Table Of Content·
John Argyris Gunter Faust
·
Maria Haase Rudolf Friedrich
An Exploration of
Dynamical Systems and Chaos
Completely Revised and Enlarged
Second Edition
ABC
JohnArgyris(Deceased) MariaHaase
InstitutfürHöchstleistungsrechnen
GunterFaust UniversitätStuttgart
Inst.StatikundDynamikder Stuttgart
Luft-undRaumfahrtkonstruktionen Germany
UniversitätStuttgart
Stuttgart RudolfFriedrich(Deceased)
Germany
ISBN978-3-662-46041-2 ISBN978-3-662-46042-9 (eBook)
DOI10.1007/978-3-662-46042-9
LibraryofCongressControlNumber:2014958980
SpringerHeidelbergNewYorkDordrechtLondon
(cid:2)c Springer-VerlagBerlinHeidelberg2015
Springer-VerlagGmbHBerlinHeidelbergispartofSpringerScience+BusinessMedia
(www.springer.com)
Preface to the Second Edition
In1994,whenthefirsteditionhasbeenpublishedinEnglishandinGerman,wedid
not anticipate that this introduction to the field of non-linear dynamics and chaos
would meet with such lively interest. Although attention to the subject has less-
ened in the media and in popular scientific publications over the last 20 years, the
fundamental ideas,the theoretical insights andthe tools developedforthe analysis
of non-linear dynamical systems have continued to spread into the most diverse
areas of science and technology as well as into the respective literature and are
nowadays part of the classical curriculum in many study programmes. Since the
Englisheditionsoldoutmanyyearsago,wedecidedtofollowuptheexpandedand
revised second German edition with another extended edition in English.
AfterJohnArgyrispassedawayin2004attheadvancedageof91,weconsideredit
ourobligationtocontinueworkonthisbookintheoriginalsense–firstinasecond
German edition – attempting to explain the complex topic of non-linear dynamics
to a wide audience as descriptively and vividly as possible.
We took the opportunity to emphasise and further clarify the differences between
purely temporal and spatio-temporal dynamics, in particular between chaos and
turbulence. The initial hope – occasionallyaccompaniedby euphoria– thatinsight
into the special features of chaotic systems meant that scientists were about to
solve the centuries-oldproblem of turbulence was not fulfilled. Without doubt, the
elucidationofcharacteristicssuchasunpredictabilityandmixinginchaoticsystems
made a substantial contribution to understanding turbulent flows, yet the funda-
mental questions concerning fully developed turbulence are unsolved to this day.
By enlarging the circle of authors to include Rudolf Friedrich, we were able to add
withChapter9acompletelynewandmoreambitiouschapteronturbulence,which
comprisesandcriticallyscrutinisesthefundamentalsofturbulentflowsandaseries
ofclassicturbulencemodels.Whilethereisamultitude ofexcellentmonographson
turbulence from the viewpoint of the engineering sciences, the new chapter mainly
discusses the fundamental questions from the physical point of view, highlighting
the common ground with concepts existing in chaos theory. To describe and inves-
tigate fully developed turbulence, it is indispensable to include probabilistic and
stochastic methods. So that the reader can understand the chapter on turbulence
better, we thought it appropriate to include two new sections in the mathematical
introduction in Chapter 3: basic concepts of probability theory and invariant mea-
sureand ergodic orbits.Ouraiminthesesectionswastopresentthebasicideasand
concepts; we did not, however, strive for a complete description, rather referring
the reader to the excellent literature on probability and stochastic processes.
In section 3.8.5, we added a brief introduction to wavelet transformation and out-
lined its relevance for time-frequency analyses of time series. The use of wavelets
is especially favourable for analysing multifractal structures as described in sec-
tion8.5.2.Insection3.9.12,wealsoincludedabriefdescriptionofMarkovanalysis,
arecentlydevelopedandfar-reachingmethodwhichallowsustoseparatethedeter-
ministic part of the dynamics from the dynamical and even measurement noise. In
section 9.6.8, this method is applied to velocity increments measured in turbulent
flows.Section8.7containsanupdatedandextendedsectiononroutesoutofchaos.
Shilnikov bifurcations and spiral chaos received much attention in various applica-
tions, from chemical reactions to the propagation of nerve impulses and epileptic
seizures. To understand such systems better, we also included a short overview in
section 10.6. In addition, the paragraph on the kinetics of chemical reactions on
surfaces has been largely revised in the hope of rendering this widely used field of
application more comprehensible to a larger audience of students, see section 10.9.
