encyclopedia of nonlinear science - Princeton Universityengine.princeton.edu/download/ju-digital... · Contents Introduction vii List of Advisers xiii List of Contributors xv Alphabetical

e n c y c l o p e d i a o f

nonlinearscience

e n c y c l o p e d i a o f

nonlinearscience

Alwyn Scott, Editor

R O U T L E DG EAn Imprint of Taylor & Francis Group

New York London

Published in 2004 by

RoutledgeAn Imprint of the Taylor & Francis Group29 West 35th StreetNew York, NY 10001www.routledge-ny.com

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Routledge is an imprint of the Taylor & Francis Group

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Contents

Introduction vii

List of Advisers xiii

List of Contributors xv

Alphabetical Entry List xxxiii

Thematic Entry List xxxix

Entries A to Z 000

Index 000

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 theoryfor theoretical chemistry, and the invention ofthe 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 (oralgebraic) 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-fivecenturies 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 seemsabhorrent to moralists. Nonlinear science suggeststhat the relation between genes and features of anadult organism are more intricate than the linearperspective assumes.--- The sad disintegration of space shuttle

Columbia on the morning of February 1, 2003 set off

vii

Introduction

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,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 (God’s 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, Newton’s 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

Newton’s laws of motion to derive nonlinear fieldequations for fluid flow, which were augmented acentury later by Louis Navier and George Stokesto include the dissipative effects of viscosity that

are present in real fluids. In their generality,these equations defied solution until the middleof the 20th century when, together with thedigital computer, elaborations of the Navier--Stokes equations provided a basis for generalmodels of the Earth’s atmosphere and oceans,with implications for the vexing question ofglobal warming. During the latter half of the19th century, however, special analytic solutionswere obtained by Joseph Boussinesq and relatedto experimental observations of hydrodynamicsolitary waves by John Scott Russell. These studies---which involved a decade of careful observationsof uniformly propagating ��heaps of water�� oncanals and in wave tanks---were among the earliestresearch programs in the area now recognizedas nonlinear science. At about the same time,Pierre Fran�cois Verhulst formulated and solved anonlinear differential equation---sometimes calledthe logistic equation---to model the populationgrowth of his native Belgium.Toward the end of the 19th century, Henri

Poincare returned to Newton’s 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�� fromLorentz’s speculation that ��the flap of a butterfly’swings in Brazil [might] set off a tornado in Texas.��)During the first half of the 20th century,

the tempo of research picked up. Although stillcarried on as unrelated activities, there appeareda notable number of experimental and theoreticalstudies now recognized as precursors of modernnonlinear science. Among others, these includeAlbert Einstein’s nonlinear theory of gravitation;nonlinear field theories of elementary particles(like the recently discovered electron) developedby Gustav Mie and Max Born; experimentalobservations of local modes in molecules byphysical chemists (for which a nonlinear theorywas developed by Reinhard Mecke in the 1930s,

viii

Introduction

forgotten, and then redeveloped in the 1970s);biological models of predator--prey populationdynamics formulated by Vito Volterra (to describeyear-to-year variations in fish catches from theAdriatic Sea); observations of a profusion oflocalized nonlinear entities in solid-state physics(including ferromagnetic domain walls, crystaldislocations, polarons, and magnetic flux vorticesin superconductors, among others); a definitiveexperimental and theoretical study of nerve impulsepropagation on the giant axon of the squid byAlan Hodgkin and Andrew Huxley; Alan Turing’stheory of pattern formation in the developmentof biological organisms; and Boris Belousov’sobservations of pattern formation in a chemicalsolution, which were at first ignored (underthe mistaken assumption that they violated thesecond law of thermodynamics) and later confirmedand extended by Anatol Zhabotinsky and ArtWinfree. Just as the invention of the laser in theearly 1960s led to numerous experimental andtheoretical studies in the new field of nonlinearoptics; thus, the steady increases in computingpower throughout the second half of the 20thcentury enabled ever more detailed numericalstudies of hydrodynamic turbulence and chaos,whittling away at the long-established Navier--Stokes equations and confirming the importanceof Poincare’s negative result on the three-bodyproblem.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 Russell’s hydrodynamic solitary wave. These��coherent 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 of the word.Interestingly, it is sometimes possible to computethe velocity of emergent entities (their speeds andshapes) from initial conditions and express themas tabulated functions (theta functions or ellipticfunctions), thereby extending the analytic reach ofnonlinear analysis. Examples of emergent entitiesinclude tornados, nerve impulses, magnetic domainwalls, tsunamis, optical solitons, Jupiter’s Great RedSpot, black holes, schools of fish, and cities, to namebut a few. A related phenomenon, exemplified bymeandering rivers, bolts of lightning, andwoodland

paths, is called filamentation, which also causesspotty output beams in poorly designed lasers.--- Surprisingly simple nonlinear systems

(Poincare’s 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 andin dissipative systems, and they are fated to wan-der unpredictably as trajectories that cannot be accu-rately extended into the future for unlimited periodsof time. As Lorenz pointed out, the chaotic behaviorof the Earth’s 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 threshold

phenomena, meaning that there is a relativelysharp boundary across which the qualitative natureof a solution changes abruptly. This is the basicproperty of an electric wall switch, the triggerof a pistol, and of the flip-flop circuit that acomputer engineer uses to store a bit of information.(Indeed, a computer can be viewed as a large,interconnected collection of threshold devices.)Sometimes called ��tipping points�� in the context ofsocial phenomena, thresholds are an important partof our daily experience, where they complicate therelationship of causality to legal responsibility. Wasit the last straw that broke the camel’s back? Or didall of the straws contribute to some degree? Shouldeach be blamed according to its weight? How doesone assign 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 producing

ix

Introduction

energy 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,hybernation 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 relatedfrom 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 gradu-ate 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 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 vii--viii), 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 439items given on pages xx--xx. 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 been

x

Introduction

working 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 contributorsand their affiliations, see pages • •, from which it isQ:1evident that they come from 25 different countries,emphasizing the international character of nonlinearscience.A proper presentation of the diverse profes-

sional perspectives that comprise 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 • •, groups entries within several cate-Q:2gories, providing a useful summary of related en-tries through which the reader can surf. Second, theentries have ``See also´´ notes, both within thetext and at the end of the entry, encouraging thereader to browse outwards from a starting node. Fi-nally, the Index contains a detailed list of topicsthat do not have their own entries but are discussedwithin the context of broader entries. If you can-not find an entry on a topic you expected to find,use the Thematic List or Index to locate the title ofthe entry that contains the item you seek. Addition-ally, all entries have selected bibliographies or sug-gestions for further reading, leading to original re-search and textbooks that augment the overview ap-proach to which an encyclopedia is necessarily lim-ited. Although much of nonlinear science evolvedfrom applied mathematics, many of the entries con-tain no equations or mathematical symbols and canbe absorbed by the general reader. Some entries are

necessarily technical, but efforts have been madeto explain all terms in simple English. Also, manyentries have either line diagrams expanding onexplanations given in the text, or photographsillustrating typical examples.The editing of thisEncyclopedia ofNonlinear Science

culminates a lifetime of study in the area, leavingmeindebted to many. First is the Acquisitions Editor,Gillian Lindsey, who conceived of the project,organized it, and carried it from its beginnings inLondon across the ocean to final publication inNew York. Without her dedication, quite simply,the Encyclopedia 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 NationalLaboratories, the University of Arizona, and theTechnical University ofDenmark,withwhom I haveinteracted over four decades. Although far toomanyto list, these collaborations are fondly remembered,and they provide the basis for much of my editorialjudgment. Finally, I express my gratitude for thegenerous financial support of research in nonlinearscience that has been provided to me since the early1960s by theNational Science Foundation (USA), theNational Institutes of Health (USA), the ConsiglioNazionale delle Ricerche (Italy), the EuropeanMolecular Biology Organization, the Departmentof Energy (USA), the Technical Research Council(Denmark), the Natural Science Research Council(Denmark), the Thomas B. Thriges Foundation, andthe Fetzer Foundation.

