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Report of the Research Bnefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity
Research Beefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity Mitchell J. Feigenbaum (Co-Chairman), Professor of Physics, Rockefeller University Martin Kruskal (Co-Chairman), Professor of Mathematics, Princeton University William A. Brock, Professor of Economics, University of Wisconsin, Madison David Campbell, Director, Center for Nonlinear Studies, Los Alamos National Laboratory lames Glimm, Professor of Mathematics, Courant institute of Science, New York, N.Y. Leo P. Kadanoff, Professor of Physics, University of Chicago Anatole Katok, Professor of Mathematics California Institute of Technology Albert Libchaber, Professor of Physics, University of Chicago Arnold Mandell, Director, Laboratory for Biological Dynamics and Theoretical Medicine, University of California, San Diego 40 Alan C. Newell, Professor of Mathematics University of Arizona, Tucson Steven Orszag, Professor of Applied and Computational Mathematics, Princeton University H. Eugene Stanley, Professor of Physics, Boston University lames Yorke, Acting Director, Institute for Physical Science and Technology, University of Maryland, College Park Staff Donalcl C. Shapero, StaffDirector Robert L. Riemer, Program Officer, Board on Physics and Astronomy, Commission on Physical Sciences, Mathematics, and Resources Allan R. Hoffman, Executive Director, Committee on Science, Engineering, and Public Policy
Report of the Research Bnefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity INTRODUCTION AND BACKGROUND Linear analysis developed as a formal mathematical discipline during the nine- teenth century, and in the intervening years its applications have achieved many spectac- ular successes throughout science and engi- neering. But in fact most phenomena ob- served in nature are nonlinear, and the linear approximations historically used to describe them are too often tacit admissions that the true problems simply cannot be solved. In some instances, including many of techno- logical importance, the effects of nonlinear- ity can be understood in terms of small per- turbations on linear behavior. in other cases, however, incorporation of the true nonlin- earities completely changes the qualitative nature of the system's possible behavior. This report focuses on several aspects of these essentially nonlinear phenomena. The difficulties posed by essential nonlin- earity can be illustrated by a familiar exam- ple. When water flows through a pipe at low velocity, its motion is laminar and is charac- teristic of linear behavior: regular, predict- able, and describable in simple mathematical terms. However, when the velocity exceeds 41 a critical value, the motion becomes turbu- lent, with eddies moving in a complicated, irregular, erratic way that typifies nonlinear behavior. Many other nonlinear phenomena exhibit sharp and unstable boundaries, er- ratic or chaotic motion, and dramatic re- sponses to very small influences. Such prop- erties typically defy full analytical treatment and make even quantitative numerical de- scription a daunting task. And yet, this task must be confronted, for the point where phe- nomena become nonlinear is often precisely where they become of interest to technology. in applications ranging from laser/plasma interactions in inertial-confinement thermo- nuclear fusion, to designs for high-perfor- mance and fuel-efficient aircraft, to ad- vanced oil recovery, nonlinearity prevails. Within the past two decades, the system- atic, coordinated investigation of nonlinear natural phenomena and their mathematical models has emerged as a powerful and excit- ing interdisciplinary subject. Studies of non- linearity seek to understand a variety of com- plicated, nonlinear problems encountered in nature and to discover their common fea- tures. The scientific methodology has de- pended on the synergetic blending of three distinct approaches:
· "Experimental mathematics," which is the use of cleverly conceived computer- based numerical simulations, typically in- volving visualization techniques such as interactive, high-quaTity graphics, to give qualitative insights into and to stimulate conjectures about analytically intractable problems; · Novel and powerful analytical mathe- matical methods to solve, for example, cer- tain nonlinear partial differential equations and to analyze nonlinear stability; and · Experimental observation of similar be- havior in natural nonlinear phenomena in many different contexts and the quantifica- tion of this similarity by high-precision ex- periments. The success of this three-prongec! attack is clearly evidenced by the remarkable progress already made toward solving many nonlinear problems Tong considered intrac- table. Essential to this progress has been the discovery that distinct nonlinear phenom- ena from many fields do indeed display com- mon features and yield to common methods of analysis. This commonality has allowed the rapid transfer of progress in one disci- pline to other fields and confirms the inher- ently interdisciplinary nature of the subject. Despite these stimulating developments, however, the present-day approach to non- linear problems is not entirely systematic. Rather it relies on the identification and ex- ploitation of paradigms, namely, unifying concepts anct associated methodologies that are broadly applicable in many different fielcls. This report focuses on three of the central paradigms of nonlinearity: coherent struc- tures, chaos, and