Ontheotherhand,wedispensedwiththesectiononcelestialmechanics;thiswould
have gone beyond the scope of this monograph.
Suddenly and unexpectedly, on 16th August 2012, Rudolf Friedrich passed away,
takenfromthemidstofhislifeandwork.Hisbroadknowledgeandinterdisciplinary
expertise made him extraordinarily inventive, not only in theoretical physics, his
specialistfield, butalso inmany otherareasofscience, including experimental and
technical applications. We hope we have been able to complete the work on this
book as he would have wished.
Withoutthevigoroussupportofnumerousfriendsandcolleaguesfrommanyscien-
tific disciplines, we would not have been able to finish this extended and updated
secondedition. To implement the originalKeTEXversionin LaTeXand to generate
andreconstructthefigureswiththeprogrammesystemAnT4.669,wereceivedpro-
fessional help. We acknowledge the valuable hints for additions and improvements
which we obtained in the course of many discussions and during proof-reading.
We wish to thank all those who constantly and sedulously helped us and are es-
pecially indebted to Viktor Avrutin, Inna Avrutina, Rolf Bader, Anton Daitche,
Martin Dziobek, Markus Eiswirth, Jan Friedrich, Jason Gallas, Svetlana Gure-
vich, Andreas Haase, Marion Hackenberg, Marcus Hauser, Oliver Kamps, David
Kleinhans, Yuri A. Kuznetsov, Bernd Lehle, Pedro Lind, Johannes Lu¨lff, Joachim
Peinke, Peter Plath, Gu¨nter Radons, Michael Schanz, Susanne Schmidt, Daniel
Stellbrink, RobertStresing,Daniel H. Sugondo,ChristianUhl, HansvandenBerg,
Judith Vogelsang, Georg Wackenhut and Michael Wilczek. Our special thanks go
to Prudence Lawday, the translator of the first English edition, for her excellent
help with this extended manuscript. We are also indebted to Michael Resch, the
director of the High PerformanceComputing Center Stuttgart (HLRS) ofthe Uni-
versityofStuttgart,forhisgenerousassistanceandtotheArgyrisFoundationforits
financial support.
Last but not least, we wish to thank the members of staff at Springer-Verlag for
theirprofessionalco-operation,andinparticularThomasDitzingerforhispatience
andcontinuousencouragementandsupport.Oursincerethanksgotoeveryonewho
supported us.
Stuttgart, October 2014 M. Haase, G. Faust
Preface to the First Edition
Chaos often breeds life,
when order breeds habit
Henry Brooks Adams, 1838-1918
Education ofHenryAdams,1907
This volume is intended as an introductory textbook on the theory of chaos and
is addressed to physicists and engineers who wish to be acquainted with this new
andexcitingscienceassociatedwithnon-lineardeterministicsystems.Mathematics
are, of course, a pre-requisitetool in such a study, and we did not shirk the task of
discussing complex mathematical issues while preferring, in general, through incli-
nationandtrainingtofocustheattentionofthereaderonaphysicalunderstanding
of phenomena.
Of course, we have to admit that a number of distinguished textbooks incorporat-
ing chaos as a primary or secondary subject have appeared in the last few years.
In particular, we draw the reader’s attention to the treatises of Moon, Thompson
& Stewart, Kreuzer, Berg´e, Pomeau & Vidal, Schuster and Nicolis & Prigogine, to
mention only six recent texts that are not primarily mathematical. The excellent
book of F. C. Moon – which has been followed by an expanded exposition – is
mainly concerned with experimental techniques and offers interesting insights into
the chaotic response of mechanical systems. The book of J. M. T. Thompson and
H. B. Stewartis, as to be expected, brilliantly written with a broadoutlookon the
subject mainly directed at mechanical systems and structures but also including
asides into subjects like the Rayleigh-B´enardconvection and the Lorenz system of
equations. E. Kreuzer’s compact book is addressed to non-linear oscillations and
mechanical systems; it is based on a sound mathematical foundation and displays
a deep knowledge of the dynamic response of non-linear systems. The book by
P. Berg´e, Y. Pomeau and C. Vidal contains inter alia an extensive investigation
of the transitions to chaos and their experimental verification on the example of
the Rayleigh-B´enard convection. H. G. Schuster’s monograph is an outstanding
exposition of chaotic manifestations in non-linear physical systems written with
precision and economy for theoretical physicists of advanced training. The last of
the aforementioned books, that of G. Nicolis and L Prigogine, offers a profound
and comprehensible study of the dynamics of non-linear systems far from ther-
modynamic equilibrium. In addition, we should like to mention the monographsof
HermannHakenonsynergeticswhich–onamathematicallydemandingbasis–deal
withthesystematicinvestigationofstructureformationinopendissipativesystems.