Alwyn ScottTucson, Arizona 2003

xi

Editorial Advisory Board

Friedrich H. BusseTheoretical Physics, Universitt Bayreuth,Germany

Antonio DegasperisDipartimento di Fisica, Universit degli Studi di Roma”La Sapienza”

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

Chris EilbeckDepartment of Mathematics, Heriot-Watt University,UK

Sergej FlachMax-Planck-Institut fuer Physik komplexer Systeme,Germany

Herman FlaschkaDepartment of Mathematics, University of Arizona,USA

Hermann HakenCenter for Synergetics, University of Stuttgart,Germany

James P. KeenerDepartment of Mathematics, University of Utah

Yuri KivsharNonlinear Physics Group, Australian National University

Yoshiki KuramotoDepartment of Physics, Kyoto University,Japan

Dave McLaughlinCourant Institute of Mathematical Sciences, New YorkUniversity,USA

Lev A. OstrovskyZel Technologies/National Oceanic &Atmospheric Admin-istration,Environmental Technology Laboratory, Boulder, Colorado,and Institute of Applied Physics,Russia

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

Art WinfreeFormerly Department of Ecology and Evolutionary Biol-ogy,University of Arizona,USA

Ludmila V. YakushevichInstitute of Cell BiophysicsRussia

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

xiv

List of Contributors

Ablowitz, Mark J.Professor, Department of Applied MathematicsUniversity of Colorado, Boulder, USAAblowitz--Kaup--Newell--Segur (AKNS) system

Aigner, AndreasResearch Associate, Department of EngineeringMathematics, University of Bristol, UKAtmospheric and ocean sciencesNavier--Stokes equationPartial differential equations, nonlinear

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

Aratyn, HenrikProfessor, Physics DepartmentUniversity of Illinois at Chicago, USADressing method

Aref, HassanDean of Engineering and Reynolds Metals ProfessorVirginia Polytechnic Institute & State University, USABernoulli’s equationChaotic advectionCluster--cluster coagulationHele--Shaw experimentNewton’s laws of motion

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

Athorne, ChristopherSenior Lecturer, Department of MathematicsUniversity of Glasgow, UKDarboux transformation

Bahr, DavidAssistant Professor, College of Engineering &Architecture, Washington State University, USAGlacial flow

Ball, RowenaDepartment of Theoretical PhysicsAustralian National University, AustraliaFairy rings of mushroomsKolmogorov cascadeSingularity theory

Barnes, HowardUnilever Research Professor of Industrial RheologyDepartment of Mathematics,University of Wales Aberystwyth, WalesRheology

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

Beck, ChristianReader in Applied MathematicsSchool of Mathematical SciencesQueen Mary & Westfield College, UKFree energyMultifractal analysisString theory

xv


Benedict, KeithSenior Lecturer, School of Physics and AstronomyUniversity of Nottingham, UKAnderson localizationFrustration

Berge, LucProfessor, Commissariat a l’Energie AtomiqueBruyeres FranceDevelopment of singularitiesFilamentationKerr effect

Bernevig, Bogdan A.Physics DepartmentMassachusetts Institute of Technology, USAHolons

Biktashev, VadimLecturer in Applied Maths, Mathematical SciencesUniversity of Liverpool, UKVortex dynamics in excitable media

Binczak, StephaneLaboratoire d’Electronique, Informatique et ImageUniversite de Bourgogne, FranceEphaptic couplingMyelinated nerves

Biondini, GinoAssistant Professor, Department of MathematicsOhio State University, USAEinstein equationsHarmonic generation

Blair, DavidProfessor, School of PhysicsThe University of Western Australia, AustraliaGravitational waves

Boardman, Alan D.Professor of Applied PhysicsInstitute for Materials ResearchUniversity of Salford, UKPolaritons

Bollt, ErikProfessor, Department of MathUnited States Naval Academy, USAMarkov partitionsOrder from chaos

Boon, J.-P.Professor, Faculte des SciencesUniversite Libre de Bruxelles, BelgiumLattice gas methods

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

Boumenir, AminDepartment of MathematicsState University of West Georgia, USAGel’fand--Levitan theory

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

Boyd, Robert W.Professor, The Institute of OpticsUniversity of Rochester, USAFrequency doubling

Bradley, ElizabethAssociate Professor, Department of Computer ScienceUniversity of Colorado, USAKirchhoff’s laws

De Bruyn, JohnProfessor, Department of Physics andPhysical Oceanography, Memorial University ofNewfoundland, CanadaPhase transitionsThermal convection

Bullough, RobinProfessor, Applied MathematicsUniversity of Manchester Institute of Science andTechnology, UKMaxwell--Bloch equationsSine-Gordon (SG) equation

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

xvi


Busse, Friedrich (Adviser)Professor, Theoretical PhysicsUniversitat Bayreuth, GermanyDynamos, homogeneousFluid dynamicsMagnetohydrodynamics

Calini, Annalisa M.Associate Professor, Department of MathematicsCollege of Charleston, USAElliptic functionsMelnikov methodJump phenomena

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

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

Chen, Wei-YinProfessor, Department of Chemical EngineeringUniversity of Mississippi, USAStochastic processes

Chernitskii, Alexander A.Department of Physical ElectronicsSt. Petersburg Electrotechnical University, RussiaBorn--Infeld equations

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

Choudhury, S. RoyProfessor, Department of MathematicsUniversity of Central Florida, USAKelvin--Helmholtz instabilityLorenz equations

Christiansen, PeterProfessor, Informatics and Mathematical ModellingTechnical University of Denmark, DenmarkSeparation of variables

Christodoulides, DemetriosProfessor, CREOL/School of OpticsUniversity of Central Florida, USAIncoherent Solitons

Coskun, TamerResearch Associate, Medical SciencesIndiana University-Purdue University IndianapolisUSAIncoherent solitons

Cruzeiro-Hansson, LeonorHonorary Fellow, Department of MathematicsHeriot-Watt University, UKDavydov soliton

Cushing, JimProfessor, Department of MathematicsUniversity of Arizona, USAPopulation dynamics

Davies, BrianDepartment of MathematicsAustralian National University, AustraliaIntegral transformsPeriod doubling

Davis, William C.Formerly, Los Alamos National LaboratoryUSAExplosions

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

Degallaix, JeromeSchool of PhysicsThe University of Western AustraliaAustraliaGravitational Waves

Deift, PercyProfessor, Department of MathematicsCourant Institute of Mathematical SciencesNew York University, USARandom matrix theory IV: Analytic methodsRiemann--Hilbert problem

Deryabin, Mikhail V.Department of MathematicsTechnical University of Denmark, DenmarkKolmogorov--Arnold--Moser theorem

xvii


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

Ding, MingzhouProfessor, Center for Complex Systems andBrain Sciences Florida Atlantic University, USAIntermittency

Degasperis, Antonio (Adviser)Professor, Dipartimento di Fisica, Universita degli Studidi Roma ”La Sapienza,” Italy

Ditto, William D. (Adviser)Applied Chaos Lab, Georgia Institute of Technology, USA

Dolgaleva, KsenaiDepartment of PhysicsM.V. Lomonosov Moscow State UniversityMoscow, Russian FederationFrequency doubling

Donoso, Jose M.Facultad de MatematicasUniversidad ComplutenseMadrid, SpainBall lightning

Doucet, ArnaudSignal Processing Group, Department of EngineeringCambidge University, UKMonte Carlo methods

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

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

Easton, RobertProfessor, Department of Applied MathUniversity of Colorado at Boulder, USAConley index

Eckhardt, BrunoProfessor, Fachbereich PhysikPhilipps Universitat Marburg, Germany

Lagrangian chaosMaps in the complex planePeriodic orbit theoryQuantum chaosRandom matrix theory I: origins andphysical applications

Shear flowSolar systemUniversality

Eilbeck, Chris (Adviser)Professor, Department of MathematicsHeriot-Watt University, UKDiscrete self-trapping system

Elgin, JohnProfessor, Maths DepartmentImperial College of ScienceTechnology and Medicine, London, UKKuramoto--Sivashinsky equation

Emmeche, ClausAssociate Professor andHead of Center for the Philosophyof Nature and Science StudiesUniversity of Copenhagen, DenmarkCausality

Enolskii, VictorProfessor, Heriot-Watt University, UKTheta functions

Falqui, GregorioLecturer, Mathematical Physics SectorInternational School for Advanced StudiesTrieste, ItalyHodagraph Transform

Falkovich, GregoryProfessor, Department of Physics of Complex SystemsWeizmann Institute of Science, IsraelMixingTurbulence

Faris, WilliamProfessor, Department of MathematicsUniversity of Arizona, USAMartingales

Feddersen, HenrikResearch Scientist, Climate Research DivisionDanish Meteorological Institute, DenmarkForecasting

xviii


Fedorenko, VladimirSenior Scientific Researcher, Institute of MathematicsNational Academic of Science of Ukraine, UkraineOne-dimensional maps

Fenimore, Paul W.Theoretical Biology and Biophysics GroupLos Alamos National Laboratory, USAProtein dynamics

Flach, Sergej (Adviser)Max-Planck-Institut fur Physik komplexer SystemeGermany.Discrete breathersSymmetry: equations vs. solutions

Flaschka, Hermann (Adviser)Professor, Department of MathematicsUniversity of Arizona, USAToda lattice

Fletcher, NevilleProfessor, Department of ElectronicMaterials EngineeringAustralian National University, AustraliaOvertones

Florιa, Luis MarioDepartment of Theory and Simulation ofComplex SystemsInstituto de Ciencia de Materiales de Aragon, SpainAubry--Mather theoryCommensurate--incommensurate transitionFrenkel--Kontorova model

Forrester, PeterDepartment of Mathematics and StatisticsUniversity of Melbourne, AustraliaRandom matrix theory II: Algebraic developments

Fowler, Beall W.Emeritus Professor, Physics DepartmentLehigh University, USAColor centers

Fraedrich, KlausProfessor, Universitat Hamburg Meteorological InstituteGermanyGeneral circulation models of the atmosphere

