Table Of ContentSpringer Complexity
Springer Complexity is a publication program, cutting across all traditional disciplines
ofsciencesaswellasengineering,economics,medicine,psychologyandcomputersci-
ences, which is aimed at researchers, students and practitioners working in the field of
complex systems. Complex Systems are systems that comprise many interacting parts
with the ability to generate a new quality of macroscopic collective behavior through
self-organization,e.g.,thespontaneousformationoftemporal,spatialorfunctionalstruc-
tures.Thisrecognition,thatthecollectivebehaviorofthewholesystemcannotbesimply
inferred from the understanding of the behavior of the individual components, has led
to various new concepts and sophisticated tools of complexity. The main concepts and
tools–withsometimesoverlappingcontentsandmethodologies–arethetheoriesofself-
organization,complexsystems,synergetics,dynamicalsystems,turbulence,catastrophes,
instabilities,nonlinearity,stochasticprocesses,chaos,neuralnetworks,cellularautomata,
adaptivesystems,andgeneticalgorithms.
The topics treated within Springer Complexity are as diverse as lasers or fluids in
physics,machinecuttingphenomenaofworkpiecesorelectriccircuitswithfeedbackin
engineering,growthofcrystalsorpatternformationinchemistry,morphogenesisinbi-
ology, brain function in neurology, behavior of stock exchange rates in economics, or
the formation of public opinion in sociology. All these seemingly quite different kinds
ofstructureformationhaveanumberofimportantfeaturesandunderlyingstructuresin
common.Thesedeepstructuralsimilaritiescanbeexploitedtotransferanalyticalmeth-
odsandunderstandingfromonefieldtoanother.TheSpringerComplexityprogramthere-
fore seeks to foster cross-fertilization between the disciplines and a dialogue between
theoreticians and experimentalists for a deeper understanding of the general structure
andbehaviorofcomplexsystems.
The program consists of individual books, books series such as “Springer Series
in Synergetics", “Institute of Nonlinear Science", “Physics of Neural Networks", and
“UnderstandingComplexSystems",aswellasvariousjournals.
Springer Series in Synergetics
SeriesEditor
HermannHaken
InstitutfürTheoretischePhysik
undSynergetik
derUniversitätStuttgart
70550Stuttgart,Germany
and
CenterforComplexSystems
FloridaAtlanticUniversity
BocaRaton,FL33431,USA
MembersoftheEditorialBoard
ÅkeAndersson,Stockholm,Sweden
GerhardErtl,Berlin,Germany
BernoldFiedler,Berlin,Germany
YoshikiKuramoto,Sapporo,Japan
JürgenKurths,Potsdam,Germany
LuigiLugiato,Milan,Italy
JürgenParisi,Oldenburg,Germany
PeterSchuster,Wien,Austria
FrankSchweitzer,Zürich,Switzerland
DidierSornette,Nice,FranceandZürich,Switzerland
ManuelG.Velarde,Madrid,Spain
SSSyn–AnInterdisciplinarySeriesonComplexSystems
ThesuccessoftheSpringerSeriesinSynergeticshasbeenmadepossiblebythecontri-
butionsofoutstandingauthorswhopresentedtheirquiteoftenpioneeringresultstothe
science community well beyond the borders of a special discipline. Indeed, interdisci-
plinarityisoneofthemainfeaturesofthisseries.Butinterdisciplinarityisnotenough:
The main goal is the search for common features of self-organizing systems in a great
varietyofseeminglyquitedifferentsystems,or,stillmorepreciselyspeaking,thesearch
forgeneralprinciplesunderlyingthespontaneousformationofspatial,temporalorfunc-
tional structures. The topics treated may be as diverse as lasers and fluids in physics,
patternformationinchemistry,morphogenesisinbiology,brainfunctionsinneurology
orself-organizationinacity.Asiswitnessedbyseveralvolumes,greatattentionisbe-
ingpaidtothepivotalinterplaybetweendeterministicandstochasticprocesses,aswell
as to the dialogue between theoreticians and experimentalists. All this has contributed
toaremarkablecross-fertilizationbetweendisciplinesandtoadeeperunderstandingof
complexsystems.Thetimelinessandpotentialofsuchanapproacharealsomirrored–
amongotherindicators–bynumerousinterdisciplinaryworkshopsandconferencesall
overtheworld.
