractor.The sampling ime t and the delay ime Tmust be
chosen ppropriately ccording o the problemunder nves-
tigation (seeReft 46 for details). When t, and also Tare too
small, then from one sample o the next there is little vari-
ationand he points (" ll lie on a diagonaln theembed-
ding space.On the other hand, when t, and T are too large,
then hepointsetA {"and heattractorobtained yconnect-
ingconsecutiveoints (" eta fuzzyappearance.n both
cases, the reconstruction and visualization of the attractor
are not very helpful. Figure 18shows n exampleof the influ-
enceof the delay time Ton the reconstructionof a calculated
chaotic ubble ttractor.The same quation s hat given n
Sec.VI hasbeenused.For the reconstruction,nly the cal-
culated radii of the bubble have been taken and embedded in
a three-dimensionaltate pace y using ourdifferent elay
timesTof)4, , }, and1 n units f theperiod Oof thedriving
sound ield.The bubblehad a radiusat restof 100/zmand
wasdrivenby a sinusoidal oundwaveof amplitude 10 kPa
and frequency 22.9 kHz. The reconstruction s best at
T= lTo. This is the samevalueas for a simpleharmonic
waveof periodTo where he elongation and the velocity
o = ./c re90*outof phase,.e.,by 4To
The chosen imension iscalled heembeddingimen-
sion. Which dimension n should be taken is not known be-
forehandwhen hesystems not knownsufficientlyas usu-
al in real experiments). ometimest happenshat a three-
dimensional tatespace s sufficient s embedding pace.
This, of course, s true for mathematical models with an a
priori hree-dimensionaltate pace s,e.g.,one-dimensional
driven oscillators (Duffing, van der Pol, Toda, Morse,
spherical bubble). In the context of acoustics,a three-di-
mensionalmbeddingpacewas oundsufficientn repre-
sentingan acoustic avitationnoiseattractor. 7'2n thisex-
periment, liquid s rradiatedwith sound f high ntensity
and the soundoutput rom the iquid s measured. he mea-
surement ieldspressure-time amples (kt,) that may be
used o construct n attractor.An examples given n Fig.
19. The reconstructed attractor in a three-dimensional era-
FIG. 18. Four reconstructions f a numerically
obtainedbubbleoscillator rom radii data only
with differentdelay imes Tof (a) To, (b)
l To, c) T, (d) 1 To,Tobeingheperiod f the
driving soundwaveof frequency 2.9 kHz and
amplitude310 kPa. The bubblehasa radiusat
rest ofR, = 100,ttm.The sampling ate t, is I
its. For bettervisualizationhe transformed o-
ordinates R * = exp(2R/R, ) have been used.
(Courtesy ofJ. Holzfuss.)
FIG. 19. Example of a reconstructedacousticcavitation noise attractor
fromsampled ressurealuesn a three-dimensionalmbeddingpace. he
attractor s shown romdifferent irectionsor visualizingtsspatial truc-
ture.Samplingime s , = i its, anddelay ime s T = 5 ts.
988 J. Acoust. Soc. Am., Vol. 84, No. 6, December 1988 W. Lauterborn and U. Parlitz: Chaos acoustics 1988
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bedding paces viewed rom differentdirectionso show ts
three-dimensionaltructurendalmostiatoverall ppear-
ance. he verypronouncedtructure uggestshat hemea-
sured cousticoisesofquite imple eterministicrigin.A
modelwith a three-dimensionaltatespace hould e suffi-
cient.To deduct he structure f the equationsrom the at-
tractor, owever,sbeyondhestate fpresent nowledge.t
is interesting o note that a very similar attractor has been
foundby Roessler" n a mathematicallyonstructed odel
of hyperchaosseeSee.X for a definition f hyperchaos).
Oncean attractorhasbeen econstructed,ts properties
can be investigated. f special nterest s the (fractal) di-
mensionf theattractor, hichmaybedeterminedsinghe
methodsf See.VIII. Indeed, s heembeddingimension
isnotknown or a realexperimentalystem nder nvestiga-
tion, n is successivelyncreased, nd the dimensionof the
point et/t " isdetermined. hen hesystemsofdetermin-
isticoriginwitha low-dimensionaltate pace,henat some
n the (fractal) dimension f/l t" will stabilizeat somedefi-
nitevalue smaller hann). The largestractaldimension f
/l t", n = 1,2.... obtainedn thisway thendetermineshe
relevant umber f (nonlinear)degrees f freedom number
of dependentariables) f thedynamic ystemnvestigated.
In this way, the dimensionof an acousticcavitation noise
attractor hasbeendetermined o be about2.5 (Ref. 47).