Freites, Juan AlfredoDepartment of Physics and AstronomyUniversity of California, Irvine, USAMolecular dynamics

Frieden, RoyOptical Sciences CenterUniversity of Arizona in Tucson, USAInformation theory

Friedrich, J.Professor, Lehrstuhl fur Physik WeihenstephanTechnische Universitat Munchen, GermanyHole burning

Fuchikami, NobukoDepartment of PhysicsTokyo Metropolitan University, JapanDripping faucet

Gallagher, MarcusSchool of Information Technology &Electrical EngineeringThe University of Queensland, AustraliaMcCulloch--Pitts networkPerceptron

Garnier, NicolasLaboratoire de PhysiqueEcole Normale Superieure de Lyon, FranceHydrothermal waves

Gaspard, PierreDepartement de MathematiqueUniversite Libre de Bruxelles, BelgiumEntropyMapsQuantum theoryRossler systems

Glass, LeonProfessor, Department of PhysiologyMcGill University, CanadaCardiac arrhythmias and electrocardiogram

Glendinning, PaulProfessor, Department of MathematicsUniversity of Manchester Institute ofScience and Technology, UKHenon mapInvariant manifolds and setsRoutes to chaos

Goriely, AlainProfessor, Department of MathematicsUniversity of Arizona, USANormal forms theory

xix


Grand, SteveDirector, Cyberlife Research Ltd., UKArtificial life

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

Grava, TamaraMaths DepartmentImperial College of ScienceTechnology and Medicine, UKHodograph transformN -soliton formulasZero-dispersion limits

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

Giuliani, AlessandroIstituto Superiore di Sanita, Rome, ItalyAlgorithmic complexity

Haken, Herman (Adviser)Professor Emeritus, Fakultet fur PhysikUniversity of Stuttgart, GermanyGestalt phenomenaSynergetics

Hallinan, JenniferInstitute for Molecular BioscienceThe University of Queensland, AustraliaGame of lifeGame theory

Hamilton, MarkProfessor, Department of Mechanical EngineeringUniversity of Texas at Austin, USANonlinear acoustics

Hamm, PeterProfessor, Physikalisch-Chemisches InstitutUniversitat Zurich, SwitzerlandFranck--Condon factorHydrogen bondPump-probe measurements

Hasselblatt, BorisProfessor, Department of MathematicsTufts University, USAAnosov and axiom A systemsMeasures Phase space

Hastings, AlanProfessor, Department of Environmental Scienceand PolicyUniversity of California, USAEpidemiology

Hawkins, JaneProfessor, Department of MathematicsUniversity of North Carolina at Chapel Hill, USAErgodic theory

Helbing, DirkInstitute for Economics and TrafficDresden University of Technology, GermanyTraffic flow

Henry, BruceDepartment of Applied MathematicsUniversity of New South Wales, AustraliaEquipartion of energyHenon--Heiles system

Henry, BryanDepartment of Chemistry and BiochemistryUniversity of Guelph, CanadaLocal modes in molecules

Hensler, GerhardProfessor, Institut fur AstronomieUniversitats-Sternwarte Wien, AustriaGalaxies

Herrmann, HansInstitute for Computational PhysicsUniversity of Stuttgart, GermanyDune formation

Hertz, JohnProfessor, Nordic Institute for Theoretical PhysicsDenmarkAttractor neural network (ANN)

Hietarinta, JarmoProfessor, Department of PhysicsUniversity of Turku, FinlandHirota’s method

xx


Hill, LarryDetonation Science & TechnologyLos Alamos National Laboratory, USAEvaporation wave

Hjorth, Poul G.Lektor, Department of MathematicsTechnical University of Denmark, DenmarkKolmogorov--Arnold-Moser theorem

Holden, ArunProfessor of Computational BiologySchool of Biomedical SciencesUniversity of Leeds, UKExcitabilityHodgkin--Huxley equationsIntegrate and fire neuronMarkin--Chizmadzhev modelPeriodic burstingSpiral wavesThreshold phenomena

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

Hone, AndrewLecturer in Applied MathematicsInstitute of Mathematics and StatisticsUniversity of Kent at Canterbury, UKExtremum principlesOrdinary differential equations, nonlinearRiccati equations

Hood, AlanProfessor, School of Mathematical andComputational SciencesUniversity of St Andrews, UKCharacteristics.

Houghton, ConorDublin School of MathematicsTrinity College, IrelandInstantonsYang--Mills theory

Howard, James E.Research Associate, Department of PhysicsUniversity of Colorado at Boulder, USANontwist mapsRegular and chaotic dynamics in atomic physics

Ivey, ThomasDepartment of MathematicsCollege of Charleston, USADifferential geometryFramed space curves

Jimenez, SalvadorProfessor, Departamento de MatematicasUniversidad Alfonso X El SabioMadrid, SpainCharge density waves

Joannopoulos, John D.Professor, Department of PhysicsMassachusetts Institute of Technology, USAPhotonic crystals

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

Johnson, Steven G.Postdoctoral Associate, Department of PhysicsMassachussetts Institute of Technology, USAPhotonic crystals

Joshi, NaliniProfessor, School of Mathematics and StatisticsUniversity of Sydney, AustraliaSolitons

Kaneko, KunihikoDepartment of Pure and Applied SciencesTokyo University, JapanCoupled map lattice

Kantz, HolgerProfessor, Max-Planck-Institut fu komplexer SystemeGermanyTime series analysis

Keener, James P. (Adviser)Professor, Department of MathematicsUniversity of Utah, USA

Kennedy, Michael PeterProfessor of Microelectronic EngineeringUniversity College, Cork, IrelandChua’s circuit

xxi


Kevrekidis, Panayotis G.Assistant Professor, Department of Mathematicsand StatisticsUniversity of Massachusetts, Amherst, USABinding energyCollisionsWave of translation

Khanin, KonstantinProfessor, Department of MathematicsHeriot-Watt University, UKDenjoy theory

King, AaronAssistant Professor, Department of Ecologyand Evolutionary BiologyUniversity of Tennessee, Knoxville, USAPhase plane

Kirby, MichaelProfessor, Department of MathematicsColorado State University, USANonlinear signal processing

Kivshar, Yuri (Adviser)Nonlinear Physics GroupAustralian National University, AustraliaOptical fiber communications

Kiyono, KenResearch Fellow of the Japan Society for thePromotion of ScienceEducational Physiology LaboratoryUniversity of Tokyo, JapanDripping Faucet

Knott, RonDepartment of MathematicsUniversity of Surrey, UKFibonacci series

Kocarev, LjupcoAssociate Research ScientistInstitute for Nonlinear ScienceUniversity of CaliforniaSan Diego, USA

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

Konotop, Vladimir V.Centro de Fιsica Teorica e ComputacionalComplexo Interdisciplinar da Universidade de LisboaPortugalWave propagation in disordered media

Kosevich, ArnoldPhysics-Engineering Institute of Low TemperaturesKharkov State Polytechnical University, UkraineBreathersDislocations in crystalsEffective massLandau-Lifshitz (LL) equationSuperfluiditySuperlattices

Kovalev, AlexanderInstitute for Low Temperature Physics and EngineeringNational Academy of Sciences of Ukraine, UkraineContinuum approximationsTopological defects

Kramer, PeterAssistant 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 MicrostructuresRussian Academy of Science, RussiaCherenkov radiation

Kuvshinov, Viatcheslav I.Professor, Institute of PhysicsBelarus Academy of Sciences, BelarusBlack holesCosmological modelsFractalsGeneral relativity

Kuznetsov, VadimAdvanced Research FellowDepartment of Applied MathematicsUniversity of Leeds, UKRotating rigid bodies

xxii


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

Lakshmanan, MuthusamyProfessor, Department of PhysicsBharathidasan University, Tiruchirapalli, IndiaEquations, nonlinearNonlinear electronics

Landa, P.S.Professor, Department of PhysicsMoscow State University, RussiaFeedbackPendulumQuasilinear analysisRelaxation oscillators

Landsberg, PeterProfessor, Faculty of Mathematical StudiesUniversity of Southampton, UKDetailed balance

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

Lee, JohnDepartment of Mechanical EngineeringMcGill University, Montreal, Quebec, CanadaFlame front

Lega, JocelineProfessor, Department of MathematicsUniversity of Arizona, USAEquilibriumFredholm theorem

Lepeshkin, NickThe Institute of OpticsUniversity of Rochester, USAFrequency doubling

Levi, DecioProfessor, Dipartimento di FisicaUniversita degli Studi di Roma III, ItalyDelay-differential equations

Lewis, KarenHydrologist, SS Papadopulos & Associates, BoulderColorado, USAGlacial flow

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

Liley, DavidSchool of Biophysical Sciences andElectrical EngineeringSwinburne University of Technology, AustraliaElectroencephalogram at mesoscopic scales

Lonngren, Karl E.Professor, Department of Electrical andComputer EngineeringUniversity of Iowa, USAPlasma soliton experiments

