THE ENCYCLOPEDIA OF NONLINEAR SCIENCE

PREFACE

Among the several advances of the twentieth century, nonlinear science is exceptional for its generality. While the invention of radio was important for communications, the discovery of DNA structure for biology, the development of quantum theory for theoretical chemistry, and the invention of the transistor for computer engineering, nonlinear science is significant in all these areas and many more. Indeed, it plays a key role in almost every branch of modern research, as this Encyclopedia of Nonlinear 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 phenomena like tsunamis (tidal waves), biological evolution, atmospheric dynamics, and the electrochemical activity of a human brain, among many others. For a research scientist, nonlinear science offers novel phenomena, including the emergence of coherent structures (an optical soliton, for example, or a nerve impulse) and chaos (characterized by the difficulties in making accurate predictions for surprisingly simple systems over extended periods of time). Both of these phenomena can be studied using mathematical methods described in this Encyclopedia. From a more fundamental perspective, a wide spectrum of applications arises because nonlinear science introduces a paradigm shift in our collective attitude about causality. What is the nature of this shift?

Consider the difference between linear and nonlinear analyses. Linear analyses are characterized by the assumption that individual effects can be unambiguously traced back to particular causes. In other words, 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 particular effect. The total effect responding to the total cause is then taken to be just the linear sum of the constituent effects.

A fundamental tenet of nonlinear science is to reject this convenient but often unwarranted assumption. Of course, the notion that components of complex causes can interact among themselves is not surprising to any thoughtful person who manages to get through an ordinary day of normal life, and it is not at all new. Twenty-five centuries ago, Aristotle described four types of cause (material, efficient, formal, and final), which overlap and intermingle in ways that were often overlooked in twentieth-century thought but now are under scrutiny. Consider some examples of linear scientific thinking that are presently being reevaluated in the context of nonlinear science.

• Around the middle of the twentieth century, behavioral psychologists took the theoretical position that human mental activity can be reduced to a sum of individual responses to specific stimuli that have been learned at earlier stages of development. Current research in neuroscience shows 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 manner to individual features of an adult organism, leading to hunts for a ``crime gene'' that seems abhorrent to moralists. Nonlinear science suggests that the relation between genes and features of an adult organism is more intricate than the linear perspective assumes.
• The sad disintegration of space shuttle Columbia on the morning of February 1st, 2003 set off a search for ``the cause of the accident,'' ignoring Aristotelian insights into the difficulties of defining such a concept, never mind sorting out the pieces. Did the mishap occur because the heat resistant tiles were timeworn (a material cause)? Or because 1.67 pounds of debris hit the left wing at 775 feet per second during takeoff (an efficient cause)? Perhaps a management culture that discounted the importance of safety measures (a formal cause) should shoulder some of the blame.
• Cultural phenomena, in turn, are often viewed as the mere sum of individual psychologies, ignoring the grim realities of war hysteria and lynch mobs, not to mention the ``tulip craze'' of seventeenth century Holland, the more recent ``dot-com bubble,'' and the outbreak of communal mourning over the death of Princess Diana.

As the practice of nonlinear science involves such abstruse issues, one might expect its history to be checkered, and indeed it is. Mathematical physics began with the seventeenth-century work of Isaac Newton, whose formulation of the laws of mechanical motion and gravitation explained how the earth moves about the sun, replacing a final cause (God's plan) with an efficient cause (the force of gravity). Because it assumed that the net gravitational force acting on any celestial body is the linear (vector) sum of individual forces, Newton's theory provides support for the linear perspective in science, as has often been emphasized. Nonetheless, the mathematical system Newton developed (calculus) is the natural language for nonlinear science, and he used this language to solve the two-body problem (collective motion of earth and moon)---the first nonlinear system to be mathematically studied. Also in the seventeenth century, Christiaan Huygens noted that two pendulum clocks (which he had recently invented) kept exactly the same time when hanging from a common support. (Confined to his room by an indisposition, Huygens observed the clocks over a period of several days, during which the swinging pendula remained in step.) If the clocks were separated to opposite sides of the room one lost several seconds a day with respect to the other. From small vibrations transmitted through the common support, he concluded, the two clocks became synchronized---a typical nonlinear phenomenon.