X. LYAPUNOV EXPONENTS AND LYAPUNOV SPECTRA
Chaoticsystems xhibitsensitive ependencen initial
conditions.his expressionas been ntroducedo denote
the propertyof a chaotic ystem,hat smalldifferencesn the
initial conditions, owever mall,arepersistently agnified
because f thedynamics f thesystem, o hat in a finite ime
thesystem ttains otallydifferent tates.t is notdifficult o
envisagehis propertywith systemshat are not bounded,
likeunstableinear ystems.utphysicalystemsre n gen-
eral bounded, nd it is not at all obvious ow a persistent
magnification f smalldifferencess brought bout. t seems
that a sensitive ependencen initial conditions an only
occur hrougha stretching nd oldingprocess f volumes f
statespace nder heactionof thedynamics. hisprocesss
depicted n Fig. 20. A persistent implestretchingwould
expand nedirectionmoreand morewithoutbounds Fig.
20(a) ]. Neighboring oints husgetmoreandmoredistant.
I
I I
I _
v(t)
r
L--4---a
f < t' V(f')
FIG. 20. Stretching nd oldingof a volume f statespace. a) Stretching
only, (b) stretching nd folding.
This expansionn onedirection an akeplace n a bounded
volumeof statespace nly whenan additional oldingpro-
cess ccurs Fig. 20 b ]. The transitionrom V(t) to V( t' ),
t'> t (see Fig. 20), may be viewedas a map, a so-called
"horseshoemap." Maps with similar propertiesare the
baker'sransformationndArnold's atmap. When terat-
ed they shouldgive the simplest xamples f a dynamical
systemwith sensitive ependencen initial conditionsn two
dimensions. his is the reasonwhy they are intensely tud-
ied.
The notionof sensitive ependencen initial conditions
is mademoreprecisehrough he ntroduction f Lyapunov
exponentsndLyapunov pectra. heir definition annicely
be llustrated.9 akea small pheren state pacencircling
a point of a given trajectory (Fig. 21). The pointsof the
spherecan be viewed as initial pointsof trajectories. his
spheres shifted n statespace nd deformed ue o the dy-
namics o hat at a later time a deformed pheres present.
To properlymakeuseof this dea,mathematically n infini-
tesimalsphereand its deformation nto an ellipsoidwith
principalaxes i (t) i = 1,2 ....rn (m = dimension f the state
space)areconsidered.he Lyapunou xponent/l may hen,
curegranosalis,be definedby
-i lim im 1 ogi(t)
-- -- (15)
,- o ,(o)-o t ri (0)
Theset Ai, i ---- ,...,m},wherebyheAusually reordered
A)A2" 2,, iscalled heLyapunoupectrum.t is to be
recalled that the ri(t) should stay infinitesimallysmall.
Then the linearized ocal dynamics pplies hat has to be
takenalonga nonlinearorbit. This is the meaning f t-- o.
A strict mathematical definition resorts to linearized flow
mapsand may be found n Ref. 15. When A > 0, a time-
dependentdirection" xists,n which he system xpands.
The systems then said o be chaotic when, additionally, t
is bounded).All alonga trajectory,neighboringrajectories
will retreat.A Lyapunovexponents a numberwhich,due o
the imit t-, oo, s a propertyof the whole rajectory.To get
an idea of how uniformlyneighboringrajectoriesecede
from a given rajectory,Lyapunov xponentsanbe defined
for pieces f trajectorieso identify he mostchaotic arts.
In dissipative ystems, he final motion takesplaceon
attractors.esideshe ractal imensionsor thescaling
spectrum), he Lyapunovspectrummay serve o character-
ize these ttractors.With the helpof the Lyapunov xpo-
nents, the motion on a chaotic attractor can be made more
precise, s he following elations nddefinitions old (con-
tinuous ystems, >A --->A,, ):
r(O) r(t)
) rz(t)
FIG. 2 I. Notions or thedefinition f Lyapunov xponents. smallsphere
in statespace s deformed o an ellipsoid ndicatingexpansion r contrac-
tion of neighboringrajectories.
1989 J. Acoust.Soc. Am., VoL 84, No. 6, December 1988 W. Lauterborn nd U. Parlitz:Chaos acoustics 1989
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A < 0--, fixedpoint;
A = 0-limit cycle 22 g2 0--, hyperchaotic ttractor
(/[3 may be), =,or
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(a)
1.0
0.5-
0.0
0.00
()
1.0
W
0.5-
0.0
0.25 0.50 0.75
1.00
1
I I I
0.00 0.25 0.50 0.75 1.00
FIG. 24. A bifurcation iagram a) anda vinding umber iagram b) of
thesinecirclemap.The windingnumberdiagram san example f a devil's
staircase. he staircases really devilishbecause etween very wo steps
thereare infinitelymany other steps nd climbingup or down the staircase
from step o stepactually s impossible.
