Table Of ContentCHAOS:
An Introduction to
Dynamical Systems
Kathleen T. Alligood
Tim D. Sauer
James A. Yorke
Springer
C H A O S An Introduction to Dynamical Systems
Springer
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C H A O S
An Introduction to Dynamical Systems
K T. A
ATHLEEN LLIGOOD
George Mason University
T D. S
IM AUER
George Mason University
J A. Y
AMES ORKE
University of Maryland
Textbooks in Mathematical Sciences
SeriesEditors:
ThomasF.Banchoff JerroldMarsden
BrownUniversity CaliforniaInstituteofTechnology
KeithDevlin StanWagon
St.Mary’sCollege MacalesterCollege
GastonGonnet
ETHZentrum,Zu¨rich
Cover: Rene Magritte, Golconde 1953. 1996 C. Herscovici, Brussels/Artists Rights
Society(ARS),NewYork.UsedbypermissionofARS.
LibraryofCongressCataloging-in-PublicationData
Alligood,KathleenT.
Chaos-anintroductiontodynamicalsystems/KathleenAlligood,
TimSauer,JamesA.Yorke.
p. cm.—(Textbooksinmathematicalsciences)
Includesbibliographicalreferencesandindex.
1. Differentiabledynamicalsystems. 2. Chaoticbehaviorin
systems. I. Sauer,Tim. II. Yorke,JamesA. III. Title. IV. Series.
QA614.8.A44 1996
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Introduction
B
ACKGROUND
Sir Isaac Newton brought to the world the idea of modeling the motion of
physical systems with equations. It was necessary to invent calculus along the
way,sincefundamentalequationsofmotioninvolvevelocitiesandaccelerations,
whicharederivativesofposition.Hisgreatestsinglesuccesswashisdiscoverythat
the motion of the planets and moons of the solar system resulted from a single
fundamentalsource:thegravitationalattractionofthebodies.Hedemonstrated
thattheobservedmotionoftheplanetscouldbeexplainedbyassumingthatthere
isagravitationalattractionbetweenanytwoobjects,aforcethatisproportional
totheproductofmassesandinverselyproportionaltothesquareofthedistance
between them. The circular, elliptical, and parabolic orbits of astronomy were
v
INTRODUCTION
nolongerfundamentaldeterminantsofmotion,butwereapproximationsoflaws
specified with differential equations. His methods are now used in modeling
motionandchangeinallareasofscience.
Subsequentgenerationsof scientistsextended the methodof usingdiffer-
ential equations to describe how physical systems evolve. But the method had
a limitation. While the differential equations were sufficient to determine the
behavior—inthesensethatsolutionsoftheequationsdidexist—itwasfrequently
difficulttofigureoutwhatthatbehaviorwouldbe.Itwasoftenimpossibletowrite
downsolutionsinrelativelysimplealgebraicexpressionsusingafinitenumberof
terms.Seriessolutionsinvolvinginfinitesumsoftenwouldnotconvergebeyond
somefinitetime.
Whensolutionscouldbefound,theydescribedveryregularmotion.Gen-
erationsofyoungscientistslearnedthesciencesfromtextbooksfilledwithexam-
plesofdifferentialequationswithregularsolutions.Ifthesolutionsremainedin
aboundedregionofspace,theysettleddowntoeither(A)asteadystate,often
due to energy loss by friction, or (B) an oscillation that was either periodic or
quasiperiodic,akintotheclocklikemotionofthemoonandplanets.(Inthesolar
system,therewereobviouslymanydifferentperiods.Themoontraveledaround
theearthinamonth,theeartharoundthesuninaboutayear,andJupiteraround
the sun in about 11.867 years. Such systems with multiple incommensurable
periodscametobecalledquasiperiodic.)
Scientistsknewofsystemswhichhadmorecomplicatedbehavior,suchas
apotofboilingwater,orthemoleculesofaircollidinginaroom.However,since
thesesystemswerecomposedofanimmensenumberofinteractingparticles,the
complexityoftheirmotionswasnotheldtobesurprising.
Around 1975, after three centuries of study, scientists in large numbers
aroundtheworldsuddenlybecameawarethatthereisathirdkindofmotion,a
type (C) motion,thatwe nowcall “chaos”. The new motionis erratic,but not
simplyquasiperiodicwith a largenumber of periods,and not necessarilydue to
alargenumberofinteractingparticles.Itisatypeofbehaviorthatispossiblein
verysimplesystems.