L.M. Pismen
Patterns and Interfaces
in Dissipative Dynamics
WithaForeword byY.Pomeau
With163Figuresand1Table
ABC
ProfessorL.M.Pismen
DepartmentofChemicalEngineering
Technion-IsraelInstituteofTechnology
TechnionCity,Haifa32000,Israel
E-mail:[email protected]
LibraryofCongressControlNumber:2005939040
ISSN0172-7389
ISBN-10 3-540-30430-4SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30430-2SpringerBerlinHeidelbergNewYork
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Foreword
Nature is full of marvels, be they in the inanimate world or in the realm of
life. Quite often their beauty and attractive power come from the regularity
of some sort of “pattern,” or – more subtly – from some obvious or hidden
irregularity (as the French poet Verlaine said, “always favour the uneven”).
Obvious natural patterns are found in the stripes of sea-shells or on the fur
of mammals, such as our domestic cats. Stones such as malachites or agathes
alsoshowfinelycraftedlayersofmatchingorcontrastingcolors.Alessobvious
case of “pattern” is seen in the opals, whose opalescence reflects wonderfully
regular arrays of spheres of silica. Civilizations of the Far East highly praised
these miraculous examples of beauty in minerals.
The way things and living beings are made or have been made began to
attracttheinterestofscientistsoftheClassicalAge,particularlyafterNewton
in the query 31 at the end of his Opticks claimed that the microscopic world,
madeofatoms,followsthesamekindoflawsastheUniverseatlarge.Thefirst
instance of a “regular pattern” produced artificially was by the spontaneous
modulation of the surface of a liquid vibrated vertically. It was first believed
that this Faraday instability was a kind of linear resonance of the capillary
waves at the frequency of the excitation. It was Rayleigh who first explained
the striking fact that actually the dominant frequency of the surface oscil-
lations is half that of the exciting acceleration, a result that is not obvious.
Rayleigh’s theory gave rise to a whole field of research, the study of ordinary
differential equations with time periodic coefficients. Once transformed into
maps of the circle this class of problems has been a major source of math-
ematical progress over the years. The next example of an artificial pattern
in experiments was the famous (and first very poorly understood) B´enard
cells. Henri B´enard, one of the first great experimentalists in fluid mechanics,
showed that a thin layer of whale oil heated from below set itself sponta-
neously in motion in a remarkably regular array of hexagonal cells. B´enard
himself did explain this phenomenon in a rather confused way and another
50 years passed before Scriven and Sterling showed that the instability is due
to the Marangoni effect, that is the temperature dependence of the surface
tension. It took another 10 years to see the explanation of the hexagonal pat-
tern by Palm, who proved that hexagons are the generic structure to expect
beyond a bifurcation yielding wavy patterns with an arbitrary orientation in
a plane. The Rayleigh–B´enard instability due to buoyancy (instead of the
VI Foreword
Marangoni effect) of a fluid layer heated from below does show rolls, because
of a particular, nongeneric symmetry.
Overtheyearsanalmostinfinitenumberof“pattern-forming”bifurcations
in physical systems have been discovered, as well as many kinds of instabil-
ities. The famous Brusselator explained the oscillating chemical reactions in
the well-known Belousov–Zhabotinsky system. The Turing instability shows
up in driven systems with widely different diffusion coefficients. Quite often
the scientists responsible for the discoveries and/or the theory of pattern-
forming systems saw in a more or less explicit way a relationship between
their observations and the structures of living beings. Such a rather close
connection surely exists for patterns on sea-shells that can be explained by
simple reaction–diffusion equations without any recourse to a detailed “pro-
gram” written in the DNA of the snails.
May be less noticed than discoveries in the experimental field, there has
beenatremendousprogressinourrationalunderstandingofwhatIwouldcall
loosely the qualitative behavior of solutions of partial differential equations
and/or ordinary differential equations. A very crucial idea, first developed by
Poincar´e and by Andronov, is the one of “robustness.” Loosely speaking, it
allows us to describe a dynamical system (= the equation(s) of motion of the
system we consider) independently of their detailed form. This is of course a
“topological theory” in the deepest sense, as it amounts to consider that two
systemsare“thesame”ifonecandeformonesmoothlyintotheother.Except
at “bifurcation” value this holds true for almost all values of the parameters
intotheequation(s)ofmotion:changingabitaparametervaluechangesonly
numerical values typical of the solutions, not their structure.
