Encyclopedia of

Nonlinear Science

Alwyn ScottEditor

Encyclopedia of

Nonlinear Science

ROUTLEDGENEW YORK AND LONDON

Published in 2005 byRoutledgeTaylor & Francis Group270 Madison AvenueNew York, NY 10016www.routledge-ny.com

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Library of Congress Cataloging-in-Publication Data

Encyclopedia of nonlinear science/Alwyn Scott, Editorp. cm.Includes bibliographical references and index.ISBN 1-57958-385-7 (hb: alk.paper)1. Nonlinear theories-Encyclopedias. 1. Scott, Alwyn, 1931--QA427, E53 2005003:75---dc22 2004011708

This edition published in the Taylor & Francis e-Library, 2006.

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Contents

Introduction vii

Editorial Advisory Board xiii

List of Contributors xv

List of Entries xxxiii

Thematic List of Entries xxxix

Entries A to Z 1

Index 1011

Introduction

Among the several advances of the 20th century,nonlinear science is exceptional for its generality.Although the invention of radio was important forcommunications, the discovery of DNA structurefor biology, the development of quantum theory fortheoretical physics and chemistry, and the inventionof the transistor for computer engineering, nonlinearscience is significant in all these areas and manymore. Indeed, it plays a key role in almost everybranch of modern research, as this Encyclopedia ofNonlinear Science shows.In simple terms, nonlinear science recognizes

that the whole is more than a sum of its parts,providing a context for consideration of phenomenalike tsunamis (tidal waves), biological evolution,atmospheric dynamics, and the electrochemicalactivity of a human brain, among many others.For a research scientist, nonlinear science offersnovel phenomena, including the emergence ofcoherent structures (an optical soliton, e.g., ora nerve impulse) and chaos (characterized bythe difficulties in making accurate predictions forsurprisingly simple systems over extended periodsof time). Both these phenomena can be studiedusing mathematical methods described in thisEncyclopedia. From amore fundamental perspective,a wide spectrum of applications arises becausenonlinear science introduces a paradigm shift inour collective attitude about causality. What is thenature of this shift?Consider the difference between linear and non-

linear analyses. Linear analyses are characterized bythe assumption that individual effects can be unam-biguously traced back to particular causes. In otherwords, a compound cause is viewed as the linear(or algebraic) sum of a collection of simple causes,each of which can be uniquely linked to a particulareffect. The total effect responding to the total causeis then considered to be just the linear sum of theconstituent effects.A fundamental tenet of nonlinear science is

to reject this convenient, but often unwarranted,

assumption. Of course, the notion that componentsof complex causes can interact among themselvesis not surprising to any thoughtful person whomanages to get through an ordinary day ofnormal life, and it is not at all new. Twenty-fourcenturies ago,Aristotle described four types of cause(material, efficient, formal, and final), which overlapand intermingle in ways that were often overlookedin 20th-century thought but are now under scrutiny.Consider some examples of linear scientific thinkingthat are presently being reevaluated in the contextof nonlinear science.--- Around the middle of the 20th century,

behavioral psychologists adopted the theoreticalposition that human mental activity can be reducedto a sum of individual responses to specificstimuli that have been learned at earlier stagesof development. Current research in neuroscienceshows this perspective to be unwarranted.--- Some evolutionary psychologists believe that

particular genes, located in the structure of DNA,can always be related in a one-to-one mannerto individual features of an adult organism,leading to hunts for a crime gene that seemabhorrent to moralists. Nonlinear science suggeststhat the relation between genes and features of anadult organism is more intricate than the linearperspective assumes.--- The sad disintegration of space shuttle

Columbia on the morning of February 1, 2003, setoff a search for the cause of the accident, ignoringAristotelian insights into the difficulties of definingsuch a concept, never mind sorting out the pieces.Did themishap occur because the heat-resistant tileswere timeworn (a material cause)? Or because 1.67pounds of debris hit the left wing at 775 ft/s duringtakeoff (an efficient cause)? Perhaps a managementculture that discounted the importance of safetymeasures (a formal cause) should shoulder some ofthe blame.--- Cultural phenomena, in turn, are often viewed

as the mere sum of individual psychologies,

vii

viii Introduction

ignoring the grim realities of war hysteria andlynch mobs, not to mention the tulip craze of17th-century Holland, the more recent dot-combubble, and the outbreak of communal mourningover the death of Princess Diana.

Evolution of the ScienceAs the practice of nonlinear science involves suchabstruse issues, one might expect its history to becheckered, and indeed it is. Mathematical physicsbegan with the 17th-century work of Isaac Newton,whose formulation of the laws ofmechanicalmotionand gravitation explained how the Earth movesabout the Sun, replacing a final cause (Gods plan)with an efficient cause (the force of gravity). Becauseit assumed that the net gravitational force actingon any celestial body is the linear (vector) sumof individual forces, Newtons theory providessupport for the linear perspective in science,as has often been emphasized. Nonetheless, themathematical system Newton developed (calculus)is the natural language for nonlinear science, and heused this language to solve the two-body problem(collective motion of Earth and Moon)---the firstnonlinear system to be mathematically studied.Also in the 17th century, Christiaan Huygens notedthat two pendulum clocks (which he had recentlyinvented) kept exactly the same time when hangingfrom a common support. (Confined to his room byan indisposition, Huygens observed the clocks overa period of several days, during which the swingingpendula remained in step.) If the clocks wereseparated to opposite sides of the room, one lostseveral seconds a daywith respect to the other. Fromsmall vibrations transmitted through the commonsupport, he concluded, the two clocks becamesynchronized---a typical nonlinear phenomenon.In the 18th century, Leonhard Euler used New-

tons laws of motion to derive nonlinear field equa-tions for fluid flow, which were augmented a cen-tury later by Louis Navier and George Stokes toinclude the dissipative effects of viscosity that arepresent in real fluids. In their generality, these equa-tions defied solution until the middle of the 20thcentury when, together with the digital computer,elaborations of the Navier--Stokes equations pro-vided a basis for general models of the Earths atmo-sphere and oceans, with implications for the vexingquestion of global warming. During the latter halfof the 19th century, however, special analytic solu-tions were obtained by Joseph Boussinesq and re-lated to experimental observations of hydrodynamicsolitary waves by John Scott Russell. These studies---which involved a decade of careful observations ofuniformly propagating heaps of water on canalsand inwave tanks---were among the earliest research

programs in the area now recognized as nonlinearscience. At about the same time, Pierre Francois Ver-hulst formulated and solved a nonlinear differentialequation---sometimes called the logistic equation---tomodel the population growth of his nativeBelgium.Toward the end of the 19th century, Henri

Poincare returned to Newtons original theme,presenting a solution of the three-body problemof celestial motion (e.g., a planet with two moons)in a mathematical competition sponsored by theKing of Sweden. Interestingly, a serious error inthis work was discovered prior to its publication,and he (Poincare, not the Swedish king) eventuallyconcluded that the three-body problem cannot beexactly solved. Now regarded by many as the birthof the science of complexity, this negative resulthad implications that were not widely appreciateduntil the 1960s,whennumerical studies of simplifiedatmospheric models by Edward Lorenz showedthat nonlinear systems with as few as threedegrees of freedom can readily exhibit the nonlinearphenomenon of chaos. (A key observation here wasof an unanticipated sensitivity to initial conditions,popularly known as the butterfly effect fromLorenzs speculation that the flap of a butterflyswings in Brazil [might] set off a tornado in Texas.)During the first half of the 20th century, the tempo

of research picked up. Although still carried on asunrelated activities, there appeared a notable num-ber of experimental and theoretical studies now rec-ognized as precursors of modern nonlinear science.Among others, these include Albert Einsteins non-linear theory of gravitation; nonlinear field theo-ries of elementary particles (like the recently discov-ered electron) developed by Gustav Mie and MaxBorn; experimental observations of local modes inmolecules by physical chemists (for which a non-linear theory was developed by Reinhard Mecke inthe 1930s, forgotten, and then redeveloped in the1970s); biological models of predator-prey popula-tion dynamics formulated by Vito Volterra (to de-scribe year-to-year variations in fish catches fromthe Adriatic Sea); observations of a profusion oflocalized nonlinear entities in solid-state physics(including ferromagnetic domain walls, crystal dis-locations, polarons, and magnetic flux vortices insuperconductors, among others); a definitive experi-mental and theoretical study of nerve impulse prop-agation on the giant axon of the squid by AlanHodgkin andAndrewHuxley; Alan Turings theoryof pattern formation in the development of biolog-ical organisms; and Boris Belousovs observationsof pattern formation in a chemical solution, whichwere at first ignored (under the mistaken assump-tion that they violated the second law of thermody-namics) and later confirmed and extended byAnatol

Introduction ix

Zhabotinsky and Art Winfree. Just as the inventionof the laser in the early 1960s led to numerous ex-perimental and theoretical studies in the new field ofnonlinear optics, the steady increases in computingpower throughout the second half of the 20th cen-tury enabled ever more detailed numerical studiesof hydrodynamic turbulence and chaos, whittlingaway at the long-established Navier--Stokes equa-tions and confirming the importance of Poincaresnegative result on the three-body problem.Thus, it was evident by 1970 that nonlinearity

manifests itself in several remarkable properties ofdynamical systems, including the following. (Thereare others, some no doubt waiting to be discovered.)--- Many nonlinear partial differential equations

(wave equations, diffusion equations, and morecomplicated field equations) are often observedto exhibit localized or lump-like solutions, similarto Russells hydrodynamic solitary wave. Thesecoherent structures of energy or activity emergefrom initial conditions as distinct dynamic entities,each having its own trajectory in space-time andcharacteristic ways of interacting with others.Thus, they are things in the normal sense ofthe word. Interestingly, it is sometimes possibleto compute the velocity of emergent entities(their speeds and shapes) from initial conditionsand express them as tabulated functions (thetafunctions or elliptic functions), thereby extendingthe analytic reach of nonlinear analysis. Examplesof emergent entities include tornadoes, nerveimpulses, magnetic domain walls, tsunamis, opticalsolitons, Jupiters Great Red Spot, black holes,schools of fish, and cities, to name but a few. Arelated phenomenon, exemplified by meanderingrivers, bolts of lightning, and woodland paths,is called filamentation, which also causes spottyoutput beams in poorly designed lasers.--- Surprisingly simple nonlinear systems

(Poincares three-body problem is the classic exam-ple) are found to have chaotic solutions, which re-main within a bounded region, while the differencebetween neighboring solution trajectories grows ex-ponentially with time. Thus, the course of a solutiontrajectory is strongly sensitive to its initial conditions(the butterfly effect). Chaotic solutions arise inboth energy-conserving (Hamiltonian) systems anddissipative systems, and they are fated to wanderunpredictably as trajectories that cannot be accu-rately extended into the future for unlimited periodsof time. As Lorenz pointed out, the chaotic behav-ior the Earths atmosphere makes detailed meteoro-logical predictions problematic, to the delight of themathematician and the despair of the weatherman.Chaotic systems also exhibit strange attractors inthe solution space, which are characterized by frac-tal (non-integer) dimensions.