In the eighteenth century, Leonhard Euler used Newton's laws of motion to derive nonlinear field equations for fluid flow, which were augmented a century later by Louis Navier and George Stokes to include the dissipative effects of viscosity that are present in real fluids. In their generality, these equations defied solution until the middle of the twentieth century when, together with the digital computer, elaborations of the NavierãStokes equations provided a basis for general models of the earth's atmosphere and oceans, with implications for the vexing question of global warming. During the latter half of the nineteenth century, however, special analytic solutions were obtained by Joseph Boussinesq and related to experimental observations of hydrodynamic solitary waves by John Scott Russell. These studies---which involved a decade of careful observations of uniformly propagating ``heaps of water'' on canals and in wave tanks---were among the earliest research programs in the area now recognized as nonlinear science. At about the same time, Pierre Francois Verhulst formulated and solved a nonlinear differential equation---sometimes called the logistic equation---to model the population growth of his native Belgium.

Toward the end of the nineteenth century Henri Poincare returned to Newton's original theme, presenting a solution of the three-body problem of celestial motion (a planet with two moons, for example) in a mathematical competition sponsored by the King of Sweden. Interestingly, a serious error in this work was discovered prior to its publication, and he (Poincare, not the Swedish king) eventually concluded that the three-body problem cannot be exactly solved. Now regarded by many as the birth of the ``science of complexity,'' this negative result had implications that were not widely appreciated until the 1960s, when numerical studies of simplified atmospheric models by Edward Lorenz showed that nonlinear systems with as few as three degrees of freedom can readily exhibit the nonlinear phenomenon of chaos. (A key observation here was of an unanticipated sensitivity to initial conditions, popularly known as the ``butterfly effect'' from Lorentz's speculation that ``the flap of a butterfly's wings in Brazil [might] set off a tornado in Texas.'')

During the first half of the twentieth century, the tempo of research picked up. Although still carried on as unrelated activities, there appeared a notable number of experimental and theoretical studies now recognized as precursors of modern nonlinear science. Among others, these include Albert Einstein's nonlinear theory of gravitation; nonlinear field theories of elementary particles (like the recently discovered electron) developed by Gustav Mie and Max Born; experimental observations of local modes in molecules by physical chemists (for which a nonlinear theory was developed by Reinhard Mecke in the 1930s, forgotten, and then redeveloped in the 1970s); biological models of predator-prey population dynamics formulated by Vito Volterra (to describe year-to-year variations in fish catches from the Adriatic Sea); observations of a profusion of localized nonlinear entities in solid-state physics (including ferromagnetic domain walls, crystal dislocations, polarons, and magnetic flux vortices in superconductors, among others); a definitive experimental and theoretical study of nerve impulse propagation on the giant axon of the squid by Alan Hodgkin and Andrew Huxley; Alan Turing's theory of pattern formation in the development of biological organisms; and Boris Belousov's observations of pattern formation in a chemical solution, which were at first ignored (under the mistaken assumption that they violated the second law of thermodynamics) and later confirmed and extended by Anatol Zhabotinsky and by Art Winfree. Just as the invention of the laser in the early 1960s led to numerous experimental and theoretical studies in the new field of nonlinear optics, so the steady increases in computing power throughout the second half of the twentieth century enabled ever more detailed numerical studies of hydrodynamic turbulence and chaos, whittling away at the long established Navier-Stokes equations and confirming the importance of Poincare's negative result on the three-body problem.

Thus it was evident by 1970 that nonlinearity manifests itself in several remarkable properties of dynamical systems, including the following. (There are others, some no doubt waiting to be discovered.)