Losert, WolfgangAssistant Professor, Institute for Plasma ResearchUniversity of Maryland, USAGranular materialsPattern formation

Luecke, ManfredInstitut fur Theoretische PhysikUniversitat des Saarlandes, Saarbrucken, GermanyThermo-diffusion effects

Ma, Wen-XiuProfessor, Department of MathematicsUniversity of South Florida, USAIntegrability

Macaskill, CharlesSchool of Mathematics and StatisticsUniversity of Sydney, AustraliaJupiter’s great red spot

Maggio, Gian MarioCenter for Wireless Communications (CWC)University of California, San Diego, USADamped driven anharmonic oscillator

xxiii


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

Mainzer, KlausDepartment of Philosophy of ScienceUniversity of Augsburg, GermanyArtificial intelligenceCellular nonlinear networksDynamical systems

Malomed, Boris A.Professor, Department of Interdisciplinary StudiesFaculty of Engineering, Tel Aviv University, IsraelComplex Ginzburg--Landau (CGL) equationConstants of motion and conservation lawsMultisoliton perturbation theoryNonlinear Schrodinger equationsPower balance

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

Manneville, PaulLaboratoire d’Hydrodynamique (LadHyX)Ecole Polytechnique in Palaiseau, FranceSpatiotemporal chaos

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

Mart�nez, Pedro JesusDepartment of Theory and Simulation ofComplex SystemsInstituto de Ciencia de Materiales de Aragon, SpainFrenkel--Kontorova model

Masmoudi, NaderAssociate Professor, Department of MathematicsCourant Institute of Mathematical SciencesNew York University, USABoundary layers

Mason, LionelMathematical Institute, Oxford University, UKTwistor theory

Mayer, AndreasInstitute for Theoretical PhysicsUniversity of Regensburg, GermanySurface waves

McKenna, JoeProfessor, Department of MathematicsUniversity of Connecticut, USATacoma narrows bridge collapse

McLaughlin, Kenneth (Adviser)Associate ProfessorDepartment of MathematicsUniversity of North Carolina at Chapel Hill, USARandom matrix theory III: combinatorics

McLaughlin, RichardAssociate Professor, Department of MathematicsUniversity of NorthCarolina, Chapel Hill, USAPlume dynamics

McMahon, BenTheoretical Biology and Biophysics GroupLos Alamos National Laboratory, USAProtein dynamicsProtein Structure

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

Miura, RobertProfessor, Department of Mathematical SciencesNew Jersey Institute of TechnologyNewark, USANonlinear toys

Moloney, Jerome V.Professor, Department of MathematicsUniversity of Arizona, USANonlinear optics

MLrk, JesterOptoelectronics, Research Center COMTechnical University of DenmarkDenmarkSemiconductor laser

xxiv


Mornev, OlegProfessor, Institute of Theoretical andExperimental Biophysics, RussiaGeometrical optics, nonlinearGradient systemNerve impulsesZeldovich--Frank-Kamenetsky equation

Mosekilde, ErikProfessor, Department of PhysicsTechnical University of Denmark, DenmarkNephron dynamics

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

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

Mygind, JesperProfessor, Department of PhysicsTechnical University of Denmark, DenmarkJosephson junctionsSuperconducting quantum interference device(SQUID)

Nakamura, YoshiharuAssociate Professor, Institute of Space andAstronautical Science, Faculty of Science and TechnologyKeio University, Yokohama, JapanPlasma soliton experiments

Natiello, MarioCentre for Mathematical SciencesLund University, SwedenWinding numbers

Newell, AlanProfessor, Department of MathematicsUniversity of Arizona, USAInverse scattering method or transform

Newton, Paul K.Professor, Department of Aerospace and MechanicalEngineeringUniversity of Southern California, USA

Berry’s phaseChaos vs. turbulence

Neyts, KristiaanProfessor, Department of Electronics andInformation SystemsGhent University, BelgiumLiquid crystals

Nicolis, G.Professor, Faculte des SciencesUniversite Libre de Bruxelles, BelgiumBrusselatorChemical kineticsNonequilibrium statistical mechanicsRecurrence

Nunez, PaulProfessor, Brain Physics GroupDepartment of Biomedical EngineeringTulane University, USAElectroencephalogram at large scales

Olsder, Geert-JanFaculty of Technical Mathematics and InformaticsDelft University of Technology, DenmarkIdempotent analysis

Olver, Peter J.Professor, School of MathematicsUniversity of Minnesota, USALie algebras and Lie groups

Ostrovsky, Lev (Adviser)Environmental Technology LaboratoryZel Technologies/National Oceanic & AtmosphericAdministration, Boulder, Colorado, USA andInstitute of Applied Physics, RussiaHurricanes and tornadoesModulated wavesNonlinear acousticsShock waves

Ott, Edward (Adviser)Distinguished University ProfessorInstitute for Research in Electronics and Applied PhysicsUniversity of Maryland, USA

Palmer, JohnProfessor, Department of MathematicsUniversity of Arizona, USAMonodromy preserving deformations

xxv


Pascual, Pedro J.Associate ProfessorDepartamento de Ingenieria InformaticaUniversidad Autonoma de Madrid, SpainCharge density waves

Pedersen, Niels FalsigDepartment of Power EngineeringTechnical University of Denmark, DenmarkLong Josephson junctionsSuperconductivity

Pelinovsky, DmitryAssistant Professor, Department of MathsMcMaster University, CanadaCoupled systems of partial differential equationsEnergy analysisGeneralized functionsLinearizationManley--Rowe relationsN -wave interactionsNumerical methodsSpectral analysis

Pelletier, JonAssistant Professor, Department of GeosciencesUniversity of Arizona, USAGeomorphology and tectonics

Pelloni, BeatriceMathematics DepartmentUniversity of Reading, UKBoundary value problemsBurgers’ equation

Petty, MichaelProfessor, Centre for Molecular andNanoscale ElectronicsUniversity of Durham, UKLangmuir--Blodgett films

Peyrard, MichelProfessor of Physics, Laboratoire de PhysiqueEcole Normale 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 and Coordinator of Undergraduate ProgramDepartment of Chemistry and BiochemistryThe University of Southern MississippiHattiesburg, USAPolymerization

Rabinovich, MikhailResident Physicist, Institute for Nonlinear ScienceUniversity of California at San Diego, USAChaotic dynamics

Ranada, Antonio F.Facultad de Fisica, Universidad ComplutenseMadrid, SpainBall lightning

Recami, ErasmoProfessore AssociateFacolta Universita Statale di Bergamo, ItalyTachyons and superluminal motion

Reucroft, StephenProfessor of Physics, Northeastern UniversityBoston, Massachusetts, USAHiggs boson

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

Robinson, JamesMathematics InstituteUniversity of Warwick, UKAttractorsDimensionsFunction SpacesFunctional analysis

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

Rogers, ColinProfessor, School of MathematicsUniversity of New South Wales, AustraliaBacklund transformations

xxvi


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

Rouvas-Nicolis, C.Climatologie DynamiqueInstitut Royal Meteorologique de BelgiqueBelgiumRecurrence

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

Sabatier, PierreProfessor, Physique MathematiqueUniversite Montpellier II, FranceInverse problems

Sakaguchi, HidetsuguDepartment of Applied Science for Electronicsand MaterialsKyushu University, JapanCoupled oscillators

Salerno, MarioDepartment of Physics ”E. R. Caianiello”Universita di Salerno, ItalyBethe ansatzSalerno equation

Sandstede, BjornAssociate Professor, Department of MathematicsOhio State University, USAEvans function

Satnoianu, RazvanDepartment of Mathematics, City University, UKFiffusionReaction diffusion systems

Sauer, TimProfessor, Department of MathematicsGeorge Mason University, USAEmbedding methods

Savin, AlexanderProfessor, Moscow Institute of Physics and TechnologyRussiaPeierls barrier

Schattschneider, DorisProfessor, Department of MathematicsMoravian College, BethlehemPennsylvania, USATessellation

Schmelcher, PeterInstitute for Physical ChemistryUniversity of Heidelberg, GermanyHartree approximation

Scholl, EckehardProfessor, Institut fur Theoretische PhysikTechnische Universitat Berlin, GermanyAvalanche breakdownDiodesDrude modelSemiconductor oscillators

Schuster, PeterInstitut fur Theoretische Chemie undMolekulare Strukturbiologie, AustriaBiological evolutionCatalytic hypercycleFitness landscape

Scott, Alwyn (Editor)Emeritus Professor of MathematicsUniversity of Arizona, USACandleDiscrete self-trapping systemDistributed oscillatorsEmergenceEuler--Lagrange equationsHierarchies of nonlinear systemsLaboratory models of nonlinear wavesLifetimeMatter, nonlinear theory ofMultiplex neuronNeuristorQuantum nonlinearityRotating wave approximation (RWA)Solitons, a brief historyState diagramsSymmetry groupsTachyons and superluminal motionWave packets, linear and nonlinear