--- Nonlinear problems often display thresholdphenomena, meaning that there is a relatively sharpboundary across which the qualitative nature of asolution changes abruptly. This is the basic propertyof an electric wall switch, the trigger of a pistol, andthe flip-flop circuit that a computer engineer usesto store a bit of information. (Indeed, a computercan be viewed as a large, interconnected collectionof threshold devices.) Sometimes called tippingpoints in the context of social phenomena,thresholds are an important part of our dailyexperience, where they complicate the relationshipof causality to legal responsibility. Was it the laststraw that broke the camels back? Or did all ofthe straws contribute to some degree? Should eachbe blamed according to its weight? How does oneassign culpability for the Murder on the OrientExpress?--- Nonlinear systems with several spatial coor-

dinates often exhibit spontaneous pattern forma-tion, examples of which include fairy rings of mush-rooms, oscillatory patterns of heart muscle activityunder fibrillation (leading to sudden cardiac arrest),weather fronts, the growth of form in a biologicalembryo, and the Gulf Stream. Such patterns can bechaotic in time and regular in space, regular in timeand chaotic in space, or chaotic in both space andin time, which in turn is a feature of hydrodyamicturbulence.--- If the input to (or stimulation of) a nonlinear

system is a single frequency sinusoid, the output (orresponse) is nonsinusoidal, comprising a spectrumof sinusoidal frequencies. For lossless nonlinearsystems, this can be an efficientmeans for producingenergy at integer multiples of the driving frequency,through the process of harmonic generation. Inelectronics, this process is widely used for digitaltuning of radio receivers. Taking advantage of thenonlinear properties of certain transparent crystals,harmonic generation is also employed in laseroptics to create light beams of higher frequency, forexample, conversion of red light to blue.--- Another nonlinear phenomenon is the syn-

chronization of weakly coupled oscillators, first ob-served by the ailing Huygens in the winter of 1665.Now recognized in a variety of contexts, this effectcrops up in the frequency locking of electric powergenerators tied to the same grid and the coupling ofbiological rhythms (circadian rhythms in humans,hibernation of bears, and the synchronized flashingof Indonesian fireflies), in addition to many appli-cations in electronics. Some suggest that neuronalfirings in the neocortex may be mutually synchro-nized.--- Shock waves are familiar to most of us as the

boom of a jet airplane that has broken the soundbarrier or the report of a cannon. Closely related

x Introduction

from a mathematical perspective are the bow waveof a speedboat, the breaking of onshore surf, andthe sudden automobile pileups that can occur on ahighway that is carrying traffic close to itsmaximumcapacity.--- More complicated nonlinear systems can be

hierarchical in nature. This comes about whenthe emergence of coherent states at one levelprovides a basis for new nonlinear dynamics at ahigher level of description. Thus, in the course ofbiological evolution, chemical molecules emergedfrom interactions among the atomic elements, andbiological molecules then emerged from simplermolecules to provide a basis for the dynamicsof a living cell. From collections of cells, multi-cellular organisms emerged, and so on up theevolutionary ladder to creatures like ourselves,who comprise several distinct levels of biologicaldynamics. Similar structures are observed in theorganization of coinage and of military units,not to mention the hierarchical arrangement ofinformation in the human brain.Often, qualitatively related behaviors---involving

one or more of such nonlinear manifestations---arefound in models that arise from different areasof application, suggesting the need for interdisci-plinary communications. By the early 1970s, there-fore, research in nonlinear science was in a statethat the physical chemists might describe as su-persaturated. Dozens of people across the globewere working on one facet or another of nonlin-ear science, often unaware of related studies in tra-ditionally unrelated fields. During the mid-1970s,this activity experienced a phase change, whichcan be viewed as a collective nonlinear effect inthe sociology of science. Unexpectedly, a numberof conferences devoted entirely to nonlinear sci-ence were organized, with participants from a vari-ety of professional backgrounds, nationalities, andresearch interests eagerly contributing. Solid-statephysicists began to talk seriously with biologists,neuroscientists with chemical engineers, and mete-orologists with psychologists. As interdisciplinarybarriers crumbled, these unanticipated interactionsled to the founding of centers for nonlinear sci-ence and the launching of several important researchjournals amid an explosion of research activity. Bythe early 1980s, nonlinear science had gained recog-nition as a key component of modern inquiry, play-ing a central role in a wide spectrum of activities.In the terminology introduced by Thomas Kuhn, anew paradigm had been established.

About this BookThe primary aim of this Encyclopedia is to providea source from which undergraduate and graduate

students in the physical and biological sciencescan study how concepts of nonlinear science arepresently understood and applied. In addition, itis anticipated that teachers of science and researchscientists who are unfamiliar with nonlinear con-cepts will use the work to expand their intellec-tual horizons and improve their lectures. Finally, itis hoped that this book will help members of theliterate public---philosophers, social scientists, andphysicians, for example---to appreciate the wealth ofnatural phenomena described by a science that doesnot discount the notion of complex causality.An early step in writing the Encyclopedia was to

choose the entry subjects---a difficult task that wasaccomplished through the efforts of a distinguishedBoard of Advisers (see page xiii), with membersfrom Australia, Germany, Italy, Japan, Russia, theUnited Kingdom, and the United States. After muchsifting and winnowing, an initial list of abouta thousand suggestions was reduced to the 438items given on pages 1--1010. Depending on thesubject matter, the entries are of several types. Someare historical or descriptive, while others presentconcepts and ideas that require notations fromphysics, engineering, or mathematics. Althoughmost of the entries were planned to be about athousand words in length, some---covering subjectsof greater generality or importance---are two or fourtimes as long.Of the many enjoyable aspects in editing

this Encyclopedia, the most rewarding has beenworking with those who wrote it---the contributors.The willing way in which these busy peopleresponded to entry invitations and their enthusiasticpreparation of assignments underscores the degreetowhich nonlinear science has become a communitywith a healthy sense of professional responsibility.In every case, the contributors have tried to presenttheir ideas as simply as possible, with a minimumof technical jargon. For a list of the contributors andtheir affiliations, see pages xv--xxxi from which itis evident that they come from about 30 differentcountries, emphasizing the international characterof nonlinear science.A proper presentation of the diverse profes-

sional perspectives that make up nonlinear sciencerequires careful organization of the Encyclopedia,which we attempt to provide. Although each entryis self-contained, the links among them can be ex-plored in several ways. First, the Thematic Liston pages xxxix--xliii groups entries within severalcategories, providing a useful summary of relatedentries through which the reader can surf. Second,the entries have See also notes, both withinthe text and at the end of the entry, encourag-ing the reader to browse outwards from a startingnode. Finally, the Index contains a detailed list of

Introduction xi

topics that do not have their own entries but arediscussed within the context of broader entries. Ifyou cannot find an entry on a topic you expectedto find, use the Thematic List or Index to lo-cate the title of the entry that contains the item youseek. Additionally, all entries have selected bibli-ographies or suggestions for further reading, lead-ing to original research and textbooks that aug-ment the overview approach to which an encyclo-pedia is necessarily limited. Although much of non-linear science evolved from applied mathematics,many of the entries contain no equations or math-ematical symbols and can be absorbed by the gen-eral reader. Some entries are necessarily technical,but efforts have been made to explain all termsin simple English. Also, many entries have eitherline diagrams expanding on explanations given inthe text or photographs illustrating typical exam-ples. Typographical errors will be posted on theencyclopedia web site at http://www.routledge-ny.com/ref/nonlinearsci/.The editing of thisEncyclopedia ofNonlinear Science

culminates a lifetime of study in the area, leavingme indebted to many. First is the Acquisitions Edi-tor, Gillian Lindsey, who conceived of the project,organized it, and carried it from its beginnings

in London across the ocean to publication in NewYork. Without her dedication, quite simply, theEncyclopedia would not exist. Equally importantto reaching the finished work were the effortsof the advisers, contributors, and referees, who,respectively, planned, wrote, and vetted the work,and to whom I am deeply grateful. On a broadertime-span are colleagues and students from theUniversity of Wisconsin, Los Alamos NationalLaboratory, the University of Arizona, and theTechnical University of Denmark, with whom Ihave interacted over four decades. Although fartoo many to list, these collaborations are fondlyremembered, and they provide the basis for muchof my editorial judgment. Finally, I express mygratitude for the generous financial support ofresearch in nonlinear science that has been providedto me since the early 1960s by the NationalScience Foundation (USA), the National Institutesof Health (USA), the Consiglio Nazionale delleRicerche (Italy), the European Molecular BiologyOrganization, the Department of Energy (USA), theTechnical Research Council (Denmark), the NaturalScience Research Council (Denmark), the ThomasB. Thriges Foundation (Denmark), and the FetzerFoundation (USA).

Alwyn ScottTucson, Arizona 2004

Editorial Advisory BoardFriedrich H. BusseTheoretical Physics, Universitat Bayreuth, Germany

Antonio DegasperisDipartimento di Fisica, Universita degli Studi di Roma La Sapienza, Italy

William D. DittoApplied Chaos Lab, Georgia Institute of Technology, USA

Chris EilbeckDepartment of Mathematics, Heriot-Watt University, UK

Sergej FlachMax Planck Institut fur Physik komplexer Systeme, Germany

Hermann FlaschkaDepartment of Mathematics, The University of Arizona, USA

Hermann HakenCenter for Synergetics, University of Stuttgart, Germany

James P. KeenerDepartment of Mathematics, University of Utah, USA

Yuri KivsharNonlinear Physics Center, Australian National University, Canberra, Australia

Yoshiki KuramotoDepartment of Physics, Kyoto University, Japan

Dave McLaughlinCourant Institute of Mathematical Sciences and Provost, New York University, USA

Lev A. OstrovskyZel Technologies/University of Colorado, Boulder, and Institute of Applied Physics,Nizhny Novgorod, Russia

Edward OttInstitute for Research in Electronics and Applied Physics, University of Maryland, USA

A.T. Winfree (deceased)Formerly Department of Ecology and Evolutionary Biology, University of Arizona, USA

Ludmila V. YakushevichInstitute of Cell Biophysics, Russian Academy of Science, Pushchino, Russia

Lai-Sang YoungCourant Institute of Mathematical Sciences, New York University, USA

xiii

List of ContributorsAblowitz, Mark J.Professor, Department of Applied Mathematics,University of Colorado, Boulder, USAAblowitz--Kaup--Newell--Segur system

Aigner, Andreas A.Research Associate, Department ofMathematical Sciences, University of Exeter, UKAtmospheric and ocean sciencesGeneral Circulation models of the atmosphereNavier--Stokes equationPartial differential equations, nonlinear

Albano, Ezequiel V.Instituto de Investigaciones Fisicoqumicas Teoricas yAplicadas (INIFTA) University of La Plata, ArgentinaForest fires