Inthis arrayoftextbooks,we nowsubmit ourtextinthe hope thataspiringphysi-
cists and engineers will find it of value in their efforts to understand and apply the
complex theory of chaos. We have endeavoured not only to expound the general
theory asfar as possible, butalso to include abroadrangeof physicalsubjects like
fluid mechanics, Rayleigh-B´enard convection, biomechanics, astronomy, physical
chemistry and other mechanical and electrical systems represented by the Duffing
and van der Pol equations; see our descriptive account in Chapter 1.
Once chaotic manifestations were first perceived consciously in meteorology, the
worldof classical science as it existed 30 yearsago seemed to fade away.Physicists
andmechanicians,guidedbytheepoch-makingwritingsofKeplerandNewton,were
unconsciouslyinfluencedovercenturiestosuchanextentthattheywereawareonly
of regular motions – whether linear or non-linear – and thus were not capable of
perceivingirregularphenomena.Asearlyastheturnofthecentury,HenriPoincar´e
hadindeeddrawnattentiontothepossibilityofirregularbehaviourindeterministic
systems;intheabsenceofcomputers,however,thiscouldnotberegistereddirectly.
WhilerecognisingthegreatcontributionsofanOsborneReynolds,wewerethusnot
inapositiontopenetratethemysteriesofturbulenceinafloworintheatmosphere
and in the oceans and hence to comprehend irregularities in natural phenomena.
For three centuries, research had been unconsciously directed at regularity.
However,intheearly1970s,aneliteofresearchersinEuropeandtheUnitedStates
initiated a concentrated effort to clear a path of understanding through all these
disorders. This groupof adventurous and unconventional scientists included physi-
cists,biologists,chemistsandmathematicians,allattemptingtoseeklinksbetween
different kinds of irregularities in animate and inanimate nature. These irregular-
ities are to be found in the dynamics of our heartbeat and in explosive variations
in certain wildlife populations as well as in the turbulence of a flow and the er-
ratic motion of a meteor. Economists were prompted to investigate the theory of
economic cycles. All these phenomena and a multitude of others, such as forks of
lightning, were observed with curiosity and analysed.
At the same time, mathematicians such as Vladimir Igorevich Arnold made new
fundamentalcontributionstoourknowledgeoflocalandglobalbifurcationsinnon-
lineardynamics. Unavoidably,scientists re-discoveredthe pioneeringworkofHenri
Poincar´e. All these efforts could not have been conceived and realised without the
revolution in science and engineering generated by the explosively growing avail-
ability and capacity of electronic computers which began a few years after World
War II.
A decade later, in the mid-Seventies, the group of scientists working on chaos had
establisheditselfasanexponentiallygrowingco-fraternitywhichwasre-shapingthe
conceptsofmodernsciencenolens volens. Wehavenowreachedthestageatwhich,
innearlyeverymajoruniversityofscientificstanding,researchersapplythemselves
tothemanifestationsofchaos,irrespectiveoftheirformalspecialisation.Indeed,at
Los Alamos, a special centre for the study of non-linear systems was created and
coordinates work on chaos and related manifestations.
Inevitably, the study of chaos has generated new advanced techniques of applying
computers and the refined graphic facilities of modern hardware. In this way, we
canview displaysofdelicate, highlyimaginativetextures whichillustrate complex-
ity in a formerly unexpected way. This new science involves the study of fractals,
bifurcations, periodicities and intermittency. All these manifestations inspire us to
anew understandingofthe conceptofmotion.Inall ourobservationsofthe world,
we now continuously discovermanifestations of chaos as, for example, in the rising
and quivering column of cigarette smoke which suddenly breaks out into a wild
disorder. Similar phenomena may be seen if we look at the complex oscillatory
response of a flag fluttering or snapping back and forth in the wind. Observing a
dripping tap, we note a transformation from a steady pattern to a random one.
Chaos, in fact, is noticed today – thanks to the revolutionary findings of Edward
Lorenzin1963whowasthefirstofallmodernscientiststocomprehendchaoticevo-
lutions and developed a simplified model of chaotic manifestation in the weather.