• Many nonlinear partial differential equations (wave equations, diffusion equations, and more complicated field equations) are often observed to exhibit localized or lump-like solutions, similar to Russell's hydrodynamic solitary wave. These ``coherent structures'' of energy or activity emerge from initial conditions as distinct dynamic entities each having its own trajectory in space-time and characteristic ways of interacting with others. Thus they are ``things'' in the normal sense of the word. Interestingly, it is sometimes possible to compute the velocity of emergent entities (their speeds and shapes) from initial conditions and express them as tabulated functions (theta functions or elliptic functions), thereby extending the analytic reach of nonlinear analysis. Examples of emergent entities include tornados, nerve impulses, magnetic domain walls, tsunamis, optical solitons, Jupiter's Great Red Spot, black holes, schools of fish, and cities, to name but a few. A related phenomenon, exemplified by meandering rivers, bolts of lightning, and woodland paths, is called filamentation, which also causes spotty output beams in poorly designed lasers.
• Surprisingly simple nonlinear systems (Poincare's three-body problem is the classic example) are found to have chaotic solutions, which remain within a bounded region while the difference between neighboring solution trajectories grows exponentially with time. Thus the course of a solution trajectory is strongly sensitive to its initial conditions (the ``butterfly effect''). Chaotic solutions arise in both energy conserving (Hamiltonian) systems and in dissipative systems, and they are fated to wander unpredictably as trajectories that cannot be accurately extended into the future for unlimited periods of time. As Lorenz pointed out, chaotic behavior of the earth's atmosphere makes detailed meteorological predictions problematic, to the delight of the mathematician and the despair of the weatherman. Chaotic systems also exhibit ``strange attractors'' in the solution space, which are characterized by fractal (non-integer) dimensions.
• Nonlinear problems often display threshold phenomena, meaning that there is a relatively sharp boundary across which the qualitative nature of a solution changes abruptly. This is the basic property of an electric wall switch, the trigger of a pistol, and of the flip-flop circuit that a computer 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 of social phenomena, thresholds are an important part of our daily experience, where they complicate the relationship of causality to legal responsibility. Was it the last straw that broke the camel's back? Or did all of the straws contribute to some degree? Should each be blamed according to its weight? How does one assign culpability for the ``Murder on the Orient Express''?
• Nonlinear systems with several spatial coordinates often exhibit spontaneous pattern formation, examples of which include fairy rings of mushrooms, oscillatory patterns of heart muscle activity under fibrillation (leading to sudden cardiac arrest), weather fronts, the growth of form in a biological embryo, and the Gulf Stream. Such patterns can be chaotic in time and regular in space, regular in time and chaotic in space, or chaotic in both space and in time, which in turn is a feature of hydrodyamic turbulence.
• If the input to (or stimulation of) a nonlinear system is a single frequency sinusoid, the output (or response) is non-sinusoidal, comprising a spectrum of sinusoidal frequencies. For lossless nonlinear systems, this can be an efficient means for producing energy at integer multiples of the driving frequency, through the process of harmonic generation. In electronics, this process is widely used for digital tuning of radio receivers. Taking advantage of the nonlinear properties of certain transparent crystals, harmonic generation is also employed in laser optics to create light beams of higher frequency: conversion of red light to blue, for example.
• Another nonlinear phenomenon is the synchronization of weakly coupled oscillators, first observed by the ailing Huygens in the winter of 1665. Now recognized in a variety of contexts, this effect crops up in the frequency locking of electric power generators tied to the same grid and the coupling of biological rhythms (circadian rhythms in humans, hybernation of bears, and the synchronized flashing of Indonesian fireflies), in addition to many applications in electronics. Some suggest that neuronal firings in the neocortex may be mutually synchronized.
• Shock waves are familiar to most of us as the boom of a jet airplane that has broken the sound barrier or the report of a cannon. Closely related from a mathematical perspective are the bow wave of a speedboat, the breaking of onshore surf, and the sudden automobile pileups that can occur on a highway that is carrying traffic near its maximum capacity.
• More complicated nonlinear systems can be hierarchical in nature. This comes about when the emergence of coherent states at one level provides a basis for new nonlinear dynamics at a higher level of description. Thus in the course of biological evolution, chemical molecules emerged from interactions among the atomic elements, and biological molecules then emerged from simpler molecules to provide a basis for the dynamics of a living cell. From collections of cells, multi-cellular organisms emerged, and so on up the evolutionary ladder to creatures like ourselves which comprise several distinct levels of biological dynamics. Similar structures are observed in the organization of coinage and of military units, not to mention the hierarchical arrangement of information in the human brain.