Aratyn, HenrikProfessor, Physics Department, University of Illinois atChicago, USADressing method

Aref, HassanDean of Engineering and Reynolds Metals ProfessorVirginia Polytechnic Institute & State University, USABernoullis equationChaos vs. turbulenceChaotic advectionCluster coagulationHele-Shaw cellNewtons laws of motion

Arrowsmith, DavidProfessor, School of Mathematical Sciences, Queen MaryUniversity of London, UKSymbolic dynamicsTopology

Athorne, ChristopherSenior Lecturer, Department of Mathematics,University of Glasgow, UKDarboux transformation

Bahr, DavidAssistant Professor, Department of Computer Science,Regis University, Colorado, USAGlacial flow

Ball, RowenaDepartment of Theoretical Physics, Australian NationalUniversity, AustraliaFairy rings of mushroomsKolmogorov cascadeSingularity theory

Barnes, HowardUnilever Research Professor of Industrial Rheology,Department of Mathematics, University of WalesAberystwyth, WalesRheology

Barthes, MarietteGroupe de Dynamique des Phases Condensees UMRCNRS 5581, Universite Montpellier 2, FranceRayleigh and Raman scattering and IR absorption

Beck, ChristianProfessor, School of Mathematical Sciences, Queen MaryUniversity of London, UKFree energyMultifractal analysisString theory

Beeckman, JeroenDepartment of Electronics and Information SystemsGhent University, BelgiumLiquid crystals

Benedict, KeithSenior Lecturer, School of Physics and Astronomy,University of Nottingham, UKAnderson localizationFrustration

xv

xvi List of Contributors

Berge, LucCommissariat a lEnergie Atomique,Bruyeres-le-Chatel, FranceDevelopment of singularitiesFilamentationKerr effect

Berland, NicoleChimie General et Organique Lycee Faidherbe de Lille,FranceBelousov--Zhabotinsky reaction

Bernevig, Bogdan A.Physics Department, Massachusetts Institute ofTechnology, USAHolons

Biktashev, Vadim N.Lecturer in Applied Maths, Mathematical Sciences,University of Liverpool, UKVortex dynamics in excitable media

Binczak, StephaneLaboratoire dElectronique, Informatique et Image,Universite de Bourgogne, FranceEphaptic couplingMyelinated nerves

Biondini, GinoAssistant Professor, Department of Mathematics,Ohio State University, USAEinstein equationsHarmonic generation

Blair, DavidProfessor, School of Physics, The University of WesternAustralia, AustraliaGravitational waves

Boardman, Alan D.Professor of Applied Physics, Institute for MaterialsResearch, University of Salford, UKPolaritons

Bollt, Erik M.Associate Professor,Departments of Mathematics & Computer Science andPhysics, Clarkson University, Potsdam, N.Y., USAMarkov partitionsOrder from chaos

Boon, J.-P.Professor, Faculte des Sciences, Universite Libre deBruxelles, BelgiumLattice gas methods

Borckmans, PierreCenter for Nonlinear Phenomena & Complex Systems,Universite Libre de Bruxelles, BelgiumTuring patterns

Boumenir, AminDepartment of Mathematics, State University of WestGeorgia, USAGelfand--Levitan theory

Bountis, TassosProfessor, Department of Mathematics andCenter for Research and Application of NonlinearSystems, University of Patras, GreecePainleve analysis

Boyd, Robert W.Professor, The Institute of Optics, University ofRochester, USAFrequency doubling

Bradley, ElizabethAssociate Professor, Department of Computer Science,University of Colorado, USAKirchhoffs laws

Bullough, RobinProfessor, Mathematical Physics, University ofManchester Institute of Science and Technology, UKMaxwell--Bloch equationsSine-Gordon equation

Bunimovich, LeonidRegents Professor, Department of Mathematics,Georgia Institute of Technology, USABilliardsDeterministic walks in random environmentsLorentz gas

Busse, Friedrich (Adviser)Professor, Theoretical Physics, University of Bayreuth,GermanyDynamos, homogeneousFluid dynamicsMagnetohydrodynamics

Calini, Annalisa M.Associate Professor, Department of Mathematics,College of Charleston, USAElliptic functionsMelnikov method

Caputo, Jean GuyLaboratoire de Mathematiques, Institut National desSciences Appliquees de Rouen, FranceJump phenomena

List of Contributors xvii

Censor, DanProfessor, Department of Electrical and ComputerEngineering, Ben-Gurion University of the Negev, IsraelVolterra series and operators

Chen, Wei-YinProfessor, Department of Chemical Engineering,University of Mississippi, USAStochastic processes

Chernitskii, Alexander A.Department of Physical Electronics, St. PetersburgElectrotechnical University, RussiaBorn--Infeld equations

Chiffaudel, ArnaudCEA-Saclay (Commissariat a lEnergie Atomique) &CNRS (Centre National de la Recherche Scientifique),FranceHydrothermal waves

Choudhury, S. RoyProfessor, Department of Mathematics, University ofCentral Florida, USAKelvin--Helmholtz instabilityLorenz equations

Christiansen, Peter L.Professor, Informatics and Mathematical Modelling andDepartment of Physics, Technical University ofDenmark, DenmarkSeparation of variables

Christodoulides, DemetriosProfessor, CREOL/School of Optics, University ofCentral Florida, USAIncoherent solitons

Coskun, TamerAssistant Professor, Department of ElectricalEngineering, Pamukkale University, TurkeyIncoherent solitons

Cruzeiro, LeonorCCMAR and FCT, University of Algarve, Campus deGambelas, Faro, PortugalDavydov soliton

Cushing, J.M.Professor, Department of Mathematics, University ofArizona, USAPopulation dynamics

Dafilis, MathewSchool of Biophysical Sciences and ElectricalEngineering, Swinbume University of Technology,AustraliaElectroencephalogram at mesoscopic scales

Davies, BrianDepartment of Mathematics, Australian NationalUniversity, AustraliaIntegral transformsPeriod doubling

Davis, William C.Formerly, Los Alamos National Laboratory USAExplosions

deBruyn, JohnProfessor, Department of Physics and PhysicalOceanography, Memorial University of Newfoundland,CanadaPhase transitionsThermal convection

Deconinck, BernardAssistant Professor, Department of Applied MathematicsUniversity of Washington, USAKadomtsev--Petviashvili equationPeriodic spectral theoryPoisson brackets

Degallaix, JeromeSchool of Physics, The University of Western Australia,AustraliaGravitational waves

Deift, PercyProfessor, Department ofMathematics, Courant Instituteof Mathematical Sciences, New York University, USARandom matrix theory IV: Analytic methodsRiemann--Hilbert problem

Deryabin, Mikhail V.Department of Mathematics, Technical University ofDenmark, DenmarkKolmogorov--Arnold--Moser theorem

Dewel, Guy (deceased)Formely Professor, Faculte des Sciences Universite Librede Bruxelles, BelgiumTuring patterns

Diacu, FlorinProfessor, Department of Mathematics and Statistics,University of Victoria, CanadaCelestial mechanicsN -body problem

Ding, MingzhouProfessor, Department of Biomedical EngineeringUniveristy of Florida, USAIntermittency

xviii List of Contributors

Dmitriev, S.V.Researcher, Institute of Industrial Science,University of Tokyo, JapanCollisions

Dolgaleva, KseniaDepartment of Physics, M.V. Lomonosov Moscow StateUniversity, Moscow andThe Institute of Optics, University of Rochester, USAFrequency doubling

Donoso, Jose M.E.T.S.I. Aeronauticos, Universidad Politecnica, Madrid,SpainBall lightning

Doucet, ArnaudSignal Processing Group, Department of Engineering,Cambridge University, UKMonte Carlo methods

Dritschel, DavidProfessor, Department of Applied Mathematics,The University of St. Andrews, UKContour dynamics

Dupuis, GerardChimie generale et organique, Lycee Faidherbe de Lille,FranceBelousov--Zhabotinsky reaction

Easton, Robert W.Professor, Department of Applied Mathematics,University of Colorado, Boulder, USAConley index

Eckhardt, BrunoProfessor, Fachbereich Physik, Philipps Universitat,Marburg, GermanyChaotic AdvectionMaps in the complex planePeriodic orbit theoryQuantum chaosRandom matrix theory I: Origins andphysical applications

Shear flowSolar systemUniversality

Efimo, I.Associate Professor of Biomedical Engineering,Stanley and Lucy Lopata Endowment,Washington University, Missouri, USACardiac muscle models

Eilbeck, Chris (Adviser)Professor, Department of Mathematics, Heriot-WattUniversity, UKDiscrete self-trapping system

Elgin, JohnProfessor, Maths Department, Imperial College ofScience, Technology and Medicine, London, UKKuramoto--Sivashinsky equation

Emmeche, ClausAssociate Professor andHead of Center for the PhilosophyofNature and Science Studies, University of Copenhagen,DenmarkCausality

Enolskii, VictorProfessor, Heriot-Watt University, UKTheta functions

Falkovich, GregoryProfessor, Department of Physics of Complex Systems,Weizmann Institute of Science, IsraelMixingTurbulence

Falqui, GregorioProfessor, Mathematical Physics Sector, InternationalSchool for Advanced Studies, Trieste, ItalyHodograph transformN -soliton formulas

Faris, William G.Professor, Department of Mathematics, University ofArizona, USAMartingales

Feddersen, HenrikResearch Scientist, Climate Research Division, DanishMeteorological Institute, DenmarkForecasting

Fedorenko, Vladimir V.Senior Scientific Researcher, Institute of Mathematics,National Academy of Science of Ukraine, UkraineOne-dimensional maps

Fenimore, Paul W.Theoretical Biology and Biophysics Group, Los AlamosNational Laboratory, USAProtein dynamics

Flach, Sergej (Adviser)Max Planck Institut fur Physik komplexer Systeme,GermanyDiscrete breathersSymmetry: equations vs. solutions

Flaschka, Hermann (Adviser)Professor, Department of Mathematics, The University ofArizona, USAToda lattice

List of Contributors xix

Fletcher, NevilleProfessor, Department of Electronic MaterialsEngineering, Australian National University, AustraliaOvertones

Flora, Luis MarioDepartment of Theory and Simulation of ComplexSystems, Instituto de Ciencia de Materiales de Aragon,SpainAubry--Mather theoryCommensurate-incommensurate transitionFrenkel--Kontorova model

Forrester, PeterDepartment of Mathematics and Statistics, University ofMelbourne, AustraliaRandom matrix theory II: Algebraic developments

Fowler, W. BeallEmeritus Professor, Physics Department, LehighUniversity, USAColor centers

Fraedrich, KlausProfessor, Meteorologisches Institut, UniversitatHamburg, GermanyAtmospheric and ocean sciencesGeneral circulation models of the atmosphere

Freites, Juan AlfredoDepartment of Physics and Astronomy, University ofCalifornia, Irvine, USAMolecular dynamics