The study of the qualitative properties of dynamical systems has expe-
rienced an explosive growth in the late 1970s and early 1980s. This is very
nicely presented in the present book. The mathematicians managed later on
to prove many results that had only been guessed by physicists in this field.
Emboldened by the success of this notion of “structural stability,” some sci-
entists, Len Pismen among them, tried to deal in the same way with the
pattern-forming systems. This amounted to summarizing the properties of
the system at hand into a set of equations as simple as possible, but with
all the relevant symmetries. This method traces back its origin in Poincar´e’s
normal forms; near a bifurcation point, one can reduce the complexity of the
equationsbykeepingasvariablesafewtime-dependentamplitudesonly,with
simple polynomial nonlinearities. This reduction of the complexity is, after
all, what we do spontaneously when we see a set of rolls in aligned clouds or
chaos and turbulence in a cloud painted by Tiepolo or Corot.
The question is posed by this reduction: Can we obtain a picture of the
phenomenageneralenoughtoyieldthemostobviousfeaturesofthesolutions
wewouldliketodescribe,althoughthemathematicsremainsimpleenoughto
beamenabletoourmethodsofanalysis?Ofcourse,suchareductionistviewis
at the heart of our physical description of the world. To take an example, the
Navier–Stokesequationsofaviscousfluidreduceallthecomplexmotionofan
Foreword VII
almost infinite number of atoms and molecules to the knowledge of the value
of the shear viscosity of this fluid and, eventually, to the pressure–density
relation.
Len Pismen starts from quite general equations, kind of generalized
reaction–diffusion equations for an arbitrary number of chemical species. He
showsmasterfullyhowtoreducethemtosimplerformsnearbifurcationpoints
(the famous Newell–Withehead–Segel amplitude equation, the Kuramoto–
Sivashinskyequation,theGinzburg–Landauequationtonamejustafew).He
then follows this general scheme of reduction, but with very little “hubris.”
Reading his text, I am amazed by the conciseness and efficiency of his pre-
sentation. Only a few lines of text and some very clever mathematical steps
bring all the necessary light in very messy questions, in the great tradition
of scientists such as Zel’dovich, Kolmogorov, and Landau. He balances very
finely the logical step, the development of a new intuition, with the help of
classicalanalysisatitsbest.Thistexthasthedoubleinteresttointroducethe
reader to a whole fascinating new field, still in a stage of fast development,
and to show how to sort out quite complex issues in an efficient way. Besides
itsobviouspedagogicalaspects,thismonumentofscholarshipisalsothevery
first one to cover in a fully unified way an extremely wide range of topics.
It reaches quite often the cutting edge of present-day research. For instance,
it gives a very thorough review of everything known about the spirals in ex-
citable media, including the instabilities of the center of the spiral that may
oscillate, meander, and so on. Although the amplitude equations have been
known for a number of years, this is – as far as I can tell – the first time all
their properties are discussed in a unified text.
As the title shows, part of the book is devoted to fronts and interfaces.
ThisisdoneinChap.2,whichisverywelcomeasitputsinasingleframework
topicssuchastheMaxwell–vanderWaalstheoryofliquid–vaporinterface,the
growth of a stable pattern in an unstable medium (the so-called KPP prob-
lem), and other “phase field models” that have been lately very thoroughly
studied.
The amplitude equation section (Chap. 4) is an exposition by a master
of the field. Everything is covered in a smooth way, including the very tricky
question of the motion of defects, where the distortion brought to the phase
field by the motion isanalyzed in depth. The latest progressin pinned fronts,
grain boundaries, etc. is well covered. The advantage of a single book is most
evident here, because the discussion relies on a rather delicate analysis of
bifurcations of trajectories in a 3D space.
Thefinalchapterisaboutwavepatternsandspirals,withalltheirinstabil-
ities, interactions, etc. This book follows another one by the same author, on
vortices in nonlinear fields, and the two bring a unified view of an extremely
active and interesting field of modern research.
Paris, France Yves Pomeau
Preface
This book involved considerable rewriting and reshuffling, despite the fact
that I thought when I started it that I knew the material well. As usual, it
wasdifficulttodecidewhattoincludeandwhatnot.Amonographshouldnot
imitate a journal review; it is too easy to send readers to find their own way
in the thick of the woods. However, the narrative may expand without limit
when attempting to be self-contained. The final compromise may sometimes
be subjective and imperfect.