Segev, MordechaiProfessor, Technion-Israel Institute of TechnologyHaifa, IsraelIncoherent solitons

xxvii


Shalfeev, VladimirHead of Department of Oscillation TheoryNizhni Novgorod State University, RussiaParametric amplification

Sharkovsky, AlexanderInstitute of MathematicsNational Academy of Sciences of Ukraine, UkraineOne-dimensional mapsTurbulence, ideal

Sharman, RobertUniversity Corporation for Atmospheric ResearchBoulder, Colorado, USAClear air turbulence

Shinbrot, TroyAssociate ProfessorDepartment of Chemical and Biochemical EngineeringRutgers University, Piscataway, USAControlling chaos

Shohet, LeonProfessor, Department of Electrical andComputer EngineeringUniversity of Wisconsin Madison, USANonlinear plasma waves

Siwak, PawelDepartment of Electrical EngineeringPoznan University of Technology, PolandIntegrable cellular automata

Smil, VaclavProfessor, Department of GeographyUniversity of Manitoba, CanadaGlobal warming

Sobell, HenryIndependent scholar, New York, USADNA premelting

Solari, Hernan GustavoDepartamento FιsicaUniversity of Buenos Aires, ArgentinaLasers

Soljacic, MarinPrincipal Research Scientist, Department of PhysicsMassachusetts Institute of Technology, USAPhotonic crystals

SLrensen, Mads PeterAssociate ProfessorInformatics and Mathematical ModelingTechnical University of Denmark, DenmarkCollective coordinatesMultiple scale analysisPerturbation theory

Sornette, DidierProfessor, Laboratoire de Physique de laMatiere CondenseeUniversite de Nice - Sophia Antipolis, FranceSandpile model

Spatschek, KarlProfessor, Institute for Theoretical Physics 1Heinrich-Heine-Universitat Dusseldorf, GermanyCenter manifold reductionDispersion management

Stauffer, DietrichInstitute for Theoretical PhysicsUniversity of Cologne, GermanyPercolation theory

Stefanovska, AnetaFaculty of Electrical EngineeringUniversity of Ljubljana, SloveniaFlip-flop circuitInhibitionNonlinearity, definition ofQuasiperiodicityWavelets

Storb, UlrichDrittmittelbeschaftigteInstitut fur Experimentelle PhysikOtto-von-Guericke-UniversitatMagdeburg, GermanyScroll waves

Strelcyn, Jean-MarieProfesseur, Departement de MathematiquesUniversite de RouenMont Saint Aignan Cedex, FrancePoincare theorems

Suris, Yuri B.Researcher, Department of MathematicsTechnische Universitat Berlin, GermanyIntegrable lattices

xxviii


Sutcliffe, PaulReader in Mathematical PhysicsInstitute of Mathematics and StatisticsUniversity of Kent at Canterbury, UKSkyrmions

Swain, JohnProfessor, Department of PhysicsNortheastern University, Boston, USADoppler shiftQuantum field theoryTensors

Tabor, MichaelProfessor, Department of MathematicsUniversity of Arizona, USAGrowth patterns

Tajiri, MasayoshiProfessor, Department of Mathematical SciencesOsaka Prefecture University, JapanSolitons, types ofWave stability and instability

Tass, PeterInstitut fur MedizinForschungszentrum Julich, GermanyStochastic analyses of neural systems

Taylor, RichardAssociate Professor, Materials Science InstituteUniversity of Oregon, USALevy flights

Teman, R.Laboratoire d’Analyse NumeriqueUniversite de Paris Sud, FranceInertial manifolds

Thompson, J.M.T.Professor, Centre for Nonlinear Dynamicsand its ApplicationsUniversity College London, UKDuffing equationStability

Tien, Ti.Professor, Membrane Biophysics LabMichigan State University, USABilayer lipid membranes

Tobias, DouglasAssociate Professor, Department of ChemistryUniversity of California at Irvine, USAMolecular dynamics

Toda, MorikazuEmeritus Professor, Tokyo University of EducationJapanNonlinear toys

Trueba, Jose L.ESCET, Universidad Rey Juan Carlos, Madrid, SpainBall lightning

Tsimring, Lev S.Research PhysicistSan Diego Institute for Nonlinear ScienceUniversity of California, USAAvalanches

Tsinober, ArkadyProfessor, Iby and Aladar Fleischman Facultyof Engineering, Tel Aviv University, IsraelHelicity

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

Tsygvintsev, AlexeiUnite de Mathematiques Pures et Appliquees Ecolenormale superieure de Lyon, FrancePoincare theorems

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

Ustinov, AlexeyPhysikalisches Institut IIIUniversity of Erlangen-Nurnberg, GermanyJosephson junction arrays

Van der Heijden, GertCentre for Nonlinear DynamicsUniversity College London, UKButterfly effectHopf bifurcation

xxix


Vazquez, LuisProfessor, Universidad Complutense de Madrid, Spain.Senior Researcher and Cofounder of the Centro deAstrobiologιaCharge density waveDispersion relationsFitzHugh--Nagumo equationVirial theoremWave propagation in disordered media

Veselov, AlexanderProfessor, Department of Mathematical SciencesLoughborough University, UKHuygens’ principle

Voiculescu, Dan-VirgilProfessor, Department of MathematicsUniversity of California, Berkeley, USAFree probability theory

Voorhees, BurtonProfessor, Department of MathematicsAthabasca University, CanadaCellular automata

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

Walter, GilbertProfessor EmeritusDepartment of Mathematical SciencesUniversity of Wisconsin-Milwaukee, USACompartmental models

Waymire, Edward C.Professor, Department of MathematicsOregon State University, USAMultiplicative processes

West, BruceDepartment of Physics, US Army Research Office, USABranching lawsFluctuation--dissipation theoremKicked rotor

Wilhelmsson, HansProfessor Emeritus of PhysicsChalmers University of TechnologyGoteborg, SwedenAlfven waves

Wilson, HughCentre for Vision ResearchYork University, CanadaNeuronsStereoscopic Vision and Binocular Rivalry

Winfree, Art (Adviser)Formerly, Department of Ecology andEvolutionary BiologyUniversity of Arizona, USADimensional analysis

Wojtkowski, MaciejProfessor, Department of MathematicsUniversity of Arizona, USALyapunov exponents

Yakushevich, Ludmilla (Adviser)Researcher, Institute of Cell BiophysicsRussian Academy of Sciences, RussiaDNA solitons

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

Yiguang Ju.Department of Mechanical Aerospace EngineeringPrinceton University, USAFlame front

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

Zabusky, NormanProfessor, Department of Mechanical andAerospace Engineering, Rutgers UniversityNew Jersey, USAVisiometricsVortex dynamics of fluids

Zbilut, Joseph P.Associate Professor, Department of Molecular Biophysicsand Physiology, Rush UniversityChicago, USAAlgorithmic complexity

xxx


Zolotaryuk, AlexBogolyubov Institute for Theoretical Physics, UkrainePolaronsRatchets

Zhou, XinProfessor, Department of MathematicsDuke University, USA

Random matrix theory IV: analytic methodsRiemann--Hilbert Problem

Zorzano, Mar�a-PazYoung Researcher, Centro de AstrobiologιaMadrid, SpainFitzHugh--Nagumo equation

xxxi

List of Entries

Ablowitz--Kaup--Newell--Segur (AKNS)system 000

Adiabatic invariants 000Alfven waves 000Algorithmic complexity 000Anderson localization 000Anosov and Axiom A systems 000Arnold diffusion 000Artificial intelligence 000Artificial life 000Atmospheric and ocean sciences 000Attractor neural network (ANN) 000Attractors 000Aubry--Mather theory 000Avalanche breakdown 000Avalanches 000Averaging methods 000Backlund transformations 000Ball lightning 000Belousov--Zhabotinsky reaction 000Bernoulli’s equation 000Berry’s phase 000Bethe ansatz 000Bifurcations 000Bilayer lipid membranes 000Billiards 000Binding energy 000Biological evolution 000Biomolecular solitons 000Bjerrum defects 000Black holes 000Born--Infeld equations 000Bose--Einstein condensation 000Boundary layers 000Boundary value problems 000Branching laws 000

Breathers 000Brownian motion 000Brusselator 000Burgers’ equation 000Butterfly effect 000Candle 000Cardiac arrhythmias and electrocardiogram 000Cardiac muscle models 000Cat map 000Catalytic hypercycle 000Catastrophe theory 000Causality 000Celestial mechanics 000Cell assemblies 000Cellular automata 000Cellular nonlinear networks 000Center manifold reduction 000Chaos vs. turbulence 000Chaotic advection 000Chaotic dynamics 000Characteristics 000Charge density wave 000Chemical kinetics 000Cherenkov radiation 000Chua’s circuit 000Clear air turbulence 000Cluster-cluster coagulation 000Coherence phenomena 000Collective coordinates 000Collisions 000Color centers 000Commensurate-incommensurate transition 000Compartmental models 000Complex Ginzburg-Landau (CGL) equation 000Conley index 000Constants of motion and conservation laws 000

xxxiii

List of Entries

Continuum approximations 000Contour dynamics 000Controlling chaos Cosmological models 000Coupled map lattice 000Coupled oscillators 000Coupled systems of partial differentialequations 000