Frieden, RoyOptical Sciences Center, University of Arizona in Tucson,USAInformation theory

Friedrich, JosephProfessor, Lehrstuhl fur Physik WeihenstephanTechnische Universitat Munchen, GermanyHole burning

Fuchikami, NobukoDepartment of Physics, Tokyo Metropolitan University,JapanDripping faucet

Gallagher, MarcusSchool of Information Technology & ElectricalEngineering, The University of Queensland, AustraliaMcCulloch--Pitts networkPerceptron

Garnier, NicolasLaboratoire de Physique, Ecole Normale Superieure deLyon, FranceHydrothermal waves

Gaspard, Pierre P.Center for Nonlinear Phenomena & Complex SystemsUniversite Libre de Bruxelles, BelgiumEntropyMapsQuantum theoryRossler systems

Gendelman, OlegFaculty of Mechanical Engineering, Israel Institute ofTechnology, IsraelHeat conduction

Giuliani, AlessandroEnvironment and Health Departmant, Istituto Superioredi Sanita, Rome, ItalyAlgorithmic complexity

Glass, LeonIsadore Rosenfeld Chair and Professor of Physiology,McGill University, CanadaCardiac arrhythmias and the electrocardiogram

Glendinning, PaulProfessor, Department of Mathematics, University ofManchester Institute of Science and Technology, UKHenon mapInvariant manifolds and setsRoutes to chaos

Goriely, AlainProfessor, Department of Mathematics, University ofArizona, USANormal forms theory

Grand, SteveDirector, Cyberlife Research Ltd., Shipham, UKArtificial life

Gratrix, SamMaths Department, Imperial College of Science,Technology and Medicine, UKKuramoto--Sivashinsky equation

Grava, TamaraMathematical Physics Sector, International school forAdvanced Studies, Trieste, ItalyHodograph transformN -soliton formulasZero-dispersion limits

xx List of Contributors

Grimshaw, RogerProfessor, Department of Mathematical Sciences,Loughborough University, UKGroup velocityKorteweg--de Vries equationWater waves

Haken, Hermann (Adviser)Professor Emeritus, Fakultat fur Physik, University ofStuttgart, GermanyGestalt phenomenaSynergetics

Halburd, Rodney G.Lecturer, Department of Mathematical Sciences,Loughborough University, UKEinstein equations

Hallinan, JenniferInstitute for Molecular Bioscience, The University ofQueensland, AustraliaGame of lifeGame theory

Hamilton, MarkProfessor, Department of Mechanical Engineering,University of Texas at Austin, USANonlinear acoustics

Hamm, PeterProfessor, Physikalisch-Chemisches Institut, UniversitatZurich, SwitzerlandFranck--Condon factorHydrogen bondPump-probe measurements

Hasselblatt, BorisProfessor, Department of Mathematics, Tufts University,USAAnosov and Axiom-A systemsMeasuresPhase space

Hastings, AlanProfessor, Department of Environmental Science andPolicy, University of California, USAEpidemiology

Hawkins, JaneProfessor, Department of Mathematics, University ofNorth Carolina at Chapel Hill, USAErgodic theory

Helbing, DirkInstitute for Economics and Traffic, Dresden Universityof Technology, GermanyTraffic flow

Henry, BruceDepartment of Applied Mathematics, University of NewSouth Wales, AustraliaEquipartition of energyHenon--Heiles system

Henry, BryanDepartment of Chemistry and Biochemistry, Universityof Guelph, CanadaLocal modes in molecules

Hensler, GerhardProfessor, Institut fur Astronomie, Universitats-Sternwarte Wien, AustriaGalaxies

Herrmann, HansInstitute for Computational Physics, University ofStuttgart, GermanyDune formation

Hertz, JohnProfessor, Nordic Institute for Theoretical Physics,DenmarkAttractor neural networks

Hietarinta, JarmoProfessor, Department of Physics, University of Turku,FinlandHirotas method

Hill, LarryTechnical Staff Member, Detonation Science &Technology, Los Alamos National Laboratory, USAEvaporation wave

Hjorth, Poul G.Associate Professor, Department of Mathematics,Technical University of Denmark, DenmarkKolmogorov--Arnold--Moser theorem

Holden, ArunProfessor of Computational Biology, School of BiomedicalSciences, University of Leeds, UKExcitabilityHodgkin--Huxley equationsIntegrate and fire neuronMarkin--Chizmadzhev modelPeriodic burstingSpiral waves

Holstein-Rathlou, N.-H.Professor, Department of Medical Physiology,University of Copenhagen, DenmarkNephron dynamics

List of Contributors xxi

Hommes, CarsProfessor, Center for Nonlinear Dynamics in Economicsand Finance, Department of Quantitative Economics,University of Amsterdam, The NetherlandsEconomic dynamics

Hone, AndrewLecturer in Applied Mathematics, Institute ofMathematics & Actuarial Science, University of Kent atCanterbury, UKExtremum principlesOrdinary differential equations, nonlinearRiccati equations

Hood, AlanProfessor, School of Mathematics and Statistics,University of St Andrews, UKCharacteristics

Houghton, ConorDepartment of Pure and Applied Mathematics, TrinityCollege Dublin, IrelandInstantonsYang--Mills theory

Howard, James E.Research Associate, Department of Physics, University ofColorado at Boulder, USANontwist mapsRegular and chaotic dynamics in atomic physics

Ivey, Thomas A.Department of Mathematics, College of Charleston, USADifferential geometryFramed space curves

Jimenez, SalvadorProfessor, Departamento de Matematicas, UniversidadAlfonso X El Sabio, Madrid, SpainCharge density wavesDispersion relations

Joannopoulos, John D.Professor, Department of Physics, MassachusettsInstitute of Technology, USAPhotonic crystals

Johansson, MagnusDepartment of Physics and Measurement Technology,Linkoping University, SwedenDiscrete nonlinear Schrodinger equations

Johnson, Steven G.Assistant Professor, Department of Mathematics,Massachussetts Institute of Technology, USAPhotonic crystals

Joshi, NaliniProfessor, School of Mathematics and Statistics,University of Sydney, AustraliaSolitons

Kaneko, KunihikoDepartment of Pure and Applied Sciences, University ofTokyo, JapanCoupled map lattice

Kantz, HolgerProfessor of Theoretical Physics, Max Planck Institut f urkomplexer Systeme, GermanyTime series analysis

Kennedy, Michael PeterProfessor of Microelectronic Engineering, UniversityCollege, Cork, IrelandChuas circuit

Kevrekidis, I.G.Professor, Department of Chemical Engineering,Princeton University, USAWave of translation

Kevrekidis, Panayotis G.Assistant Professor, Department of Mathematics andStatistics, University of Massachusetts, Amherst, USABinding energyCollisionsWave of translation

Khanin, KonstantinProfessor, Department of Mathematics, Heriot-WattUniversity, UKDenjoy theory

Khovanov, Igor A.Department of Physics, Saratov State University, RussiaQuasiperiodicity

Khovanova, Natalya A.Department of Physics, Saratov State University, RussiaQuasiperiodicity

King, AaronAssistant Professor, Department of Ecology andEvolutionary Biology, University of Tennessee,Knoxville, USAPhase plane

Kirby, Michael J.Professor, Department of Mathematics, Colorado StateUniversity, USANonlinear signal processing

xxii List of Contributors

Kirk, EdilbertMeteorologisches Institut, UniversitatHamburg, GermanyGeneral circulation models of the atmosphere

Kivshar, Yuri (Adviser)Nonlinear Physics Center, Australian NationalUniversity, AustraliaOptical fiber communications

Kiyono, KenResearch Fellow of the Japan Society for the Promotion ofScience, Educational Physiology Laboratory, Universityof Tokyo, JapanDripping faucet

Knott, RonDepartment of Mathematics, University of Surrey, UKFibonacci series

Kocarev, LiupcoAssociate Research Scientist, Institute for NonlinearScience, University of California, San Diego, USADamped-driven anharmonic oscillator

Konopelchenko, Boris G.Professor, Dipartimento di Fisica, University of Lecce,ItalyMultidimensional solitons

Konotop, Vladimir V.Centro de Fsica Teorica e Computacional ComplexoInterdisciplinar da Universidade de Lisboa, PortugalWave propagation in disordered media

Kosevich, ArnoldB. Verkin Institute for Low Temperature Physics andEngineering, National Academy of Sciences of Ukraine,Kharkov, UkraineBreathersDislocations in crystalsEffective massLandau--Lifshitz equationSuperfluiditySuperlattices

Kovalev, Alexander S.Institute for Low Temperature Physics and Engineering,National Academy of Sciences of Ukraine, UkraineContinuum approximationsTopological defects

Kramer, Peter R.Assistant Professor, Department of MathematicalSciences, Rensselaer Polytechnic Institute, USABrownian motionFokker--Planck equation

Krinsky, ValentinProfessor, Institut Non-Lineaire de Nice, FranceCardiac muscle models

Kuramoto, Yoshiki (Adviser)Department of Physics, Kyoto University, JapanPhase dynamics

Kurin, V.Institute for Physics of Microstructures, RussianAcademy of Science, RussiaCherenkov radiation

Kuvshinov, Viatcheslav I.Professor, Institute of Physics, Belarus Academy ofSciences, BelarusBlack holesCosmological modelsFractalsGeneral relativity

Kuzmin, AndreiProfessor, Institute of Physics, Belarus Academy ofSciences, BelarusFractals

Kuznetsov, VadimAdvanced Research Fellow, Department of AppliedMathematics, University of Leeds, UKRotating rigid bodies

LaBute, Montiago X.Theoretical Biology and Biophysics Group,Los Alamos National Laboratory, USAProtein structure

Lakshmanan, MuthusamyProfessor, Department of Physics, BharathidasanUniversity, Tiruchirapalli, IndiaEquations, nonlinearNonlinear electronicsSpin systems

Landa, Polina S.Professor, Department of Physics, Moscow StateUniversity, RussiaFeedbackPendulumQuasilinear analysisRelaxation oscillators

Landsberg, PeterProfessor, Faculty of Mathematical Studies, University ofSouthampton, UKDetailed balance

List of Contributors xxiii

Lansner, AndersDepartment of Numerical Analysis and ComputerScience (NADA), Royal Institute of Technology (KTH),SwedenCell assembliesNeural network models

Lee, JohnProfessor, Department of Mechanical Engineering,McGill University, CanadaFlame front

Lega, JocelineAssociate Professor, Department of Mathematics, Uni-versity of Arizona, USAEquilibriumFredholm theorem

Lepeshkin, NickThe Institute of Optics, University of Rochester, USAFrequency doubling

Levi, DecioProfessor, Dipartimento di Ingegneria Electronica,Universita degli Studi Roma tre, ItalyDelay-differential equations

Lichtenberg, Allan J.Professor, Department of Electrical Engineering andComputer Science, University of California at Berkeley,USAArnold diffusionAveraging methodsElectron beam microwave devicesFermi acceleration and Fermi mapFermi--Pasta--Ulam oscillator chainParticle acceleratorsPhase-space diffusion and correlations