complex configurations and pattern selection. The following sections cover recent progress in research and future opportunities for research and technological applications of these paradigms, the interna- tional standing of U. S. work in the field, and administrative strategies for enhancing progress in this important interdisciplinary subject. PARADIGMS OF NONLINEARITY: DEFINITIONS, OPPORTUNITIES, AND APPLICATIONS COHERENT STRUCTURES AND ORDER From the Red Spot of Jupiter, to clumps of electromagnetic radiation in turbulent plas- mas, to microstructures on the atomic scale, Tong-lived, spatially localized, collective ex- citations abound in nonlinear systems. These coherent structures show a surprising order in the midst of complex nonlinear be- havior and often represent the natural modes for expressing the dynamics. Thus, for example, isolated coherent structures may dominate Tong-time behavior, and anal- ysis of their interactions may explain the ma- jor aspects of the dynamical evolution. Rec- ognition of these possibilities constitutes a fundamental change in the approach to non- linear systems and has opened up a range of new analytical and computational tech- niques that yield deep insights into nonTin- ear natural phenomena. Although the importance of vortices and eciclies in turbulent fluid flows has been ap- preciated since ancient times, the critical event in the modern concept of coherent structures was the discovery in 1965 of the remarkable "soliton" behavior of localized nonlinear waves governed by the Korte- weg-deVries equation, which describes waves in a shallow, narrow channel of water (e.g., a canal) and in many other physical media. Solitons represent coherent struc- tures in the purest sense in that their form is exactly restored after temporary distortion during interactions. Surprisingly, many equations, of wide applicability, have turned out to support solitons, and a major mathe- maticalL success has been the revelation that most of these equations can be solved explic- itly and systematically by a novel analytical technique known as the inverse spectral transform. These developments have drawn upon and greatly stimulated several branches of 42
ORDER, CHAOS, AND PATTERNS: ASPECTS OF NONLINEARITY pure mathematics, including infinite- climensional analysis, algebraic geometry, partial differential equations, and dynamic systems theory. For instance, soliton equa- tions have been shown to correspond to a very special subclass of those nonlinear dy- namic systems that have an infinite number of independent parts. Technically, the num- ber of parts is referred to as the number of degrees of freedom or the phase-space dimension. The special characteristic of a so- liton equation is that it describes a HamiTto- nian dynamic system of infinite phase-space dimension that is, in technical parlance, completely integrable. The Hamiltonian consequently possesses infinitely many in- dependent conservation laws, which deter- mine its behavior. The existence of individ- ual solitons can be understood as a delicate balance between nonlinear focusing and dis- persive broadening, while the invariance of solitons under interactions is a consequence of the many conservation laws. A wide variety of soliton equations has been discovered, allowing a broad range of applications to natural phenomena. In fiber optics, Tosephson transmission lines, con- ducting polymers and other chainlike solids, and plasma ''cavitons," the prevailing mathematical moclels are slight modifica- tions of soliton equations. Thus, with sys- tematic approximations, the behavior of real physical systems can be described quite ac- curately. An example of potential technolog- ical significance can be ctrawn from nonlin- ear optics. In this discipline, as the name suggests, nonlinear phenomena, including self-induced transparency, optical phase conjugation, and optical bistability, are dom- inant. Considerable recent research has in- vestigated the prospect of using solitons to improve long-distance communications in optical fibers. At Tow intensities, light pulses in optical fibers propagate linearly and tend to disperse, degrading the signal. To com- pensate for this and reconstruct the pulse, repeaters must be added to the fiber at regu- lar intervals. If the light intensity is increased 43 into the nonlinear regime, soliton pulses can be formed, the nonlinearity compensating for dispersion. in the iclealized limit of no dissipative energy Toss, the solitons propa- gate without degradation of shape; they are indeed the natural, stable, localized modes for propagation in the fiber. Further, realistic theoretical estimates suggest that a soliton- based system could have an information rate one order of magnitude greater than that of conventional linear systems. Although de- tailed questions of practical implementation remain (primarily costs), the prospects for using optical solitons in long-distance com- munication are real. In the more general case, coherent struc- tures interact strongly and do not necessarily maintain their form or even their separate identities for all times. Instabilities generat- ing fluid vortices can lead to vortex pairs, and a pair may merge to form a single coher- ent structure equivalent to a new and larger vortex. Interactions among shock waves give rise to diffraction patterns of incident, reflected, and transmitted waves. Bubbles and droplets interact through merging and splitting. Significantly, physical examples of these more general coherent structures are nearly universal and, apart