Critical phenomena 000Damped driven anharmonic oscillator 000Darboux transformation 000Davydov soliton 000Delay-differential equations 000Denjoy theory 000Derrick--Hobart theorem 000Detailed balance 000Determinism 000Deterministic walks in random environments 000Development of singularities 000Differential geometry 000Diffusion 000Dimensional analysis 000Dimensions 000Diodes 000Discrete breathers 000Discrete nonlinear Schrodinger equations 000Discrete self-trapping system 000Dislocations in crystals 000Dispersion management 000Dispersion relations 000Distributed oscillators 000DNA premelting 000DNA solitons 000Domain walls 000Doppler shift 000Dressing method 000Dripping faucet 000Drude model 000Duffing equation 000Dune formation 000Dynamical systems 000Dynamos, homogeneous 000Economic system dynamics 000Effective mass 000Einstein equations 000Electroencephalogram at large scales 000Electroencephalogram at mesoscopic scales 000Electron beam microwave devices 000Elliptic functions 000Embedding methods 000Emergence 000Energy analysis 000Entropy 000Ephaptic coupling 000Epidemiology 000Equations, nonlinear 000

Equilibrium 000Equipartion of energy 000Ergodic theory 000Euler--Lagrange equations 000Evans function 000Evaporation wave 000Excitability 000Excitons 000Explosions 000Extremum principles 000Fairy rings of mushrooms 000Feedback 000Fermi acceleration and Fermi map 000Fermi--Pasta--Ulam (FPU) oscillator chain 000Ferromagnetism and ferroelectricity 000Fibonacci series 000Filamentation 000Fitness landscape 000FitzHugh--Nagumo equation 000Flame front 000Flip-flop circuit 000Fluctuation-dissipation theorem 000Fluid dynamics 000Fokker--Planck equation 000Forecasting 000Forest fires 000Fractals 000Framed space curves 000Franck--Condon factor 000Fredholm theorem 000Free energy 000Free probability theory 000Frenkel--Kontorova model 000Frequency doubling 000Frohlich theory 000Frustration 000Function spaces 000Functional analysis 000Galaxies 000Game of life 000Game theory 000Gel’fand--Levitan theory 000General circulation models of the atmosphere 000General relativity 000Generalized functions 000Geometrical optics, nonlinear 000Geomorphology and tectonics 000Gestalt phenomena 000Glacial flow 000Global warming 000Gradient system 000Granular materials 000Gravitational waves 000Group velocity Growth patterns 000Henon map 000

xxxiv

List of Entries

Henon--Heiles system 000Hamiltonian systems 000Harmonic generation 000Hartree approximation 000Heat conduction 000Hele--Shaw cell 000Helicity 000Hierarchies of nonlinear systems 000Higgs boson 000Hirota’s method 000Hodgkin--Huxley equations 000Hodograph transform 000Hole burning 000Holons 000Hopf bifurcation 000Horseshoes and hyperbolicity in dynamicalsystems 000

Hurricanes and tornadoes 000Huygens’ principle 000Hydrogen bond 000Hydrothermal waves 000Hysteresis 000Idempotent analysis 000Incoherent solitons 000Inertial manifolds 000Information theory 000Inhibition 000Instantons 000Integrability 000Integrable cellular automata 000Integrable lattices 000Integral transforms 000Integrate and fire neuron 000Intermittency 000Invariant manifolds and sets 000Inverse problems 000Inverse scattering method or transform 000Ising model 000Josephson junction arrays 000Josephson junctions 000Jump phenomena 000Jupiter’s Great Red Spot 000Kadomtsev--Petviashvili equation 000Kelvin--Helmholtz instability 000Kerr effect 000Kicked rotor 000Kirchhoff’s laws 000Knot theory 000Kolmogorov cascade 000Kolmogorov--Arnold--Moser theorem 000Korteweg--de Vries equation 000Kuramoto--Sivashinsky equation 000Laboratory models of nonlinear waves 000Lagrangian chaos 000Landau--Lifshitz (LL) equation 000

Langmuir--Blodgett films 000Lasers 000Lattice gas methods 000Levy flights 000Lie algebras and Lie groups 000Lifetime 000Linearization 000Liquid crystals 000Local modes in molecular crystals 000Local modes in molecules 000Long Josephson junctions 000Lorentz gas 000Lorenz equations 000Lyapunov exponents 000Magnetohydrodynamics 000Manley--Rowe relations 000Maps 000Maps in the complex plane 000Markin--Chizmadzhev model 000Markov partitions 000Martingales 000Matter, nonlinear theory of 000Maxwell--Bloch equations 000McCulloch--Pitts network 000Measures 000Mechanics of solids 000Melnikov method 000Mixing 000Modulated waves 000Molecular dynamics 000Monodromy preserving deformations 000Monte Carlo methods 000Morphogenesis, biological 000Multidimensional solitons 000Multifractal analysis 000Multiple scale analysis 000Multiplex neuron 000Multiplicative processes 000Multisoliton perturbation theory 000Myelinated nerves 000Navier--Stokes equation 000N -body problem 000Nephron dynamics 000Nerve impulses 000Neural network models 000Neuristor 000Neurons 000Newton’s laws of motion 000Nonequilibrium statistical mechanics 000Nonlinear acoustics 000Nonlinear electronics 000Nonlinear optics 000Nonlinear plasma waves 000Nonlinear Schrodinger equations 000Nonlinear signal processing 000

xxxv

List of Entries

Nonlinear toys 000Nonlinearity, definition of 000Nontwist maps 000Normal forms theory 000N -soliton formulas 000Numerical methods 000N -wave interactions 000One-dimensional maps 000Optical fiber communications 000Order from chaos 000Order parameters 000Ordinary differential equations,nonlinear 000

Overtones 000Painleve analysis 000Parametric amplification 000Partial differential equations, nonlinear 000Particle accelerators 000Particles and antiparticles 000Pattern formation 000Peierls barrier 000Pendulum 000Perceptron 000Percolation theory 000Period doubling 000Periodic bursting 000Periodic orbit theory 000Periodic spectral theory 000Perturbation theory 000Phase dynamics 000Phase plane 000Phase space 000Phase space diffusion and correlations 000Phase transitions 000Photonic crystals 000Plasma soliton experiments 000Plume dynamics 000Poincare theorems 000Poisson brackets 000Polaritons 000Polarons 000Polymerization 000Population dynamics 000Power balance 000Protein dynamics 000Protein structure 000Pump-probe measurements 000Quantum chaos 000Quantum field theory 000Quantum groups 000Quantum inverse scattering method 000Quantum nonlinearity 000Quantum theory 000Quasilinear analysis 000Quasiperiodicity 000

Random matrix theory I: origins and physicalapplications 000

Random matrix theory II: algebraicdevelopments 000

Random matrix theory III: combinatorics 000Random matrix theory IV: analytic methods 000Random walks 000Ratchets 000Rayleigh and Raman scattering and IR absorption 000Rayleigh--Taylor instability 000Reaction-diffusion systems 000Recurrence 000Regular and chaotic dynamics in atomic physics 000Relaxation oscillators 000Renormalization groups 000Rheology 000Riccati equations 000Riemann--Hilbert problem 000Rossler systems 000Rotating rigid bodies 000Rotating wave approximation 000Routes to chaos 000Salerno equation 000Sandpile model 000Scheibe aggregates 000Scroll waves 000Semiconductor laser 000Semiconductor oscillators 000Separation of variables 000Shear flow 000Shock waves 000Sinai--Ruelle--Bowen measures 000Sine-Gordon (SG) equation 000Singularity theory 000Skyrmions 000Solar system 000Solitons 000Solitons, a brief history 000Solitons, types of 000Spatiotemporal chaos 000Spectral analysis 000Spin systems 000Spiral waves 000Stability 000Standard map 000State diagrams 000Stereoscopic Vision and Binocular Rivalry 000Stochastic analyses of neural systems 000Stochastic processes 000String theory 000Structural complexity 000Superconducting quantum interference device(SQUID) 000

Superconductivity 000Superfluidity 000

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List of Entries

Superlattices 000Surface waves 000Symbolic dynamics 000Symmetry groups 000Symmetry: equations vs. solutions 000Symplectic maps 000Synchronization 000Synergetics 000Tachyons and superluminal motion 000Tacoma Narrows Bridge collapse 000Taylor--Couette flow 000Tensors 000Tessellation 000Thermal convection 000Thermo-diffusion effects 000Theta functions 000Threshold phenomena 000Time series analysis 000Toda lattice 000Topological defects 000Topology 000Traffic flow 000