Liley, DavidSchool of Biophysical Sciences and ElectricalEngineering, Swinburne University of Technology,AustraliaElectroencephalogram at mesoscopic scales

Lonngren, Karl E.Professor, Department of Electrical and ComputerEngineering, University of Iowa, USAPlasma soliton experiments

Losert, WolfgangAssistant Professor, Department of Physics, IPST andIREAP, University of Maryland, USAGranular materialsPattern formation

Lotric, Maja-BracicFaculty of Electrical Engineering, University ofLiubljana, SloveniaWavelets

Luchinsky, Dmitry G.Department of Physics, Lancaster University, UKNonlinearity, definition of

Lucke, ManfredInstitut fur Theoretische Physik, Universitat desSaarlandes, Saarbrucken, GermanyThermo-diffusion effects

Lunkeit, FrankMeteorologisches Institut, UniversitatHamburg, GermanyGeneral circulation models of the atmosphere

Ma, Wen-XiuDepartment ofMathematics, University of South Florida,USAIntegrability

Macaskill, CharlesAssociate Professor, School ofMathematics and Statistics,University of Sydney, AustraliaJupiters Great Red Spot

MacClune, Karen LewisHydrologist, SS Papadopulos & Associates, Boulder,Colorado, USAGlacial flow

Maggio, Gian MarioST Microelectronics and Center for WirelessCommunications (CWC), University of California atSan Diego, USADamped-driven anharmonic oscillator

Maini, Philip K.Professor, Centre for Mathematical Biology,Mathematical Institute, University of Oxford, UKMorphogenesis, biological

Mainzer, KlausProfessor, Director of the Institute of InterdisciplinaryInformatics, Department of Philosophy of Science,University of Augsburg, GermanyArtificial intelligenceCellular nonlinear networksDynamical systems

Malomed, Boris A.Professor, Department of Interdisciplinary Studies,Faculty of Engineering, Tel Aviv University, Israel

xxiv List of Contributors

Complex Ginzburg--Landau equationConstants of motion and conservation lawsMultisoliton perturbation theoryNonlinear Schrodinger equationsPower balance

Manevitch, LeonidProfessor, Institute of Chemical Physics, RussiaHeat conductionMechanics of solidsPeierls barrier

Manneville, PaulLaboratoire dHydrodynamique (LadHyX), EcolePolytechnique, Palaiseau, FranceSpatiotemporal chaos

Marklof, JensSchool of Mathematics, University of Bristol, UKCat map

Marsden, Jerrold E.Professor of Control and Dynamical SystemsCalifornia Institute of Technology, Pasadena, USABerrys phase

Martnez, Pedro JesusDepartment of Theory and Simulation of ComplexSystems, Instituto de Ciencia de Materiales de Aragon,SpainFrenkel--Kontorova model

Masmoudi, NaderAssociate Professor, Department of Mathematics,Courant Institute of Mathematical Sciences, New YorkUniversity, USABoundary layersRayleigh--Taylor instability

Mason, LionelMathematical Institute, Oxford University, UKTwistor theory

Mayer, AndreasInstitute for Theoretical Physics, University ofRegensburg, GermanySurface waves

McKenna, JoeProfessor, Department of Mathematics, University ofConnecticut, USATacoma Narrows Bridge collapse

McLaughlin, KennethAssociate Professor, Department of Mathematics,University of North Carolina at Chapel Hill, USARandom matrix theory III: Combinatorics

McLaughlin, RichardAssociate Professor, Department of Mathematics,University of North, Carolina, Chapel Hill, USAPlume dynamics

McMahon, BenTheoretical Biology and Biophysics Group, Los AlamosNational Laboratory, USAProtein dynamicsProtein structure

Meiss, JamesProfessor, Department of Applied Mathematics,University of Colorado at Boulder, USAHamiltonian systemsStandard mapSymplectic maps

Minkevich, AlbertProfessor of Theoretical Physics, Belorussian StateUniversity, Minsk, BelarusCosmological modelsGeneral relativity

Miura, RobertProfessor, Department of Mathematical Sciences,New Jersey Institute of Technology, USANonlinear toys

Moloney, Jerome V.Professor, Department of Mathematics, University ofArizona, USANonlinear optics

Moore, Richard O.Assistant Professor, Departmant of MathematicalSciences, New Jersey, Institute of Technology, USAHarmonic generation

MLrk, JesperProfessor, Optoelectronics, Research Center COM,Technical University of Denmark, DenmarkSemiconductor laser

Mornev, OlegSenior Researcher, Institute of Theoretical andExperimental Biophysics, RussiaGeometrical optics, nonlinearGradient systemZeldovich--Frank-Kamenetsky equation

Mosekilde, E.Professor, Department of Physics, Technical University ofDenmark, DenmarkNephron dynamics

List of Contributors xxv

Mueller, Stefan C.Department of Biophysics, Otto-von-Guericke-Universitat Magdeburg, GermanyScroll waves

Mullin, TomProfessor of Physics and Director of Manchester Centrefor Nonlinear Dynamics, University of Manchester, UKBifurcationsCatastrophe theoryTaylor--Couette flow

Mygind, JesperProfessor, Department of Physics, Technical University ofDenmark, DenmarkJosephson junctionsSuperconducting quantum interference device

Nakamura, YoshiharuAssociate Professor, Institute of Space and AstronauticalScience, Kanagawa, JapanPlasma soliton experiments

Natiello, MarioCentre for Mathematical Sciences, Lund University,SwedenLasersWinding numbers

Newell, AlanProfessor, Department of Mathematics, University ofArizona, USAInverse scattering method or transform

Newton, Paul K.Professor, Department of Aerospace and MechanicalEngineering, University of Southern California, USABerrys phaseChaos vs. turbulence

Neyts, KristiaanProfessor, Department of Electronics and InformationSystems, Ghent University, BelgiumLiquid crystals

Nicolis, G.Professor, Faculte des Sciences, Universite Libre deBruxelles, BelgiumBrusselatorChemical kineticsNonequilibrium statistical mechanicsRecurrence

Nunez, PaulProfessor, Brain Physics Group, Department ofBiomedical Engineering, Tulane University, USAElectroencephalogram at large scales

Olsder, Geert JanFaculty of Technical Mathematics and Informatics, DelftUniversity of Technology, The NetherlandsIdempotent analysis

Olver, Peter J.Professor, School of Mathematics, University ofMinnesota, USALie algebras and Lie groups

Ostrovsky, Lev (Adviser)Professor, Zel Technologies/Univeristy of Colorado,Boulder, Colorado, USA, and Institute of AppliedPhysics, Nizhny Novgorod, RussiaHurricanes and tornadoesModulated wavesNonlinear acousticsShock waves

Ottova-Leitmannova, AngelicaDepartment of Physiology, Michigan StateUniversity, USABilayer lipid membrance

Palmer, JohnProfessor, Department of Mathematics, University ofArizona, USAMonodromy preserving deformations

Pascual, Pedro J.Associate Professor, Departamento de IngenieriaInformatica, Universidad Autonoma de Madrid, SpainCharge density waves

Pedersen, Niels FalsigProfessor, Department of Power Engineering, TechnicalUniversity of Denmark, DenmarkLong Josephson junctionsSuperconductivity

Pelinovsky, DmitryAssociate Professor, Department of Mathematics,McMaster University, CanadaCoupled systems of partial differential equationsEnergy analysisGeneralized functionsLinearizationManley--Rowe relationsNumerical methodsN -wave interactionsSpectral analysis

Pelletier, Jon D.Assistant Professor, Department of Geosciences,University of Arizona, USAGeomorphology and tectonics

xxvi List of Contributors

Pelloni, BeatriceMathematics Department, University of Reading, UKBoundary value problemsBurgers equation

Petty, MichaelProfessor, Centre for Molecular and NanoscaleElectronics, University of Durham, UKLangmuir--Blodgett films

Peyrard, MichelProfessor of Physics, Laboratoire de Physique, EcoleNormale Superieure de Lyon, FranceBiomolecular solitons

Pikovsky, ArkadyDepartment of Physics Universitat Potsdam, GermanySynchronizationVan der Pol equation

Pitchford, JonLecturer, Department of Biology, University of York, UKRandom walks

Pojman, John A.Professor, Department of Chemistry and Biochemistry,The University of Southern Mississippi, USAPolymerization

Pumiri, A.Directeur de Recherche, Institut Non-Lineaire de Nice,FranceCardiac muscle models

Pushkin Dmitri O.Department of Theoretical and Applied Mechanics,University of IIIinois, Urbana--Champaign, USACluster coagulation

Rabinovich, MikhailResearch Physicist, Institute for Nonlinear Science,University of California at San Diego, USAand Institute of Applied Physics, Russian Academy ofSciencesChaotic dynamics

Ranada, Antonio F.Facultad de Fisica, Universidad Complutense, Madrid,SpainBall lightning

Recami, ErasmoProfessor of Physics, Faculty of Engineering, BergamoState University, Bergamo, ItalyTachyons and superluminal motion

Reucroft, StephenProfessor of Physics, Northeastern University, Boston,USAHiggs boson

Ricca, Renzo L.Professor, Dipartimento di Matematica e Applicazioni,Universita di Milano-Bicocca, Milan, ItalyKnot theoryStructural complexity

Robinson, James C.Mathematics Institute, University of Warwick, UKAttractorsDimensionsFunction spacesFunctional analysis

Robnik, MarkoProfessor, Center for Applied Mathematics andTheoretical Physics, University of Maribor, SloveniaAdiabatic invariantsDeterminism

Rogers, ColinProfessor, Australian Research Council Centre ofExcellence for Mathematics and Statistics of ComplexSystems, School of Mathematics, University of NewSouth Wales, AustraliaBacklund transformations

Romanenko, ElenaSenior Scientific Researcher, Institute of Mathematics,National Academy of Science of Ukraine, UkraineTurbulence, ideal

Rosenblum, MichaelDepartment of Physics, University of Potsdam, GermanySynchronizationVan der Pol equation

Rouvas-Nicolis, C.Climatologie Dynamique, Institut Royal Meteorologiquede Belgique, BelgiumRecurrence

Ruijsenaars, SimonCenter for Mathematics and Computer Science,The NetherlandsDerrick--Hobart theoremParticles and antiparticles

Rulkov, NikolaiInstitute for Nonlinear Science, University of Californiaat San Diego, USAChaotic dynamics

List of Contributors xxvii

Sabatier, PierreProfessor, Physique Mathematique, UniversiteMontpellier II, FranceInverse problems

Sakaguchi, HidetsuguDepartment of Applied Science for Electronics andMaterials, Kyushu University, JapanCoupled oscillators

Salerno, MarioProfessor, Departimento di Fisica E.R. Caianiello,Universita degli Studi, Salerno, ItalyBethe ansatzSalerno equation