As mentioned bove,windingnumbers an only be de-
fined n thosecaseswhere he trajectory s part of an invari-
ant torus.But there are many systems,ike the periodically
drivenDuffingoscillator r the bubbleoscillator,where he
existenceof such an invariant torus can be definitely ex-
cluded.This means hat the definitionof the windingnum-
bergiven boveannot eapplied. e hereforentroduced
a similar uantity alled eneralizedinding umber7'37o
classifyhe resonancesf this ypeof systems. s in the case
of the Lyapunov xponents, e consider trajectory ' that
starts n the vicinityof a givenorbit 7/.But now we are not
interestedn the divergence r convergencef these rajec-
tories,but in the way they are twisted around eachother. A
frequency maybe attachedo the orbit7/thatgiveshe
meannumberof twistsof y' abouty per unit time. To com-
pute this torsionrequency t, the linearizeddynamics long
the whole orbit has to be considered (for details, see Ref.
67). If we choose sunit time the periodTo of the oscillation,
the number of twists is called the torsionnumber n of the
closedorbit. Torsionnumbersmay be used o classify eson-
ances nd bifurcationcurves n the parameterspaceof non-
linearoscillators.7'38f thesolution ecomesperiodic,he
torsionnumbercannot be definedanymore,but in this case
theratio to --- l/co of the torsion requencyl and he driving
frequency oof the oscillatorstill exists.We call this ratio to
the (generalized) winding number, because t equals he
winding number introduced above n thosecaseswhere the
trajectory s part of an invariant two-dimensional orus. For
period-doublingascades,wo recursion chemes xist for
the windingnumbersw to: w3 .... at the period-doubling
bifurcationoints f hecontrol arameter.7'38hewinding
numberof a chaoticsolutiondescribes omeaspects f the
foldedgeometryof the strangeattractor. Detailsof the pro-
cedureo computewindingnumbers f nonlinear scillators
are given n Refs.37 and 67.
Xlll. CONCLUDING REMARKS
The main newmethods f chaosphysics avebeenpre-
sented n a tutorial manner and illustratedwith examples
from acoustics, speciallydriven bubble oscillationsand
acoustic cavitation noise. The methods described have been
invented o characterizerregularmotion rom deterministic
systemsmorespecificallyhan by, for instance,heir Fourier
spectra nd correlationswhich are intrinsically inear con-
cepts. n this context,chaosphysics uggestshat the notion
ofa degreeffreedorhust e evised.n conventionalhys-
ics,a degree f freedom s connected ith a linearmode (a
harmonicoscillator).A Fourier spectrumwith many ines s
interpreted scoming rom a systemwith asmanyharmonic
oscillators nd thusas many degrees f freedom.One of the
resultsof chaosphysics s that there are nonlinearsystems
with just a three-dimensional tate space, or instance he
driven bubble oscillator, that will give rise to broadband
Fourier spectra (a sign of their irregular behavior). Thus
theyobviouslyaveust hree degrees'ofreedom"nstead
of infinitelymanyas suggestedy the Fourierspectrum e-
cause nly threecoordinates re needed o specify heir state
completely.
Chaosphysics lsoaffects he notionof randomness.
Randomnesss no onger domainof high-dimensionalys-
tems oo arge o beproperly escribedya setof determinis-
tic equationsnd nitialconditions. random-looking o-
tion may well be the outcome f a deterministic ystemwith
a low-dimensionaltatespace.Deterministic quations re
thus ar morecapablen describing ature hanpreviously
thought rovidedhat heyarenonlinear.ndeed, onlinear-
ity is thenecessaryasic ngredientor thiscapability.
As nonlinear oscillators are the natural extension of the
harmonic scillatorhatplays fundamentalole n physics,
chaosphysics ill find oneof its main applicationsn the
area of nonlinearoscillatorysystems. he investigations
there will center around the laws that are valid in the fully
nonlinear case. A few universal laws have already been
found,andmanymoreare waiting or their discovery. n a
localscalen parameterpace,hemostprominent niversal
phenomenons perioddoubling.On a global cale,t is o be
expectedhat the bifurcation uperstructure'32'33'38f
resonances of nonlinear oscillators is of universal nature.
Chaos hysicssa rapidly rowingieldwithapplicationsll
over the differentareasof physics nd even extending o
chemistry, iology,medicine, cology, nd economy. he
methods escribedn thisarticlemay help n the dissemina-
tion of these deas,and the authorswouldbe proud f some
readerswould be attracted o the fascinating nd rewarding
field of chaoticdynamics.
ACKNOWLEDGMENTS
The authors thank the membersof the Nonlinear Dy-
namicsGroupat theThird Physicalnstitute,University f
1991 J. Acoust.ec.Am.,Vol. 4,No.6, December988 W.LauterbornndU. Parlitz: haos coustics 1991
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Gbttingen,or manystimulating iscussions,. Englischor
supplying swith a beautiful xample f basins f attraction,
and J. Holzfuss or letting us use his reconstructed ttrac-
tors.The computations avebeencarriedout on a SPERRY
1100 and a VAX 8650 of the Gesellschaft ftir wissenschaft-
licheDatenverarbeitung, /Sttingen,ndon a CRAY X-MP
of the Konrad-Zuse-Zentrum fir Informationstechnik, Ber-
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