Turbulence 000Turbulence, ideal 000Turing patterns 000Twistor theory 000Universality 000Van der Pol equation 000Virial theorem 000Visiometrics 000Volterra series and operators 000Vortex dynamics in excitable media 000Vortex dynamics of fluids 000Water waves 000Wave of translation 000Wave packets, linear and nonlinear 000Wave propagation in disordered media 000Wave stability and instability 000Wavelets 000Winding numbers 000Yang--Mills theory 000Zeldovich--Frank-Kamenetsky equation 000Zero-dispersion limits 000

xxxvii

Thematic list of entries

HISTORY OF NONLINEAR SCIENCE

Butterfly effect, Candle, Integrability, Celestial me-chanics, Davydov soliton, Determinism, Feedback,Fermi--Pasta--Ulam oscillator chain, Fibonacci se-ries, Hodgkin--Huxley equations, Lorenz equations,Manley--Rowe relations, Markin--Chizmadzhevmodel, Martingales, Matter, nonlinear theory of,Poincare theorems, Preface, Solar system, Soli-ton, a brief history, Tacoma Narrows Bridge col-lapse, Van der Pol equation, Zeldovich--Frank-Kamenetsky equation

COMMON EXAMPLES OFNONLINEAR PHENOMENA

Avalanches, Ball lightning, Brownian motion, But-terfly effect, Candle, Clear air turbulence, Drippingfaucet, Dune formation, Explosions, Fairy rings ofmushrooms, Filamentation, Flame front, Fluid dy-namics, Forest fires, Glacial flow, Global warming,Hurricanes and tornadoes, Jupiter’s Great Red Spot,Nonlinear toys, Order from chaos, Pendulum, Phasetransitions, Plume dynamics, Solar system, TacomaNarrows Bridge collapse, Traffic flow, Waterwaves

ANALYTICAL METHODS

Backlund transformations, Bethe ansatz, Center-manifold reduction, Characteristics, Collective coor-dinates, Continuum approximations, Dimensional

analysis, Dispersion relations, Dressing method,Elliptic functions, Energy analysis, Evans function,Fredholm theorem, Gel’fand--Levitan theory, Gen-eralized functions, Hamiltonian systems, Hirota’smethod, Hodograph transform, Idempotent anal-ysis, Inverse scattering method, Integrable trans-forms, Kirchhoff laws, Multiple scale analysis, Mul-tisoliton perturbation theory, Non-equilibrium sta-tistical mechanics, Normal forms theory, N -solitonformulas, Painleveanalysis, Periodic spectral the-ory, Perturbation theory, Phase dynamics, Poissonbrackets, Power balance, Quantum inverse scat-tering method, Quasilinear analysis, Riccati equa-tions, Rotating wave approximation, Separationof variables, Spectral analysis, Stability, State dia-grams, Synergetics, Tensors, Theta functions, Timeseries analysis, Volterra series, Zero-dispersionlimits

COMPUTATIONAL METHODS

Averaging methods, Cellular automata, Cellularnonlinear networks, Characteristics, Compartmen-tal models, Contour dynamics, Embedding meth-ods, Extremum principles, Fitness landscape, Fore-casting, Framed space curves, Hartree approxima-tion, Integrability, Inverse problems, Lattice gasmethods, Linearization, Maps, Martingales, Monte--Carlo methods, Numerical methods, Recurrence,Theta functions, Time series analysis, Visiometrics,Volterra series, Wavelets

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Thematic List of Entries

TOPOLOGICAL METHODS

Conley index, Cat map, Darboux transformation,Denjoy theory, Derrick--Hobart theorem, Differ-ential geometry, Extremum principles, Functionalanalysis, Horseshoes and hyperbolicity, Huygens’principle, Inertial manifolds, Invariant manifoldsand sets, Knot theory, Kolmogorov--Arnold--Mosertheory, Lie algebras and Lie groups, Maps, Mea-sures, Monodromy-preserving deformations, Mul-tifractal analysis, Nontwist maps, One-dimensionalmaps, Periodic orbit theory, Phase space, Renorma-lization groups, Riemann--Hilbert problem, Singu-larity theory, Symbolic dynamics, Symmetry groups,Topology, Virial theorem, Winding 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 theo-rem, Fokker--Planck equation, Free probability the-ory, Frustration, Hele--Shaw cell, Horseshoes andhyperbolicity, Lagrangian chaos, Levy flights, Lya-punov exponents, Martingales, Melnikov method,Order from chaos, Percolation theory, Phase spaceanalysis, Quantum chaos, Random matrix theory,Random walks, Routes to chaos, Spatiotemporalchaos, Stochastic processes, Turbulence, Turbu-lence, ideal

COHERENT STRUCTURES

Biomolecular solitons, Black holes, Breathers, Cellassemblies, Davydov soliton, Discrete breathers,Dislocations in crystals, DNA solitons, Domainwalls, Dune formation, Emergence, Fairy rings ofmushrooms, Flame front, Higgs boson, Holons,Hurricanes and tornadoes, Instantons, Jupiter’sGreat Red Spot, Local modes in molecular crys-tals, Local modes in molecules, Multidimensionalsolitons, Nerve impulses, Polariton, Polaron, Shockwaves, Skyrmions, Solitons, types of, Spiral waves,Tachyons and superluminal motion, Turing pat-terns, Wave of translation

DYNAMICAL SYSTEMS

Anosov/axiom A system, Arnold diffusion, At-tractors, Aubry--Mather theory, Bifurcations, Bil-

liards, Butterfly effect, Catastrophe theory, Catmap, Center-manifold reduction, Chaotic dynam-ics, Coupled map lattice, Deterministic walks inrandom environment, Development of singular-ities, Dynamical systems, Equilibrium, Ergodictheory, Fitness landscape, Framed space curves,Function spaces, Gradient system, Hamiltoniansystems, Henon map, Hopf bifurcation, Horse-shoes and hyperbolicity, Inertial manifolds, In-termittency, Kicked rotor, Kolmogorov--Arnold--Moser theory, Lyapunov exponents, Maps, Mea-sures, Melnikov method, One-dimensional maps,Pattern formation, Periodic orbit theory, Phaseplane, Phase space, Phase space diffusion and cor-relations, Poincaretheorems, Reaction diffusion sys-tems, Rossler systems, Rotating rigid bodies, Routesto chaos, Sinai--Ruelle--Bowen measure, Standardmap, Stochastic processes, Symbolic dynamics,Synergetics, Universality, Visiometrics, Windingnumbers

GENERAL PHENOMENA

Adiabatic invariants, Algorithmic complexity, An-derson loccalization, Arnold diffusion, Attractors,Berry’s phase, Bifurcations, Binding energy, Bound-ary layers, Branching laws, Breathers, Brownianmotion, Butterfly effect, Causality, Chaotic dy-namics, Characteristics, Clusterr coagulation, Co-herence phenomena, Collisions, Critical phenom-ena, Detailed balance, Determinism, Diffusion, Do-main walls, Doppler shift, Effective mass, Emer-gence, Entropy, Equilibria, Equipartition of energy,Excitability, Explosions, Feedback, Filamentation,Fractals, Free energy, Frequency doubling, Frus-tration, Gestalt phenomena, Group velocity, Har-monic generation, Helicity, Hopf bifurcation, Hys-teresis, Incoherent solitons, Inhibition, Integrabil-ity, Intermittency, Jump phenomena, Kolmogorovcascade, Levy flights, Lifetime, Mixing, Modu-lated waves, Multiplicative processes, Nonlinear-ity, definition of, N -wave interactions, Order fromchaos, Order parameters, Overtones, Pattern for-mation, Period doubling, Periodic bursting, Powerbalance, Quantum chaos, Quantum nonlinearity,Quasiperiodicity, Recurrence, Routes to chaos,Scroll waves, Shear flow, Solitons, Spiral waves,Structural complexity, Symmetry: equations vs. so-lutions, Synergetics, Tessellation, Thermal convec-

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tion, Threshold phenomena, Turbulence, Universal-ity, Wave packets, linear and nonlinear, Wave prop-agation in disordered media, Wave stability andinstability

MAPS

Aubry--Mather theory, Backlund transforms, Catmap, Coupled map lattice, Darboux transformation,Denjoy theory, Embedding methods, Fermi acceler-ation and Fermi map, Henon map, Maps, Maps inthe complex plane, Monodromy-preserving defor-mations, Nontwist maps, One-dimensional maps,Periodic orbit theory, Recurrence, Renormaliza-tion groups, Singularity theory, Standard map,Symplectic maps