Sandstede, BjornAssociate Professor, Department of Mathematics, OhioState University, USAEvans function

Satnoianu, RazvanCentre for Mathematics, School of Engineering andMathematical Sciences, City University, UKDiffusionReaction-diffusion systems

Sauer, TimProfessor, Department of Mathematics, George MasonUniversity, USAEmbedding methods

Savin, AlexanderProfessor, Moscow Institute of Physics and Technology,RussiaPeierls barrier

Schaerf, TimothySchool of Mathematics and Statistics, University ofSydney, AustraliaJupiters Great Red Spot

Schattschneider, DorisProfessor, Department of Mathematics, MoravianCollege, Pennsylvania, USATessellation

Schirmer, JochenProfessor, Institute for Physical Chemistry, Heidelberg,GermanyHartee approximation

Schmelcher, PeterInstitute for Physical Chemistry, University ofHeidelberg, GermanyHartree approximation

Scholl, EckehardProfessor, Institut fur Theoretische Physik, TechnischeUniversitat Berlin, GermanyAvalanche breakdownDiodesDrude modelSemiconductor oscillators

Schuster, PeterInstitut fur Theoretische Chemie und MolekulareStrukturbiologie, AustriaBiological evolutionCatalytic hypercycleFitness landscape

Scott, Alwyn (Editor)Emeritus Professor of Mathematics, University ofArizona, USACandleDiscrete self-trapping systemDistributed oscillatorsEmergenceEuler--Lagrange equationsHierarchies of nonlinear systemsLaboratory models of nonlinear wavesLifetimeMatter, nonlinear theories ofMultiplex neuronNerve impulsesNeuristorQuantum nonlinearityRotating-wave approximationSolitons, a brief historyState diagramsSymmetry groupsTachyons and superluminal phenomenaThreshold phenomenaWave packets, linear and nonlinear

Segev, MordechaiProfessor, Technion-Israel Institute of Technology, Haifa,IsraelIncoherent solitons

Shalfeev, VladimirHead of Department of Oscillation Theory, NizhniNovgorod State University, RussiaParametric amplification

Sharkovsky, Alexander N.Institute of Mathematics, National Academy of Sciencesof Ukraine, UkraineOne-dimensional mapsTurbulence, ideal

Sharman, RobertNational Center for Atmospheric Research,Boulder, Colorado, USAClear air turbulence

xxviii List of Contributors

Shinbrot, TroyAssociate Professor, Department of Chemical andBiochemical Engineering, Rutgers University, USAControlling chaos

Shohet, J. LeonProfessor, Department of Electrical and ComputerEngineering, University of Wisconsin-Madison, USANonlinear plasma waves

Siwak, PawelDepartment of Electrical Engineering, PoznanUniversity of Technology, PolandIntegrable cellular automata

Skufca, Joe D.Department of Mathematics, US Naval Academy,USAMarkov partition

Skufca, JosephCenter for Computational Science and MathematicalModelling, University of Maryland, USAMarkov partitions

Smil, VaclavProfessor, Department of Environment, University ofManitoba, CanadaGlobal warming

Sobell, Henry M.Independent scholar, New York, USADNA premelting

Solari, Hernan GustavoDepartamento Fsica, University of Buenos Aires,ArgentinaLasersWinding numbers

Soljacic, MarinPrincipal Research Scientist, Research Laboratory ofElectronics, Massachusetts Institute of Technology, USAPhotonic crystals

SLrensen, Mads PeterAssociate Professor, Department of Mathematics,Technical University of Denmark, DenmarkCollective coordinatesMultiple scale analysisPerturbation theory

Sornette, DidierProfessor, Laboratoire de Physique de la MatiereCondensee, Universite de Nice - Sophia Antipolis, FranceSandpile model

Sosnovtseva, O.Lecturer, Department of Physics, Technical University ofDenmark, DenmarkNephron dynamics

Spatschek, KarlProfessor, Institut fur Theoretische Physics 1,Heinrich-Heine-Universitat Dusseldorf, GermanyCenter manifold reductionDispersion management

Stadler, Michael A.Professor, Institut fur Physchologie andKognitionsforschung, Bremen, GermanyGestalt phenomena

Stauffer, DietrichInstitute for Theoretical Physics, University of Cologne,GermanyPercolation theory

Stefanovska, AnetaHead, Nonlinear Dynamics and Synergetics GroupFaculty of Electrical Engineering, University ofLjubljana, SloveniaFlip-flop circuitInhibitionNonlinearity, definition ofQuasiperiodicityWavelets

Storb, UlrichInstitut fur Experimentelle Physik, Otto-von-Guericke-Universitat, Magdeburg, GermanyScroll waves

Strelcyn, Jean-MarieProfesseur, Departement de Mathematiques, Universitede Rouen, Mont Saint Aignan Cedex, FrancePoincare theorems

Suris, Yuri B.Department of Mathematics, Technische UniversitatBerlin, GermanyIntegrable lattices

Sutcliffe, PaulProfessor of Mathematical Physics, Institute ofMathematics & Acturial Science, University of Kent atCanterbury, UKSkyrmions

Sverdlov, MashaTEC High School, Newton, Massachusetts, USAHurricanes and Tornadoes

List of Contributors xxix

Swain, John DavidProfessor, Department of Physics, NortheasternUniversity, Boston, USADoppler shiftQuantum field theoryTensors

Tabor, MichaelProfessor, Department of Mathematics, University ofArizona, USAGrowth patterns

Tajiri, MasayoshiEmeritus Professor, Department of MathematicalSciences, Osaka Prefecture University, JapanSolitons, types ofWave stability and instability

Tass, PeterProfessor, Institut fur Medizin, ForschungszentrumJulich, GermanyStochastic analysis of neural systems

Taylor, RichardAssociate Professor, Materials Science Institute,University of Oregon, USALevy flights

Teman, RogerLaboratoire dAnalyse Numerique, Universite de ParisSud, FranceInertial manifolds

Thompson, MichaelEmeritus Professor (UCL) and Honorary Fellow,Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, UKDuffing equationStability

Tien, H. Ti (deceased)Formerly Professor, Membrane Biophysics Laboratory,Michigan State University, USABilayer lipid membranes

Tobias, Douglas J.Associate Professor, Department of Chemistry,University of California at Irvine, USAMolecular dynamics

Toda, MorikazuEmeritus Professor, Tokyo University of Education,JapanNonlinear toys

Trueba, Jose L.Departmento di Mathematicas, y Fisica Aplicadas yCiencias de la Natura, Universidad Rey Juan Carlos,Mostoles, SpainBall lightning

Tsimring, Lev S.Research Physicist, Institute for Nonlinear Science,University of California, San Diego USAAvalanches

Tsinober, ArkadyProfessor, Iby and Aladar Fleischman Faculty ofEngineering, Tel Aviv University, IsraelHelicity

Tsironis, Giorgos P.Department of Physics, University of Crete, GreeceBjerrum defectsExcitonsIsing modelLocal modes in molecular crystals

Tsygvintsev, AlexeiMaitre de Conferences, Unite de Mathematiques Pures etAppliquees, Ecole Normale Superieure de Lyon, FrancePoincare theorems

Tuszynski, JackDepartment of Physics, University of Alberta, CanadaCritical phenomenaDomain wallsFerromagnetism and ferroelectricityFrohlich theoryHysteresisOrder parametersRenormalization groupsScheibe aggregates

Ustinov, Alexey V.Physikalisches Institut III, University ofErlangen-Nurnberg, GermanyJosephson junction arrays

van der Heijden, GertCentre for Nonlinear Dynamics, University CollegeLondon, UKButterfly effectHopf bifurcation

Vazquez, LuisProfessor, Faculted de Informatica, UniversidadComplutense de Madrid, Spain. Senior Researcher andCofounder of the Centro de Astrobiologa, InstituoNacional de Tecnica Aeroespacial, Madrid, Spain

xxx List of Contributors

Charge density wavesDispersion relationsFitzHugh--Nagumo equationVirial theoremWave propagation in disordered media

Verboncoeur, John P.Associate Professor, Nuclear Engineering Department,University of California, Berkeley, USAElectron beam microwave devices

Veselov, AlexanderProfessor, Department of Mathematical Sciences,Loughborough University, UKHuygens principle

Vo, Ba-NguElectrical and Electronic Engineering Department,The Univeristy of Melbourne, Victoria, AustraliaMonte Carlo methods

Voiculescu, Dan-VirgilProfessor, Department of Mathematics, University ofCalifornia at Berkeley, USAFree probability theory

Voorhees, Burton H.Professor, Department of Mathematics, AthabascaUniversity, CanadaCellular automata

Wadati, M.Professor, Department of Physics, University of Tokyo,JapanQuantum inverse scattering method

Walter, Gilbert G.Professor Emeritus, Department of MathematicalSciences, University of Wisconsin-Milwaukee, USACompartmental models

Waymire, Edward C.Professor, Department of Mathematics, Oregon StateUniversity, USAMultiplicative processes

West, Bruce J.Chief Scientist, Mathematics, US Army Research Office,North Carolina, USABranching lawsFluctuation-dissipation theoremKicked rotor

Wilhelmsson, HansProfessor Emeritus of Physics, Chalmers University ofTechnology, SwedenAlfven waves

Wilson, Hugh R.Centre for Vision Research, York University, CanadaNeuronsStereoscopic vision and binocular rivalry

Winfree, A.T. (Adviser) (deceased)Formerly, Department of Ecology and EvolutionaryBiology, University of Arizona, USADimensional analysis

Wojtkowski, Maciej P.Professor, Department of Mathematics, University ofArizona, USALyapunov exponents

Yakushevich, Ludmilla (Adviser)Researcher, Institute of Cell Biophysics, RussianAcademy of Sciences, RussiaDNA solitons

Young, Lai-Sang (Adviser)Professor, Courant Institute of Mathematical Sciences,New York University, USAAnosov and Axiom-A systemsHorseshoes and hyperbolicity in dynamical systemsSinai--Ruelle--Bowen measures

Yiguang, JuAssistant Professor, Department of Mechanical andAerospace Engineering, Princeton University,USAFlame front

Yukalov, V.I.Professor, Bogolubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, RussiaBose--Einstein condensationCoherence phenomena

Zabusky, Norman J.Professor, Department of Mechanical and AerospaceEngineering, Rutgers University, USAVisiometricsVortex dynamics of fluids

Zbilut, Joseph P.Professor, Department of Molecular Biophysics andPhysiology, Rush University, USAAlgorithmic complexity

Zhou, XinProfessor, Department of Mathematics, Duke University,USARandom matrix theory IV: Analytic methodsRiemann--Hilbert problem

List of Contributors xxxi

Zolotaryuk, Alexander V.Bogolyubov Institute for Theoretical Physics,UkrainePolaronsRatchets

Zorzano, Mara-PazYoung Researcher, Centro de Astrobiologa, InstitutoNacional de Tecnica Aeroespacial, Madrid, SpainFitzHugh--Nagumo equationsVirial Theorem