Few appreciated the hard labor of Procrustes, who had to fit all kinds
of strangers into his infamous bed. The author finds himself in a somewhat
similar position, with an additional constraint on the use of violent means
and an obligation to keep his customers alive and well, while resizing them to
common logic and notation. This is not easy even when processing one’s own
workofyesteryear.Onewishestogiveupandwriteanotherpaperinstead,but
thisbecomesimpossibleafteracriticalpointhasbeenpassedandtheinvested
effort has grown too heavy to discard. The result is before the reader.
Patterns and interfaces live in space; I could not avoid, nevertheless, to
include a long chapter on dynamical systems, which live in time only, but
provide both technical means and examples to follow for more robust spatial
structures. Ardent students of dynamical systems may find it almost sacrile-
gious to see a variety of bifurcations and endless abyss of chaos crowded into
a few dozen pages, still shared with the dynamics of regular patterns (which
imitate far simpler systems by forfeiting their imperfections and natural vari-
ability). These are, of course, pure essentials that are needed to understand
further material.
The core of the book lies in various applications of perturbation theory,
which attempts to tackle nonlinear spatiotemporal structures that, at first
sight, are likely to break our modest analytical tools. A modern paradox is
thatthesestructuresareofteneasiertosimulatenumericallythantocarvean-
alytically.Simulationis,however,notasubstituteforunderstanding,andone
hopesthatanalytical skillswillnotfadeawaywiththechangeofgenerations.
X Preface
Just imagine a feeble knight-errant, armed only with a paper sword –
and, perhaps, a laptop with symbolic computation software: can he dare to
overcomeahordeofnonlinearbeasts?Thereare,essentially,twotricksknown
from ancient times: separate them or try to surprise them while they are still
waking up. The first method is called scale separation and is the principal
tool in Chaps. 2 and 3, where we study first fronts, and then structures built
up by their combinations in reaction–diffusion and related equations. The
second method, called amplitude expansion, deriving and analyzing universal
equations near symmetry-breaking transitions, prevails in Chaps. 4 and 5.
Reaction–diffusion systems, with their plethora of chemical and biological
applicationsandcommonbutversatilestructure,providethebulkofmaterial
ofthebook.Othercommonpattern-formingsystemsrootedinfluidmechanics
and nonlinear optics are not considered explicitly, but they converge to the
same universal equations of amplitude and phase dynamics.
I am indebted to many colleagues and friends of our “nonlinear commu-
nity”, which has forged during the late decades of 20th century our basic
understanding of pattern formation far from equilibrium. The new century,
that seems to be interested, in its youth, in the particular more than in the
universal, may still find this knowledge useful for building up future biomor-
phic technologies based on bottom-up self-organization rather than top-down
manufacturing.
Technion – Israel Institute of Technology
Haifa, November 2005 L.M. Pismen
Contents
Introduction................................................... 1
Quest for Complexity .................................... 1
Conservative and Dissipative Systems ...................... 2
Closed and Open Systems ................................ 3
Field Variables .......................................... 4
Historical and Bibliographical Notes ....................... 6
1 Dynamics, Stability and Bifurcations ...................... 9
1.1 Basic Models ........................................... 9
1.1.1 Dynamical Systems ............................... 9
1.1.2 Reaction–Diffusion System ......................... 13
1.1.3 Vertical Structure and Representative Equations ...... 15
1.1.4 Spectral Decompositions ........................... 17
1.1.5 Trajectories and Attractors ........................ 19
1.2 Linear Analysis ......................................... 21
1.2.1 Instabilities of Stationary States .................... 21
1.2.2 Turing and Hopf Instabilities ....................... 23
1.2.3 Dispersion Relations .............................. 24
1.2.4 Instabilities of Periodic Orbits ...................... 25
1.3 Weakly Nonlinear Analysis ............................... 27
1.3.1 Multiscale Expansion .............................. 27
1.3.2 Bifurcation of Stationary States .................... 29
1.3.3 Derivation of Amplitude Equations ................. 32
1.3.4 Hopf Bifurcation .................................. 35
1.3.5 Degenerate Bifurcations ........................... 38
1.4 Global Bifurcations ...................................... 39
1.4.1 Global Dynamics: An Overview ..................... 39
1.4.2 Systems with Separated Time Scales ................ 42
1.4.3 Almost Hamiltonian Dynamics...................... 45
1.4.4 Bifurcation Diagrams ............................. 46
1.5 Deterministic Chaos ..................................... 51