MATHEMATICAL MODELS

Ablowitz--Kaup--Newell--Segur system, Attractorneural networks, Billiards, Boundary-value prob-lems, Brusselator, Burgers equation, Cat map, Cel-lular automata, Compartmental models, ComplexGinzburg--Landau equation, Continuum approxi-mations, Coupled map lattice, Coupled systemsof partial differential equations, Delay-differentialequations, Difference equations, Discrete nonlinearSchrodinger equations, Discrete self-trapping sys-tem, Duffing equation, Equations, nonlinear, Euler--Lagrange equations, Fitzhugh--Nagumo equa-tion, Fokker--Planck equation, Frenkel--Kontorovamodel, Game of life, General circulation modelsof the atmosphere, Henon--Heiles system, Inte-grable cellular automata, Integrable lattices, Isingmodel, Kadomtsev--Petviashvili equation, Knottheory, Korteweg--de Vries equation, Kuramoto--Sivashinsky equation, Landau--Lifshitz equation,Lattice gas methods, Lie algebras and Lie groups,Lorenz model, Markov partitions, Martingales,Maxwell--Bloch system, McCulloch--Pitts network,Navier--Stokes equation, Neural network models,Newton’s laws of motion, Nonlinear Schrodingerequations, One-dimensional maps, Ordinary differ-ential equations, nonlinear, Partial differential equa-tions, nonlinear, Random walks, Riccati equations,Salerno equation, Sandpile model, Sine-Gordonequation, Spin systems, Stochastic processes, Sym-bolic dynamics, Synergetics, Toda lattice, Vander Pol equation, Zeldovich--Frank-Kamenetskyequation

STABILITY

Attractors, Bifurcations, Butterfly effect, Catastro-phe theory, Controlling chaos, Development of sin-gularities, Dispersion management, Dispersion re-lations, Emergence, Equilibrium, Excitability, Feed-back, Growth patterns, Hopf bifurcation, Lyapunovexponents, Nonequilibrium statistical mechanics,Stability

ASTRONOMY AND ASTROPHYSICS

Alfv�enwaves, Black holes, Celestialmechanics, Cos-mological models, Einstein equations, Galactic dy-namics, Gravitational waves, Henon--Heiles system,Jupiter’s Great Red Spot, N -body problem, Solarsystem

BIOLOGY

Artificial life, Bilayer lipid membranes, Biologicalevolution, Biomolecular solitons, Cardiac musclemodels, Catalytic hypercycle, Compartmental mod-els, Davydov soliton, DNA premelting, DNA soli-tons, Cardiac arrhythmias and electrocardiogram,Cardiac muscle models, Epidemiology, Excitabil-ity, Fairy rings of mushrooms, Fibonacci series,Fitness landscape, Frohlich theory, Game of life,Growth patterns, Holons, Morphogenesis, Biologi-cal, Nephron dynamics, Protein dynamics, Proteinstructure, Scroll waves, Turing patterns, Synerget-ics

CHEMISTRY

Belousov--Zhabotinsky reaction, Biomolecular soli-tons, Brusselator, Candle, Catalytic hypercycle,Chemical kinetics, Flame front, Franck--Condonfactor, Hydrogen bond, Langmuir--Blodgett films,Molecular dynamics, Oregonator, Polymerization,Protein structure, Reaction--diffusion systems, Scheibeaggregates, Vortex dynamics in excitable media

CONDENSED MATTER ANDSOLID-STATE PHYSICS

Anderson localization, Avalanche breakdown, Bjer-rum defects, Bose--Einstein condensation, Chargedensity wave, Cherenkov radiation, Color cen-ters, Commensurate--incommensurate transition,Discrete breathers, Dislocations in crystals, Domain

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walls, Drude model, Effective mass, Excitons, Fer-roelectricity and ferromagnetism, Franck--Condonfactor, Frenkel--Kontorova model, Frustration, Heatconduction,Hydrogenbond, Isingmodel, Langmuir--Blodgett films, Liquid crystals, Local modes inmolecular crystals, Mechanics of solids, Nonlin-ear acoustics, Peierls barrier, Percolation theory,Regular and chaotic dynamics in atomic physics,Scheibe aggregates, Semiconductor oscillators, Spinsystems, Superconductivity, Superfluidity

EARTH SCIENCE

Alfv�en waves, Angle of repose, Atmospheric andocean sciences, Avalanches, Ball lightning, Butterflyeffect, Clear air turbulence, Dune formation, Fairyrings of mushrooms, Forest fires, General circula-tionmodels of the atmosphere, Geomorphology andtectonics, Glacial flow, Global warming, Hurricanesand tornadoes, Kelvin--Helmholtz instability, Sand-pile model, Water waves

ENGINEERING

Artificial intelligence, Cellular automata, Cellu-lar nonlinear networks, Chua’s circuit, Controllingchaos, Coupled oscillators, Diodes, Dispersionman-agement, Dynamos, homogeneous, Electron beamdevices, Explosions, Feedback, Flip-flop circuit,Frequency doubling, Information theory, Hystere-sis, Josephson junction arrays, Josephson junctions,Langmuir--Blodgett films, Lasers, Long Joseph-son junction, Manley--Rowe relations, Neuristor,Nonlinear electronics, Nonlinear optics, Nonlinearsignal processing, Optical fiber communications,Parametric amplification, Particle accelerators, Ratch-ets, Relaxation oscillators, Semiconductor laser,Semiconductor oscillators, Superconducting quan-tum interference device, Synchronization, TacomaNarrows Bridge collapse

FLUIDS

Alfv�en waves, Atmospheric and ocean sciences,Bernoulli’s equation, Clear-air turbulence, Con-tour dynamics, Electron beam devices, Evapora-tion wave, Fluid dynamics, Forecasting, General cir-culation models of the atmosphere, Glacial flow,Hurricanes and tornadoes, Hydrothermal waves,Jump phenomena, Jupiter’s Great Red Spot, Kelvin--

Helmholtz instability, Laboratory models of non-linear waves, Lattice gas methods, Liquid crys-tals, Lorentz gas, Magnetohydrodynamics, Navier--Stokes equation, Nonlinear plasma waves, Plasmasoliton experiments, Plume dynamics, Rayleigh--Taylor instability, Shear flow, Shock waves, Super-fluidity, Surface waves, Taylor--Couette flow, Ther-mal convection, Thermo-diffusion effects, Trafficflow, Turbulence, Turbulence, ideal, Visiometrics,Vortex dominated flows, Water waves

NEUROSCIENCE

Artificial intelligence, Attractor neural network, Cellassemblies, Compartmental models, Depth percep-tion and binocular rivalry, Electroencephalogram atlarge scales, Electroencephalogram at mesoscopicscales, Ephaptic coupling, Evans function, Gestaltphenomena, Hodgkin--Huxley equations, Integrateand fire neuron, Multiplex neuron, Myelinatednerves, Nerve impulses, Neural network models,Neurons, Pattern formation, Perceptron, Stochasticanalyses of neural systems, Synergetics

NONLINEAR OPTICS

Damped driven anharmonic oscillator, Cherenkovradiation, Color centers, Dispersion management,Distributed oscillators, Excitons, Filamentation, Ge-omentrical optics, nonlinear, Harmonic generation,Hole burning, Kerr effect, Lasers, Liquid crystals,Maxwell--Bloch system, Nonlinear optics, Opticalfiber communications, Photonic crystals, Polariton,Polaron, Pump-probe measurements, Rayleigh andRaman scattering and IR absorption, Semiconductorlaser, Tachyons and superluminal motion

PLASMA PHYSICS

Alfv�en waves, Ball lightning, Charge-density wave,Drude model, Dynamos, homogeneous, Electronbeam devices, Magnetohydrodynamics, Nonlinearplasma waves, Particle accelerators, Plasma solitonexperiments

SOCIAL SCIENCE

Economic dynamics, Epidemiology, Game theory,Hierarchies of nonlinear systems, Population dy-namics, Traffic flow

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SOLID MECHANICS ANDNONLINEAR VIBRATIONS

Angle of repose, Avalanche breakdown, Bilayerlipid membranes, Bjerrum defects, Charge-densitywave, Cluster coagulation, Color centers, Detailedbalance, Dislocations in crystals, Domain walls,Frustration, Glacial flow, Granular media, Growthpatterns, Heat conduction, Hydrogen bond, Isingmodel, Kerr effect, Langmuir--Blodgett films, Liquidcrystals, Localmodes inmolecular crystals,Mechan-ics of solids, Molecular dynamics, Nonlinear acous-tics, Protein dynamics, Ratchets, Rheology, Sandpilemodel, Scheibe aggregates, Shock waves, Spin sys-tems, Superlattices, Tessellation, Topological defects

THEORETICAL PHYSICS

Berry’s phase, Black holes, Born--Infeld equation,Celestial mechanics, Cherenkov radiation, Coner-

vationlaws and constants of motion, Cosmologicalmodels, Critical phenomena, Derrick--Hobart theo-rem, Detailed balance, Einstein equations, Entropy,Equipartition of energy, Fluctuation-dissipation the-orem, Fokker--Planck equation, Free energy, Galax-ies, General relativity, Gravitational waves, Hamil-tonian systems, Higgs boson, Instantons, N-bodyproblem, Newton’s laws of motion, Nonlinear the-ory of matter, Particles and antiparticles, Quantumfield theory, Quantum theory, Regular and chaoticdynamics in atomic physics, Rotating rigid bodies,Skyrmions, String theory, Tachyons and superlumi-nal motion, Twistor theory, Virial theorem, Yang--Mills theory

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Documents

encyclopedia of nonlinear science - Princeton Universityengine.princeton.edu/download/ju-digital... · Contents Introduction vii List of Advisers xiii List of Contributors xv Alphabetical