List of EntriesAblowitz--Kaup--Newell--Segursystem

Adiabatic invariantsAlfven wavesAlgorithmic complexityAnderson localizationAnosov and Axiom-A systemsArnold diffusionArtificial intelligenceArtificial lifeAtmospheric and ocean sciencesAttractor neural networkAttractorsAubry--Mather theoryAvalanche breakdownAvalanchesAveraging methods

Backlund transformationsBall lightningBelousov--Zhabotinsky reactionBernoullis equationBerrys phaseBethe ansatzBifurcationsBilayer lipid membranesBilliardsBinding energyBiological evolutionBiomolecular solitonsBjerrum defectsBlack holesBorn--Infeld equationsBose--Einstein condensationBoundary layersBoundary value problemsBranching lawsBreathers

Brownian motionBrusselatorBurgers equationButterfly effect

CandleCardiac arrhythmias and the electrocardiogramCardiac muscle modelsCat mapCatalytic hypercycleCatastrophe theoryCausalityCelestial mechanicsCell assembliesCellular automataCellular nonlinear networksCenter manifold reductionChaos vs. turbulenceChaotic advectionChaotic dynamicsCharacteristicsCharge density wavesChemical kineticsCherenkov radiationChuas circuitClear air turbulenceCluster coagulationCoherence phenomenaCollective coordinatesCollisionsColor centersCommensurate-incommensurate transitionCompartmental modelsComplex Ginzburg--Landau equationConley indexConstants of motion and conservation lawsContinuum approximationsContour dynamics

xxxiii

xxxiv List of Entries

Controlling chaosCosmological modelsCoupled map latticeCoupled oscillatorsCoupled systems of partial differentialequations

Critical phenomena

Damped-driven anharmonic oscillatorDarboux transformationDavydov solitonDelay-differential equationsDenjoy theoryDerrick--Hobart theoremDetailed balanceDeterminismDeterministic walks in random environmentsDevelopment of singularitiesDifferential geometryDiffusionDimensional analysisDimensionsDiodesDiscrete breathersDiscrete nonlinear Schrodinger equationsDiscrete self-trapping systemDislocations in crystalsDispersion managementDispersion relationsDistributed oscillatorsDNA premeltingDNA solitonsDomain wallsDoppler shiftDressing methodDripping faucetDrude modelDuffing equationDune formationDynamical systemsDynamos, homogeneous

Economic system dynamicsEffective massEinstein equationsElectroencephalogram at large scalesElectroencephalogram at mesoscopic scalesElectron beam microwave devicesElliptic functionsEmbedding methodsEmergenceEnergy analysisEntropyEphaptic couplingEpidemiologyEquations, nonlinearEquilibrium

Equipartition of energyErgodic theoryEuler--Lagrange equationsEvans functionEvaporation waveExcitabilityExcitonsExplosionsExtremum principles

Fairy rings of mushroomsFeedbackFermi acceleration and Fermi mapFermi--Pasta--Ulam oscillator chainFerromagnetism and ferroelectricityFibonacci seriesFilamentationFitness landscapeFitzHugh--Nagumo equationFlame frontFlip-flop circuitFluctuation-dissipation theoremFluid dynamicsFokker--Planck equationForecastingForest firesFractalsFramed space curvesFranck--Condon factorFredholm theoremFree energyFree probability theoryFrenkel--Kontorova modelFrequency doublingFrohlich theoryFrustrationFunction spacesFunctional analysis

GalaxiesGame of lifeGame theoryGelfand--Levitan theoryGeneral circulation models ofthe atmosphere

General relativityGeneralized functionsGeometrical optics, nonlinearGeomorphology and tectonicsGestalt phenomenaGlacial flowGlobal warmingGradient systemGranular materialsGravitational wavesGroup velocityGrowth patterns

List of Entries xxxv

Hamiltonian systemsHarmonic generationHartree approximationHeat conductionHele-Shaw cellHelicityHenon mapHenon--Heiles systemHierarchies of nonlinear systemsHiggs bosonHirotas methodHodgkin--Huxley equationsHodograph transformHole burningHolonsHopf bifurcationHorseshoes and hyperbolicity in dynamicalsystems

Hurricanes and tornadoesHuygens principleHydrogen bondHydrothermal wavesHysteresis

Idempotent analysisIncoherent solitonsInertial manifoldsInformation theoryInhibitionInstantonsIntegrabilityIntegrable cellular automataIntegrable latticesIntegral transformsIntegrate and fire neuronIntermittencyInvariant manifolds and setsInverse problemsInverse scattering method or transformIsing model

Josephson junction arraysJosephson junctionsJump phenomenaJupiters Great Red Spot

Kadomtsev--Petviashvili equationKelvin--Helmholtz instabilityKerr effectKicked rotorKirchhoffs lawsKnot theoryKolmogorov cascadeKolmogorov--Arnold--Moser theoremKorteweg--de Vries equationKuramoto--Sivashinsky equation

Laboratory models of nonlinear wavesLandau--Lifshitz equationLangmuir--Blodgett filmsLasersLattice gas methodsLevy flightsLie algebras and Lie groupsLifetimeLinearizationLiquid crystalsLocal modes in molecular crystalsLocal modes in moleculesLong Josephson junctionsLorentz gasLorenz equationsLyapunov exponents

MagnetohydrodynamicsManley--Rowe relationsMapsMaps in the complex planeMarkin--Chizmadzhev modelMarkov partitionsMartingalesMatter, nonlinear theory ofMaxwell--Bloch equationsMcCulloch--Pitts networkMeasuresMechanics of solidsMelnikov methodMixingModulated wavesMolecular dynamicsMonodromy preserving deformationsMonte Carlo methodsMorphogenesis, biologicalMultidimensional solitonsMultifractal analysisMultiple scale analysisMultiplex neuronMultiplicative processesMultisoliton perturbation theoryMyelinated nerves

Navier--Stokes equationN -body problemNephron dynamicsNerve impulsesNeural network modelsNeuristorNeuronsNewtons laws of motionNonequilibrium statistical mechanicsNonlinear acousticsNonlinear electronicsNonlinear opticsNonlinear plasma waves

xxxvi List of Entries

Nonlinear Schrodinger equationsNonlinear signal processingNonlinear toysNonlinearity, definition ofNontwist mapsNormal forms theoryN -soliton formulasNumerical methodsN -wave interactions

One-dimensional mapsOptical fiber communicationsOrder from chaosOrder parametersOrdinary differential equations,nonlinear

Overtones

Painleve analysisParametric amplificationPartial differential equations, nonlinearParticle acceleratorsParticles and antiparticlesPattern formationPeierls barrierPendulumPerceptronPercolation theoryPeriod doublingPeriodic burstingPeriodic orbit theoryPeriodic spectral theoryPerturbation theoryPhase dynamicsPhase planePhase spacePhase-space diffusion and correlationsPhase transitionsPhotonic crystalsPlasma soliton experimentsPlume dynamicsPoincare theoremsPoisson bracketsPolaritonsPolaronsPolymerizationPopulation dynamicsPower balanceProtein dynamicsProtein structurePump-probe measurements

Quantum chaosQuantum field theoryQuantum inverse scattering methodQuantum nonlinearityQuantum theory

Quasilinear analysisQuasiperiodicity

Random matrix theory I: Origins and physicalapplications

Random matrix theory II: Algebraicdevelopments

Random matrix theory III: CombinatoricsRandom matrix theory IV: Analytic methodsRandom walksRatchetsRayleigh and Raman scattering and IR absorptionRayleigh--Taylor instabilityReaction-diffusion systemsRecurrenceRegular and chaotic dynamics in atomic physicsRelaxation oscillatorsRenormalization groupsRheologyRiccati equationsRiemann--Hilbert problemRossler systemsRotating rigid bodiesRotating-wave approximationRoutes to chaos

Salerno equationSandpile modelScheibe aggregatesScroll wavesSemiconductor laserSemiconductor oscillatorsSeparation of variablesShear flowShock wavesSinai--Ruelle--Bowen measuresSine-Gordon equationSingularity theorySkyrmionsSolar systemSolitonsSolitons, a brief historySolitons, types ofSpatiotemporal chaosSpectral analysisSpin systemsSpiral wavesStabilityStandard mapState diagramsStereoscopic vision and binocular rivalryStochastic analysis of neural systemsStochastic processesString theoryStructural complexitySuperconducting quantum interferencedevice

List of Entries xxxvii

SuperconductivitySuperfluiditySuperlatticesSurface wavesSymbolic dynamicsSymmetry groupsSymmetry: equations vs. solutionsSymplectic mapsSynchronizationSynergetics

Tachyons and superluminal motionTacoma Narrows Bridge collapseTaylor--Couette flowTensorsTessellationThermal convectionThermo-diffusion effectsTheta functionsThreshold phenomenaTime series analysisToda latticeTopological defectsTopologyTraffic flowTurbulence

Turbulence, idealTuring patternsTwistor theory

Universality

Van der Pol equationVirial theoremVisiometricsVolterra series and operatorsVortex dynamics in excitable mediaVortex dynamics of fluids

Water wavesWave of translationWave packets, linear and nonlinearWave propagation in disordered mediaWave stability and instabilityWaveletsWinding numbers

Yang--Mills theory

Zeldovich--Frank-Kamenetsky equationZero-dispersion limits

Thematic List of Entries

General

HISTORY OF NONLINEAR SCIENCE

Bernoullis equation, Butterfly effect, Candle, Ce-lestial mechanics, Davydov soliton, Determinism,Feedback, Fermi--Pasta--Ulam oscillator chain, Fi-bonacci series, Hodgkin--Huxley equations, Intro-duction, Integrability, Lorenz equations, Manley--Rowe relations, Markin--Chizmadzhev model, Mar-tingales, Matter, nonlinear theory of, Poincaretheorems, Solar system, Solitons, a brief his-tory, Tacoma Narrows Bridge collapse, Vander Pol equation, Zeldovich--Frank-Kamenetskyequation

COMMON EXAMPLES OFNONLINEAR PHENOMENA

Avalanches, Ball lightning, Brownian motion, But-terfly effect, Candle, Clear air turbulence, Diffusion,Dripping faucet, Dune formation, Explosions, Fairyrings of mushrooms, Filamentation, Flame front,Fluid dynamics, Forest fires, Glacial flow, Globalwarming, Hurricanes and tornadoes, Jupiters GreatRed Spot, Nonlinear toys, Order from chaos, Pendu-lum, Phase transitions, Plume dynamics, Solar sys-tem, Tacoma Narrows Bridge collapse, Traffic flow,Water waves