Indeed, in this weather model, he noted the extreme sensitivity of the response
arising from small changes in the initial conditions and mentioned the so-called
butterfly effect first. But chaos is also contained latently in the response of an
airplane in flight and arises inter alia through turbulent boundary layers and sep-
aration effects as well as other chaotic manifestations. If a dense stream of cars
chokesamotorway,thisisalsoanexampleofachaoticresponse.Irrespectiveofthe
medium in which these chaotic outbursts take place, the behaviour obeys certain
common general rules.
Our incidental remarks are intended to demonstrate that chaos poses problems in
most realms of present-day researchthat do not fit into the traditional patterns of
scientific thinking. Incontrast,the imaginativestudy ofchaosallowsus to discover
the universal characteristics of the response of complex non-linear systems. The
first chaos researchers who initiated this discipline shared certain preoccupations.
They were, for example, fascinated by patterns, especially those that emerge and
repeat themselves at the same time on different scales. To the initiated researcher,
odd questions like how long the jagged coast of Great Britain is became part of a
fundamental enquiry. These early scientists had a talent for exploring complexity
and studying jagged edges and sudden leaps. Inevitably, these apostles in chaos
speculated about determinism and free will and about the nature of conscious in-
telligence.
The most profound thinkers of the new science asserted, and still assert today,
that 20th century science will be remembered for three great scientific philosophi-
cal concepts: relativity,quantum mechanics andchaos. We goone step further and
think that the explorationof chaos will determine the mainstream of scientific dis-
covery in the 21st century and shape the evolution of physics, mechanics and also
chemistry; naturally, this will also affect engineering. In this way, the preachers
of the new science also believe that chaos erodes principles of Newton’s physics.
As one eminent physicist puts it: relativity eliminated the Newtonian illusion of
absolute space and time; the quantum theory eliminated the Newtonian dream
of a controllable procedure of measurement and, ultimately, chaos eliminated the
Laplacian phantasy (or pipe dream) of deterministic predictability. Of these three
scientific revolutions, the revolution in chaos applies to the whole universe as we
comprehendandobserveitandaffectsusthroughmanifestationsonahumanscale.
Forallthebrilliantachievementsofgreatphysicists,wehavetoaskourselvestoday
how this edifice of physics could have evolvedoverso many yearsinto a phantastic
missionwithoutprovidingmeansofansweringsomeofthe mostfundamental ques-
tions about nature. How does life begin and what is the mystery of turbulence? In
a universe ruled by entropy and drawing inexorably towards greater and greater
disorder, how does order establish itself? Moreover, objects of everyday life like
fluidsandnon-linearmechanicalsystemscametobe consideredassoordinarythat
physicistscouldassumethattheywerewellunderstood,atleastbyengineers.Until
thirty years or so ago, the reason for our ignorance of such systems was caused by
the absence of computer and graphic equipment. In the meantime, experimental
and numerical computation has proved that our ignorance was profound.
As the revolution in chaos evolves in a cascade of new surprises, leading physicists
find it quite natural to return without embarrassment to problems immediately
related to our human nature. They are just as happy studying puffed clouds as
galaxies.Leadingjournalspublisharticlesonthe strangedynamicsofaballbounc-
ing on a table as well as on esoteric quantum physics. The simplest non-linear
systems–andpracticallyallsystemsintherealworldarenon-linear–seemtogen-
erate extremely difficult problems of predictability. At the same time, order may
suddenly succeedchaosandviceversa.Inmostsystems,wecanobservethe cohab-
itation of chaos and order. We are looking at a new kind of science with which we
expect to bridge the gulf between knowledge of what one single item can do – e.g.
a watermolecule orone cell of a hearttissue – andwhat anassembly ofmillions of
them can create in co-operation. This is one of the main topics to which Hermann
Haken dedicated his work.
In the past, physicists traditionally deduced from complex results a hypothesis on
complexcauses.Whentheyobservedarandomrelationshipbetweenwhatgoesinto
a system and what emerges from it, they assumed that they would have to intro-
duce randomness into a realistic theory by adding the effects of noise or error. In
contrast,themodernstudyofchaoswasgeneratedtherealisationintheearly1960s
that quite simple, but still non-linear deterministic mathematical equations could
generate responses every bit as surprising as the phantasmagoric turbulence of a
waterfall. Small differences in initial conditions in the input are quickly magnified
and produce large differences in the output. Thanks to Lorenz, we now assign this
phenomenon to an extremely sensitive dependence on initial conditions. To take
the classicexampleofweather,sucheffects maybe expressedwithamockingsmile
as part of the butterfly effect. Thus, a butterfly stirring the air in Beijing might
generate stormy weather the following month in New York.