A small number of mathematicians and physicists were familiar with the
existenceofathirdtypeofmotionpriortothistime.JamesClerkMaxwell,who
studiedthemotionofgasmoleculesinabout1860,wasprobablyawarethateven
a system composed of two colliding gas particles in a box would have neither
motiontypeAnorB,andthatthelongtermbehaviorofthemotionswouldfor
all practical purposes be unpredictable. He was aware that very small changes
in the initial motion of the particles would result in immense changes in the
trajectoriesofthemolecules,eveniftheywerethoughtofashardspheres.
vi
INTRODUCTION
Maxwell began his famous study of gas laws by investigating individual
collisions.Considertwoatomsofequalmass,modeledashardspheres.Givethe
atomsequalbutoppositevelocities,andassumethattheirpositionsareselected
atrandominalargethree-dimensionalregionofspace.Maxwellshowedthatif
theycollide,alldirectionsoftravelwillbe equallylikely afterthecollision.He
recognized that small changes in initial positions can result in large changes in
outcomes. In a discussion of free will, he suggested that it would be impossible
totestwhetheraleopardhasfreewill,becauseonecouldnevercomputefroma
studyofitsatomswhattheleopardwoulddo.Butthechaosofitsatomsislimited,
for,asheobserved,“Noleopardcanchangeitsspots!”
Henri Poincare´ in 1890 studied highly simplified solar systems of three
bodiesandconcludedthatthemotionsweresometimesincrediblycomplicated.
(See Chapter 2). His techniques were applicable to a wide variety of physical
systems.ImportantfurthercontributionsweremadebyBirkhoff,Cartwrightand
Littlewood,Levinson,Kolmogorovandhisstudents,amongothers.Bythe1960s,
there were groups of mathematicians, particularly in Berkeley and in Moscow,
striving to understand this third kind of motion that we now call chaos. But
only with the advent of personal computers, with screens capable of displaying
graphics,havescientistsandengineersbeenabletoseethatimportantequations
intheirownspecialtieshadsuchsolutions,atleastforsomerangesofparameters
thatappearintheequations.
Inthepresentday,scientistsrealizethatchaoticbehaviorcanbeobserved
inexperimentsandincomputermodelsofbehaviorfromallfieldsofscience.The
keyrequirementisthatthesysteminvolveanonlinearity.Itisnowcommonfor
experiments whose previous anomalous behavior was attributed to experiment
error or noise to be reevaluated for an explanation in these new terms. Taken
together,thesenewtermsformasetofunifyingprinciples,oftencalleddynamical
systemstheory,thatcrossmanydisciplinaryboundaries.
The theory of dynamical systems describes phenomena that are common
to physical and biological systems throughout science. It has benefited greatly
from the collision of ideas from mathematics and these sciences. The goal of
scientists and applied mathematicians is to find nature’s unifying ideas or laws
andtofashionalanguagetodescribetheseideas.Itiscriticaltotheadvancement
of science that exacting standards are applied to what is meant by knowledge.
Beautifultheoriescanbeappreciatedfortheirown sake,butscienceisasevere
taskmaster. Intriguing ideas are often rejected or ignored because they do not
meetthestandardsofwhatisknowledge.
Thestandardsofmathematiciansandscientistsareratherdifferent.Mathe-
maticiansprovetheorems.Scientistslookatrealisticmodels.Theirapproachesare
vii
INTRODUCTION
somewhatincompatible.Thefirstpapersshowingchaoticbehaviorincomputer
studiesofverysimplemodelsweredistastefultobothgroups.Themathematicians
fearedthatnothingwasprovedsonothingwaslearned.Scientistssaidthatmodels
withoutphysicalquantitieslikecharge,mass,energy,oraccelerationcouldnotbe
relevanttophysicalstudies.Butfurtherreflectionledtoachangeinviewpoints.
Mathematiciansfoundthatthesecomputerstudiescouldleadtonewideasthat
slowlyyieldednewtheorems.Scientistsfoundthatcomputerstudiesofmuchmore
complicatedmodels yielded behaviorssimilar to thoseof thesimplisticmodels,
andthatperhapsthesimplermodelscapturedthekeyphenomena.