Often qualitatively related behaviors---involving one or more of such nonlinear manifestations---are found in models that arise from different areas of application, suggesting the need for interdisciplinary communications. By the early 1970s, therefore, research in nonlinear science was in a state that the physical chemists might describe as ``supersaturated.'' Dozens of people across the globe were working on one facet or another of nonlinear science, often unaware of related studies in traditionally unrelated fields. During the mid-1970s, this activity experienced a ``phase change'' which can be viewed as a collective nonlinear effect in the sociology of science. Unexpectedly, there were organized a number of conferences devoted entirely to nonlinear science, with participants from a variety of professional backgrounds, nationalities and research interests eagerly contributing. Solid-state physicists began to talk seriously with biologists, neuroscientists with chemical engineers, and meteorologists with psychologists. As interdisciplinary barriers crumbled, these unanticipated interactions led to the founding of centers for nonlinear science and the launching of several important research journals amid an explosion of research activity. By the early 1980s, nonlinear science had become recognized as a key component of modern inquiry, playing a central role in a wide spectrum of activities. In the terminology introduced by Thomas Kuhn, a new paradigm had been established.

The primary aim of this Encyclopedia is to provide a source from which undergraduate and graduate students in the physical and biological sciences can study how concepts of nonlinear science are presently understood and applied. In addition, it is anticipated that teachers of science and research scientists who are unfamiliar with nonlinear concepts will use the work to expand their intellectual horizons and improve their lectures. Finally, it is hoped that this book will help members of the literate public---philosophers, social scientists, and physicians, for example---to appreciate the wealth of natural phenomena described by a science that does not discount the notion of causality.

An early step in writing the Encyclopedia was to choose the entry subjects---a difficult task that was accomplished through the efforts of a distinguished Board of Advisers with members from Australia, Germany, Italy, Japan, Russia, the United Kingdom, and the United States. After much sifting and winnowing, an initial list of about a thousand suggestions was reduced to 440 items. Depending on the subject matter, the entries are of several types. Some are historical or descriptive, while others present concepts and ideas that require notations from physics, engineering, or mathematics. Although most of the entries were planned to be about a thousand words in length, some---covering subjects of greater generality or importance---are two or four times as long.

Of the many enjoyable aspects in editing this Encyclopedia, the most rewarding has been working with those who wrote it---the contributors. The willing way in which these busy people responded to entry invitations and their enthusiastic preparation of assignments underscores the degree to which nonlinear science has become a community with a healthy sense of professional responsibility. In every case, the contributors have tried to present their ideas as simply as possible, with a minimum of technical jargon.

A proper presentation the diverse professional perspectives that comprise nonlinear science requires careful organization of the Encyclopedia, which we attempt to provide. Although each entry is self-contained, the links among them can be explored in several ways. First, the Thematic List groups entries within several categories, providing a useful summary of related entries through which the reader can surf. Second, the entries have ``See also'' notes both within the text and at the end of the entry, encouraging the reader to browse outwards from a starting node. Finally the Index contains a detailed list of topics that do not have their own entries but are discussed within the context of broader entries. If you can't find an entry on a topic you expected to find, use the Thematic List or Index to locate the title of the entry that contains the item you seek. Additionally, all entries have selected bibliographies or suggestions for further reading, leading to original research and textbooks that augment the overview approach to which an encyclopedia is necessarily limited. Although much of nonlinear science grew out of applied mathematics, many of the entries contain no equations or mathematical symbols and can be absorbed by the general reader. Some entries are necessarily technical, but efforts have been made to explain all terms in simple English. Also, many entries have either line diagrams expanding on explanations given in the text, or photographs illustrating typical examples.

The editing of this Encyclopedia of Nonlinear Science culminates a lifetime of study in the area, leaving me indebted to many. First is the Acquisitions Editor, Gillian Lindsey, who conceived of the project, organized it, and carried it from its beginnings in London across the ocean to final publication in New York. Without her dedication, quite simply, the Encyclopedia wouldn't exist. Equally important to reaching the finished work were the efforts of the advisers, contributors and referees, who respectively planned, wrote, and vetted the work, and to whom I am deeply grateful. On a broader time-span are colleagues and students from the University of Wisconsin, Los Alamos National Laboratories, the University of Arizona, and the Technical University of Denmark, with whom I have interacted over four decades. Although far too many to list, these collaborations are fondly remembered, and they provide the basis for much of my editorial judgment. Finally, I express thanks for the generous financial support of research in nonlinear science that has been provided to me since the early 1960s by the National Science Foundation (USA), the National Institutes of Health (USA), the Consiglio Nazionale delle Ricerche (Italy), the European Molecular Biology Organization, the Department of Energy (USA), the Technical Research Council (Denmark), the Natural Science Research Council (Denmark), the Thomas B. Thriges Foundation, and the Fetzer Foundation.