from the struc- tures already mentioned, include elastoplas- tic waves and shear bands, chemical-reac- tion waves and nonlinear diffusion fronts, phase boundaries, and dislocations in metals. There is a deep mathematical basis to this universality. In a first approximation, these nonlinear wavelike phenomena are subject to conservation laws. In contrast to the soliton case, there are usually only a few conserved quantities (e.g., mass, energy, and momentum). Nonetheless, these few conservation laws strongly restrict the possi- ble behavior of the system. Nonlinearity im- plies that the speed of a wave depends on the amplitude of the wave itself. As a result, the conservation laws lead to focusing and defo- cusing of waves. The defocused waves dis- perse, while the focused waves become co- herent structures, the nonlinear modes in
which the dynamics is naturally described. They may dominate the long-time behavior of the system, engage in complicated mo- tions and interactions, or organize into com- plex configurations and patterns. Fluid vortices a classic example of which is provided by the Red Spot of Jupiter (Figure 1) cars be used to illustrate the essential role of general coherent structures in nonlinear systems. The existence and stability of the Red Spot of Jupiter have been confirmed since the seventeenth century. A more mod- ern example is the vortex pattern formed in the wake of an airfoil. These vortices are of sufficient size anc! importance that they gov- ern the allowed spacing between aircraft at landing and thus limit the efficiency of air port utilization. Similarly, the manner in which vortices are shed from the airfoil strongly affects fuel efficiency and is essen- tial in designing high-performance aircraft. Specifically, vortices are microstructures that make up the critical turbulent boundary layer at the wing surface. More generally, an understanding of the highly nonlinear dy- namics of vortices is one of the central prob- lems of applied fluid dynamics. Further examples of dominant coherent structures can be drawn from almost any field of the natural sciences or engineering. Chemical-reaction fronts are important in many situations anct, in flame fronts and in- ternal combustion engines, are coupled strongly to fluid modes. Concentration Figure 1 A close-up of the giant Red Spot of Jupiter, a coherent structure that exists in the turbulent shear flow in the Southern Hemisphere. Note the coexistence of this large vortex with smaller eddies on many different scales. A1- though it is not apparent from this single image, the series of time-lapse photographs taken by the Voyager spacecraft shows that the Red Spot is highly dynamic, spinning rapidly and moving westwardly at 11 km/hr. (Courtesy National Aeronautics and Space Administration, let Propulsion Laboratory) 44
ORDER, CHAOS, AND PATTERNS: ASPECTS OF NONLINEARITY fronts arise in the leaching of minerals from ore beds. Fronts between infected and unin- fected individuals can be identified in the ep- idemiology of diseases such as rabies. In ge- ology, elastoplastic waves are important in the slow, long-time deformation of struc- tures. For example, salt domes are formed by a gravitational instability in which the flow of rock layers occurs on geological time scales. Understanding the development of such ge- ological formations is important both theo- retically and in the evaluation of potential oil reservoirs. Finally, at the microscopic level the nonlinear dynamics of dislocations may lead to novel effects crucial for interpreting the behavior of materials subjected to high strain rates, and transport phenomena in certain classes of quasi-one-climensional ma- terials may be controlled by the nonlinear co- herent structures they support. A final example with potential major tech- nological implications is the recent identifi- cation of new types of coherent structures and interactions in wave phenomena in oil reservoirs. The essential discovery is that when the speeds of two families of nonlinear waves coincide, a type of nonlinear reso nance may give rise to a surprising range of novel wave phenomena. it has recently been shown that nonlinear resonance of this type must occur in three-phase flow in oil reser- voirs, and a systematic program is well un- der way to identify and classify all possible types of nonlinear wave interaction and to assess their importance for oil recovery methods. Given the ubiquity and importance of co- herent structures in nonlinear phenomena, it is gratifying that recent years have wit- nessed remarkable progress in studying them and that there is great promise for still deeper insights. Significantly, this progress has been achieved by precisely the synergy among computation, theory, and experi- ment that characterizes nonlinear science. in particular, experimental mathematics has been essential to the understanding of co- herent structures and their interactions. 