Finally, laboratory experiments began to be carried out that showed un-
equivocal evidence of unusual nonlinear effects and chaotic behavior in very
familiarsettings.Thenewdynamicalsystemsconceptsshowedupinmacroscopic
systemssuchasfluids,commonelectroniccircuitsandlow-energylasersthatwere
previouslythoughttobefairlywellunderstoodusingtheclassicalparadigms.In
thissense,thechaoticrevolutionisquitedifferentthanthatofrelativity,which
showsitseffectsathighenergiesandvelocities,andquantumtheory,whoseeffects
aresubmicroscopic.Manydemonstrationsofchaoticbehaviorinexperimentsare
notfarfromthereader’sexperience.
Inthisbookwestudythisfieldthatistheuncomfortableinterfacebetween
mathematicsandscience.Wewilllookatmanypicturesproducedbycomputers
andwetrytomakemathematicalsenseofthem.Forexample,acomputerstudyof
thedrivenpenduluminChapter2revealsirregular,persistent,complexbehavior
fortenmillionoscillations.Doesthisbehaviorpersistforonebillionoscillations?
Theonlywaywecanfindoutistocontinuethecomputerstudylonger.However,
even if it continues its complex behavior throughout our computer study, we
cannot guarantee it would persist forever. Perhaps it stops abruptly after one
trillion oscillations;we do not know for certain. We can prove that there exist
initial positions and velocities of the pendulum that yield complex behavior
forever,butthesechoicesareconceivablyquiteatypical.Thereareevensimpler
models where we know that such chaotic behavior does persist forever. In this
world,pictureswithuncertainmessagesremainthemediumofinspiration.
There is a philosophy of modeling in which we study idealized systems
thathavepropertiesthatcan becloselyapproximatedbyphysicalsystems.The
experimentalist takes the view that only quantities that can be measured have
meaning.Yetwecanprovethattherearebeautifulstructuresthataresoinfinitely
intricate that they can never be seen experimentally. For example, we will see
immediatelyinChapters1and2thewaychaosdevelopsasaphysicalparameter
likefrictionisvaried.Weseeinfinitelymanyperiodicattractorsappearingwith
infinitely many periods. This topic is revisited in Chapter 12, where we show
viii
INTRODUCTION
how this rich bifurcation structure, called a cascade, exists with mathematical
certainty in many systems. This is a mathematical reality that underlies what
the experimentalist can see. We know that as the scientist finds ways to make
thestudyofaphysicalsystemincreasinglytractable,moreofthismathematical
structurewillberevealed.Itisthere,butoftenhiddenfromviewbythenoiseof
theuniverse.Allscienceisofcoursedependentonsimplisticmodels.Ifwestudy
a vibrating beam, we will generally not model the atoms of which it is made.
If we model the atoms, we will probably not reflect in our model the fact that
the universe has a finite age and that the beam did not exist for all time. And
we do not include in our model (usually) the tidal effects of the stars and the
planetsonourvibratingbeam.Weignorealltheseeffectssothatwecanisolate
theimplicationsofaverylimitedlistofconcepts.
Itisourgoaltogiveanintroductiontosomeofthemostintriguingideasin
dynamics,theideaswelovemost.Justaschemistryhasitselementsandphysics
hasitselementaryparticles,dynamicshasitsfundamentalelements:withnames
likeattractors,basins,saddles,homoclinicpoints,cascades,andhorseshoes.The
ideasinthisfieldarenottransparent.Asareader,yourabilitytoworkwiththese
ideaswillcomefromyourowneffort.Wewillconsiderourjobtobeaccomplished
ifwecanhelpyoulearnwhattolookforinyourownstudiesofdynamicalsystems
oftheworldanduniverse.
A B
BOUT THE OOK
As we developed the drafts of this book, we taught six one semester classes at
GeorgeMasonUniversityandtheUniversityofMaryland.Thelevelisaimedat
undergraduates and beginning graduate students. Typically, we have used parts
ofChapters1–9asthecoreofsuchacourse,spendingroughlyequalamountsof
timeoniteratedmaps(Chapters1–6)anddifferentialequations(Chapters7–9).
Someofthemapsweuseasexamplesintheearlychapterscomefromdifferential
equations,sothattheirimportanceinthesubjectisstressed.Thetopicsofstable
manifolds,bifurcations,andcascadesareintroducedinthefirsttwochaptersand
thendevelopedmorefullyintheChapters10,11,and12,respectively.Chapter
13ontimeseriesmaybeprofitablyreadimmediatelyafterChapter4onfractals,
althoughthe concepts of periodicorbit (of a differentialequation)and chaotic
attractorwillnotyethavebeenformallydefined.
The impetus for advances in dynamical systems has come from many
sources:mathematics,theoreticalscience,computersimulation,andexperimen-
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