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Springer 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,morphogenesisinbiology, 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 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorandtechbooksusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:10822705 54/techbooks 543210 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 mathematical progress over the years. The next example of an artificial pattern in experiments was the famous (and first very poorly understood) Bénard cells. Henri Bénard, one of the first great experimentalists in fluid mechanics, showed that a thin layer of whale oil heated from below set itself spontaneously in motion in a remarkably regular array of hexagonal cells. Bénard 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 pattern 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–Bénard 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 instabilities. 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 “program” 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 Poincaré 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 scientists, 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 Poincaré’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 community”, 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