1988 Methods of Chaos Physics and Their Application to Acoustics

7/24/2019 1988 Methods of Chaos Physics and Their Application to Acoustics

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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.

1975 J. Acoust.Soc. Am. 84 (6), December 1988 0001-4966/88/121975-19500.80 @ 1988 AcousticalSociety of America 1975


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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.

1976 J. Acoust. Soc. Am., Vol. 84, No. 6, December 1988 W. Lauterborn and U. Parlitz: Chaos acoustics 1976


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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.

1977 J. Acoust. Soc. Am., Vol. 84, No. 6, December 1988 W. Lauterborn and U. Parritz: Chaos acoustics 1977


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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.

1978 J. Acoust. oc.Am.,Vol.84, No.6, December 988 W. LauterbornndU. Parlitz:Chaos coustics 1978


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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.).

1979 J. Acoust. SOC_Am., Vol. 84. No. 6, December 1988 W. Lauterbom and U. Parlitz: Chaos acoustics 1979


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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


7/19

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


8/19

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


9/19

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


10/19

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


11/19

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

1985 J. Acoust. Soc. Am., Vol. 84, No. 6, December 1988 W. Lauterbom and U. Parlitz: Chaos acoustics 1985


12/19

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


13/19

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)


14/19

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.



15/19

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


16/19

A < 0--, fixedpoint;

A = 0-limit cycle 22 g2 0--, hyperchaotic ttractor

(/[3 may be), =,or


17/19

(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


18/19

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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1988 Methods of Chaos Physics and Their Application to Acoustics

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