45 Typically, the forms of the coherent struc- tures are not immediately obvious from the underlying nonlinear equations. Hence vi- sualizations of flow patterns and dynamics using interactive graphics will play an in- creasingly important role. in summary, coherent structures reflect an essential paradigm of nonlinear science, pro- viding a unifying concept and an associated methodology at the theoretical, computa- tional, and experimental levels. Their impor- tance for technological applications, as well as their inherent interest for fundamental science, guarantees their central role in all fu- ture research in this subject. CHAO S The appearance of irregular, aperiodic, in- tricately detailed, unpredictable motion in deterministic systems is a truly nonlinear ef- fect. Loosely termed chaos, it is remote from linear phenomena. Although chaotic motion is observed, the processes are strictly deter- ministic: sufficiently accurate knowledge of an initial state allows arbitrarily accurate pre- ctictions but only over a limited interval of time. In particular, it is not necessary to drive a process randomly to observe motion of a stochastic character. indeed, attempting to model "deterministically chaotic" systems as responding to random forces fails to cap- ture their true behavior. While the mathematical seeds had already been planted by Poincare at the turn of the century, they have germinated only in the past three decades, with the advances in in- teractive computation that we have termed experimental mathematics playing an essen- tial role. One striking recent development has been the recognition that certain chaotic motions unfold themselves with a total lack of regard for the specific mechanisms at work: objects exhibiting certain complex mo- tions follow similar destinies independent of whether their microscopic behavior is gov- erned by equations derived from the theory
of chemical interactions, or fluids, or electro- magnetism. The discovery of this universal- ity and its application to experiments on the transition to turbulence is one of the tri- umphs of nonlinear science. The field of chaotic dynamics continues to undergo explosive growth, with many ad- vances and applications being made across a broad spectrum of disciplines, including physics, chemistry, engineering, fluid me- chanics, ecology, and economics. Chaotic systems can be observed in both experimen- tal data and numerical models. Examples in- clude the weather, chemical systems, and beating chicken hearts. The dripping of household faucets can be chaotically irregu- lar, while it has been argued that the satellite Hyperion of Saturn tumbles chaotically in its eccentric elliptical orbit, having no fixed axis because it is constantly kicked by the varying tidal pulls of Saturn. Medical research has revealed that many physiological parameters vary chaotically in the healthy individual, while more regular- ity can be a sign of pathology. For example, the familiar pattern of the beating heart is subtly irregular under close examination, and the absence of chaotic components seems to occur in pathological conditions. Similarly, the normally chaotic oscillations of red and white blood cell densities become periodic in some leukemias and anemias. There are many similar examples including periodic catatonias and manic-depressive disorders. Recent research suggests possible applica- tions to realistic economic models. General equilibrium-theory models have been con- structed that are chaotic, but with parameter values that do not mesh well enough with empirical studies to be persuasive. On the other hand economists, motivated by the ideas of chaotic dynamics, have developed new and powerful statistical tests for analyz- ing time series, which may be useful in other areas of nonlinear science. As this brief listing suggests, deterministic chaos is essential to the understanding of 46 many reai-worId nonlinear phenomena. To indicate further aspects of our present un- derstancling, more technical detail is neces- sary. The concept of the phase-space dimen- sion of a dynamic system was discussed ear- lier. For a complex object, this dimension is a priori quite high; for a continuous system, such as a fluid, it is in fact infinite. However, if many parts are effectively locked together, as in a coherent structure like a fluid vortex, the effective dimension is reduced, perhaps drastically. This general phenomenon is re- ferred to as mode reduction. As the character of the system's motion changes, so will the number of reduced modes and hence the ef- fective dimension. In the example of pipe flow quoted in the introduction, as veloc- ity increases, the fluid motion becomes suddenly more complex. Such sudden tran- sitions to qualitatively new motions are re- lated to the mathematical phenomenon of bi- furcations. Recent advances in the study of bifurcations provide an understanding of the mechanism leading from ordered to cha- otic behavior. More specifically, transitions in the behavior of physical systems can arise through an infinite cascade of bifurcations, the best known and first isolated of which is period doubling. This period-doubling cas- cade is controlled by a special behavior (with certain scaling properties) just at the point of transition, which fully organizes both the or- derliness prior to transition and the chaotic behavior after it. Significantly, theory shows that this behavior is correctly expressed by a very low-dimensional, mode-reduced dy- namics, independent of the original phase- space dimension of the system. Even more important, the behavior is universal: what- ever the system, the properties exhibited are identical. Recent experimental confirmation of these theoretical predictions in systems from convecting fluids to nonlinear elec- tronic circuits is one of the triumphs of non . . linear science. Once it is recognized that the original equations contain superfluous information because of mode reduction, it becomes im