We conclude our prologue by offering five very brief illustrations of the revolution
that shaped the new theory of chaos.
i. EdwardLorenzwasthefirstinspiredscientistwho,throughhisnumericalcom-
putational experiments, perceived the essence of chaos. He workedas a meteo-
rologist in 1961 at the MIT on an ungainly primitive computer known by the
impressive name of RoyalMcBee. By selecting a simple model ofthe Rayleigh-
B´enard convection in a layer of air activated by a difference of temperature,
he proposedto representits complex responseby aninnocuous systemofthree
non-linear ordinary differential equations. With this model, he hoped to study
the problemof weather forecasting.He useda primitive printing device to pro-
duce a graph of the direction and speed of wind. In this way, he made the
epoch-making discovery that the smallest difference in the initial conditions
may produce diverging courses and patterns which drift apart until they bear
no resemblance.This was the clue to the butterfly effect and the ultimate real-
isation of the unpredictability of weather over longer periods of time. Lorenz’s
findings were revolutionary and initiated the search for the nature of chaos.
Lorenz also discovereda partial view of the corresponding strange attractor in
the phase space, but did not name it in this way. Edward Lorenz’s work and
its influence on the sciences can be illustrated by a quotation from the New
Testament:
(cid:0)(cid:2)(cid:3)(cid:4)(cid:6)(cid:5) (cid:8)(cid:7)(cid:9)(cid:10)(cid:11)(cid:12)(cid:4)(cid:13) (cid:14)(cid:15)(cid:6)(cid:16) (cid:8)(cid:7)(cid:9)(cid:10)(cid:11)(cid:12)(cid:8)(cid:13) (cid:7)(cid:6)(cid:10)(cid:9)(cid:8)(cid:13) (cid:17)(cid:5)(cid:13)(cid:17)(cid:10)(cid:14)(cid:18)i(cid:19)i(cid:11)i(cid:13), 5
(cid:20)(cid:17)(cid:11)(cid:13)(cid:8)(cid:5) (cid:3)(cid:11)(cid:17)(cid:21)(cid:8)(cid:10)(cid:12)(cid:8)(cid:22) (cid:0)(cid:23)(cid:14)(cid:11)(cid:24)(cid:18)(cid:4)(cid:9)(cid:8)(cid:5) (cid:0)(cid:2)(cid:17)(cid:12)(cid:25)(cid:10)(cid:26)(cid:4)(cid:6)
Behold, how great a matter a little fire kindleth.
General Epistle of James, iii,5
ii. Mitchell Feigenbaum joined the Los Alamos National Laboratory in 1974.
Feigenbaum brought to Los Alamos a conviction that the understanding of
non-linearproblemswaspracticallynon-existent.One ofhis firstinvestigations
concerned a most elementary logistic map with a quadratic form depending
on a single parameter. He discovered that this primitive mathematical system
produced not only the expected steady answers but also, through a cascade of
bifurcations, periodic and also doubly and higher periodic responses leading to
chaotic manifestations beyond certain values of the parameter. But this chaos
wasagaininterruptedby windowsofregularresponse.Feigenbaumwenta step
furtherandprovedtheuniversalityofhisfindingswhichapplywithasurprising
similarity to different and more complex mathematical expressions.
iii. Let us next consider some of the finer points of the work of the highly gifted
BenoˆıtMandelbrot with particular reference to the Mandelbrotset. This set is
possibly the most complex object existing in mathematics. Benoˆıt Mandelbrot
initiatedhissearchin1979onageneralisationofacertainclassofshapesknown
asJuliasets.ThesewereoriginallyinventedbytwodistinguishedFrenchmath-
ematicians,GastonJuliaandPierreFatou, duringWorldWar I inFrance.Had
these two researchers possessed computers and graphics, they would no doubt
have been co-discoverers of chaos. The French mathematician Adrien Douady
described the Julia sets as follows: “You obtain an incredible variety of Julia
sets: some are a fatty cloud, others are a skinny bush of brambles, some look
like meandering sparks which float in the air after a firework has gone off. One
has the shape of a rabbit but lots of them have sea-horse tails”.
In 1979, Mandelbrot ingeniously created one image in a complex plane that
could serve as a dictionary or catalogue of all Julia sets. Scientists like Ju-
lia, Fatou, Hubbard, Barnsley and Mandelbrot invented novel rules on how to
construct extravagant geometrical shapes that are now called fractals and are
ruled by the principle of self-similarity. They produce strange ethereal pictures
of great beauty which are seemingly dotted with separate, solitary islands, all