Methods and Models

ANALYTICAL METHODS

Backlund transformations, Bethe ansatz, Center-manifold reduction, Characteristics, Collective coor-dinates, Continuum approximations, Dimensionalanalysis, Dispersion relations, Dressing method,Elliptic functions, Energy analysis, Evans function,Fredholm theorem, Gelfand--Levitan theory, Gen-eralized functions, Hamiltonian systems, Hirotasmethod, Hodograph transform, Idempotent analy-sis, Integral transforms, Inverse scattering methodor transform, Kirchhoffs laws, Multiple scaleanalysis, Multisoliton perturbation theory, Non-equilibrium statistical mechanics, Normal forms

theory, N -soliton formulas, Painleve analysis, Peri-odic spectral theory, Perturbation theory, Phase dy-namics, Phase plane, Poisson brackets, Power bal-ance, Quantum inverse scattering method, Quasi-linear analysis, Riccati equations, Rotating-waveapproximation, Separation of variables, Spectralanalysis, Stability, State diagrams, Synergetics, Ten-sors, Theta functions, Time series analysis, Volterraseries, Wavelets, Zero-dispersion limits

COMPUTATIONALMETHODS

Averaging methods, Cellular automata, Cellularnonlinear networks, Characteristics, Compartmen-

xxxix

xl Thematic List of Entries

tal models, Contour dynamics, Embeddingmethods, Extremum principles, Fitness landscape,Forecasting, Framed space curves, Hartree approxi-mation, Integrability, Inverse problems, Lattice gasmethods, Linearization, Maps, Martingales, Monte--Carlo methods, Numerical methods, Recurrence,Theta functions, Time series analysis, Visiometrics,Volterra series and operators, Wavelets

TOPOLOGICAL METHODS

Backlund transformations, Cat map, Conley index,Darboux transformation, Denjoy theory, Derrick--Hobart theorem, Differential geometry, Extremumprinciples, Functional analysis, Horseshoes and hy-perbolicity in dynamical systems, Huygens prin-ciple, Inertial manifolds, Invariant manifolds andsets, Knot theory, Kolmogorov--Arnold--Moser the-orem, Lie algebras and Lie groups, Maps, Mea-sures, Monodromy-preserving deformations, Mul-tifractal analysis, Nontwist maps, One-dimensionalmaps, Periodic orbit theory, Phase plane, Phasespace, Renormalization groups, Riemann--Hilbertproblem, Singularity theory, Symbolic dynamics,Symmetry groups, Topology, Virial theorem,Wind-ing numbers

CHAOS, NOISE AND TURBULENCE

Attractors, Aubry--Mather theory, Butterfly effect,Chaos vs. turbulence, Chaotic advection, Chaoticdynamics, Clear air turbulence, Dimensions, En-tropy, Ergodic theory, Fluctuation-dissipation the-orem, Fokker--Planck equation, Free probabilitytheory, Frustration, Hele-Shaw cell, Horseshoesand hyperbolicity in dynamical systems, Levyflights, Lyapunov exponents, Martingales, Mel-nikov method, Order from chaos, Percolation the-ory, Phase space, Quantum chaos, Random matrixtheory, Random walks, Routes to chaos, Spatiotem-poral chaos, Stochastic processes, Turbulence, Tur-bulence, ideal

COHERENT STRUCTURES

Biomolecular solitons, Black holes, Breathers, Cellassemblies, Davydov soliton, Discrete breathers,Dislocations in crystals, DNA solitons, Domain

walls, Dune formation, Emergence, Fairy rings ofmushrooms, Flame front, Higgs boson, Holons,Hurricanes and tornadoes, Instantons, JupitersGreat Red Spot, Local modes in molecular crys-tals, Local modes in molecules, Multidimen-sional solitons, Nerve impulses, Polaritons, Po-larons, Shock waves, Skyrmions, Solitons, typesof, Spiral waves, Tachyons and superluminalmotion, Turbulence, Turing patterns, Wave oftranslation

DYNAMICAL SYSTEMS

Anosov and axiom-A systems, Arnold diffusion,Attractors, Aubry--Mather theory, Bifurcations, Bil-liards, Butterfly effect, Cat map, Catastrophe the-ory, Center manifold reduction, Chaotic dynamics,Coupledmap lattice, Deterministicwalks in randomenvironments, Development of singularities, Dy-namical systems, Equilibrium, Ergodic theory,Fitness landscape, Framed space curves, Func-tion spaces, Gradient system, Hamiltonian sys-tems, Henon map, Hopf bifurcation, Horse-shoes and hyperbolicity in dynamical systems,Inertial manifolds, Intermittency, Kicked rotor,Kolmogorov--Arnold--Moser theorem, Lyapunovexponents, Maps, Measures, Melnikov method,One-dimensional maps, Pattern formation, Peri-odic orbit theory, Phase plane, Phase space, Phase-space diffusion and correlations, Poincare theo-rems, Reaction-diffusion systems, Rossler systems,Rotating rigid bodies, Routes to chaos, Sinai--Ruelle--Bowen measures, Standard map, Stochasticprocesses, Symbolic dynamics, Synergetics, Univer-sality, Visiometrics, Winding numbers

GENERAL PHENOMENA

Adiabatic invariants, Algorithmic complexity, An-derson localization, Arnold diffusion, Attractors,Berrys phase, Bifurcations, Binding energy, Bound-ary layers, Branching laws, Breathers, Brownianmotion, Butterfly effect, Causality, Chaotic dy-namics, Characteristics, Cluster coagulation, Co-herence phenomena, Collisions, Critical phenom-ena, Detailed balance, Determinism, Diffusion,Domain walls, Doppler shift, Effective mass,Emergence, Entropy, Equilibrium, Equipartitionof energy, Excitability, Explosions, Feedback,

Thematic List of Entries xli

Filamentation, Fractals, Free energy, Frequencydoubling, Frustration, Gestalt phenomena, Groupvelocity, Harmonic generation, Helicity, Hopf bi-furcation, Huygens principle, Hysteresis, Inco-herent solitons, Inhibition, Integrability, Intermi-ttency, Jump phenomena, Kolmogorov cascade,Levy flights, Lifetime, Mixing, Modulated waves,Multiplicative processes, Nonlinearity, definitionof, N -wave interactions, Order from chaos, Orderparameters, Overtones, Pattern formation, Perioddoubling, Periodic bursting, Power balance, Quan-tum chaos, Quantum nonlinearity, Quasiperiodic-ity, Recurrence, Routes to chaos, Scroll waves, Shearflow, Solitons, Spiral waves, Structural complex-ity, Symmetry: equations vs. solutions, Synergetics,Tachyons and superluminal motion, Tessellation,Thermal convection, Threshold phenomena, Turbu-lence,Universality,Wavepackets, linear andnonlin-ear, Wave propagation in disordered media, Wavestability and instability

MAPS

Aubry--Mather theory, Backlund transformations,Cat map, Coupledmap lattice, Darboux transforma-tion, Denjoy theory, Embedding methods, Fermi ac-celeration and Fermi map, Henonmap, Maps, Mapsin the complex plane,Monodromypreservingdefor-mations, Nontwist maps, One-dimensional maps,Periodic orbit theory, Recurrence, Renormalizationgroups, Singularity theory, Standardmap, Symplec-tic maps

MATHEMATICAL MODELS

Ablowitz--Kaup--Newell--Segur system, Attractorneural network, Billiards, Boundary value prob-lems, Brusselator, Burgers equation, Cat map,

Cellular automata, Compartmental models, Com-plex Ginzburg--Landau equation, Continuum ap-proximations, Coupled map lattice, Coupledsystems of partial differential equations, Delay-differential equations, Discrete nonlinear Schrod-inger equations, Discrete self-trapping system,Duffing equation, Equations, nonlinear, Euler--Lagrange equations, Fitzhugh--Nagumo equation,Fokker--Planck equation, Frenkel--Kontorovamodel,Game of life, General circulation models of theatmosphere, Henon--Heiles system, Integrable cel-lular automata, Integrable lattices, Ising model,Kadomtsev--Petviashvili equation, Knot theory,Korteweg--de Vries equation, Kuramoto--Sivashin-sky equation, Landau--Lifshitz equation, Lattice gasmethods, Lie algebras and Lie groups, Lorenz equa-tions, Markov partitions, Martingales, Maxwell--Bloch equation, McCulloch--Pitts network, Navier--Stokes equation, Neural network models, New-tons laws of motion, Nonlinear Schrodingerequations, One-dimensional maps, Ordinarydifferential equations, nonlinear, Partial differen-tial equations, nonlinear, Random walks, Riccatiequations, Salerno equation, Sandpile model, Sine-Gordon equation, Spin systems, Stochastic pro-cesses, Structural complexity, Symbolic dynamics,Synergetics, Toda lattice, Van der Pol equation,Zeldovich--Frank-Kamenetsky equation

STABILITY

Attractors, Bifurcations, Butterfly effect, Catastro-phe theory, Controlling chaos, Development ofsingularities, Dispersion management, Dispersionrelations, Emergence, Equilibrium, Excitability,Feedback, Growth patterns, Hopf bifurcation,Lyapunov exponents, Nonequilibrium statisticalmechanics, Stability

Disciplines

ASTRONOMY AND ASTROPHYSICS

Alfven waves, Black holes, Celestial mechanics,Cosmological models, Einstein equations, Galax-ies, Gravitational waves, Henon--Heiles system,Jupiters Great Red Spot, N -body problem, Solarsystem

BIOLOGY

Artificial life, Bilayer lipid membranes, Biologicalevolution, Biomolecular solitons, Cardiac arrhyth-mias and electro cardiogram, Cardiac muscle mod-els, Catalytic hypercycle, Compartmental models,Davydov soliton, DNA premelting, DNA solitons,

xlii Thematic List of Entries

Epidemiology, Excitability, Fairy rings of mush-rooms, Fibonacci series, Fitness landscape, Frohlichtheory, Game of life, Growth patterns, Morpho-genesis, biological, Nephron dynamics, Protein dy-namics, Protein structure, Scroll waves, Turingpatterns

CHEMISTRY

Belousov--Zhabotinsky reaction, Biomolecular soli-tons, Brusselator, Candle, Catalytic hypercycle,Chemical kinetics, Cluster coagulation, Flame front,Franck--Condon factor, Hydrogen bond, Langmuir--Blodgett films, Molecular dynamics, Polymeriza-tion, Protein structure, Reaction-diffusion systems,Scheibe aggregates, Turing patterns, Vortexdynamics in excitable media

CONDENSEDMATTER ANDSOLID-STATE PHYSICS

Anderson localization, Avalanche breakdown, Bjer-rum defects, Bose--Einstein condensation, Chargedensity waves, Cherenkov radiation, Color cen-ters, Commensurate-incommensurate transition,Discrete breathers, Dislocations in crystals, Do-main walls, Drude model, Effective mass, Exci-tons, ferromagnetism and Ferroelectricity, Franck--Condon factor, Frenkel--Kontorova model, Frus-tration, Heat conduction, Hydrogen bond, Isingmodel, Langmuir--Blodgett films, Liquid crystals,Local modes in molecular crystals, Mechanics ofsolids, Nonlinear acoustics, Peierls barrier, Percola-tion theory, Regul