ORDER, CHAOS, AND PATTERNS: ASPECTS OF NONLINEARITY portent to deduce the actual number of effec- tive equations that is, the dimension of the reduced system and then to determine the form of the equations. The first part of this program has been well implemented in the last few years by so-called phase-space re- construction techniques. Provided that the data support a dimension of below, say, 10, that number can be extracted reliably. Tn- deed, ideas from thermodynamics provide a graphic depiction that can quickly illuminate some cletaiTs of the nature of the excitations as well as the dimension. These methods, however, must be refined. The second part of this program has rarely been accomplished and then only on a case- by-case basis. In some instances, assumed forms can be fit to the data. At this point an easily simulated simple set of equations completely replaces the original ones. For example, three first-order ordinary differen- tial equations exactly replace the full fluid equations throughout a certain regime of motion. Now a real payoff accrues: the model system can easily be time-depen- clently forced, in contrast to an actual experi- mental fluid with its physically imposed exi- gencies, such as boundaries. This can lead to insights of profound technological impor- tance. A recent Soviet effort has apparently succeeded by just this program in forestall- ing the onset of turbulence in a nozzle flow by imposing periodic stress; clearly such suppression (or enhancement) of turbulence could have many vital applications. More generally, away from transition regions, the specific forms of the mode-recluced equa- tions may play a role. In this regard, an im- portant and generally open problem is to es- tablish the relation, if any, between coherent structures observed in a given motion and the recluced modes that in principle charac- terize the motion. In certain specific prob- lems, notably perturbed soliton equations and models for chemical-diffusion fronts, progress has been made, but much further research is required. To delve still creeper into current progress 47 anct to indicate what may lie ahead, it is nec- essary to introduce some additional termi- nology. For dissipative systems (e.g., those with friction) a wide class of initial motions may in the long-time limit approach some set of phase-space points, which is then called an attractor. Very commonly an attractor is a single point or a closed curve. However, sometimes the attracting set is much more ir- regular, and for a "strange attractor" the di- mension need not even be an integer. This concept of fractional dimension, related to mathematical work begun in the 1920s, has recently become more widely appreciated through the development and application of the theory of such "fractal" objects. KnowI- edge of fractals is essential to understanding modern nonlinear dynamic systems theory. For example, in a cleterministically chaotic system, the attracting set can be a chaotic strange attractor, on which two initially very close points begin to separate exponentially fast. This yields an exquisite sensitivity to initial conditions, for tiny initial uncertain- ties later produce profound ones. in general, a complicated physical system may contain several attractors, each with its own basin of attraction. A subtle further consequence of nonlinear dynamics is that the boundaries between these basins of attraction can them- seIves be extraordinarily complex and, in fact, fractal. These fractal basin boundaries mean that totally different Tong-time behav- ior can result from indistinguishably close initial configurations. An illustration of these concepts is pro- vided by weather forecasting. A chaotic dy- namic model, based on a crude approxima- tion of atmospheric fluid flow, explains why weather prediction works only for short pe- riods of time. Since small uncertainties grow so rapidly, there is a limit on how far ahead one can predict whether it will rain on a given day, no matter how large and fast the computer that is used to forecast. At the same time, specific familiar local weather patterns for example, summer thunder- showers in the mountains can be uncler
stood in terms of attractors in local models of weather. Figure 2 depicts a strange attractor found in a mode! simulation of the behavior of an optical switch. The sequence shown reveals the persistence of the attractor's convoluted structure at successively greater magnifica- tions. This nontrivial structure appearing on all scales correctly suggests that the object does not fill the two-dimensional surface on which it lies, but rather is a fractal with di- mension between ~ (a smooth curve) and 2 (a smooth surface). in fact it has dimension I. 7. An unmistakable property of the sequence of Figure 2 is that the very small details are reminiscent of the entire object. This prop- erty is called scaling, the formal theory of which allows the construction of fine detail from crude features. Thus, a conceptually new means of describing complicated ob- jects has emerged from these studies. The systematic classification of the strange sets that arise in Tow-climensional chaotic mo- tions remains one of the challenges of cur- rent studies in nonlinear dynamics. The impact of deterministic chaos is only now beginning to be felt throughout science. The recognition that even simple systems Figure 2 The trajectory traced out by the time evolution of a nonlinear dynamic system model- ing the behavior of an optical switch. The com- plicated path never closes on itself and hence the motion never exactly repeats: the trajectory is a "strange attractor." As the three successive magnifications (top right, lower left, lower right) suggest, the intricate detail persists, in slightly modified form, on all length scales. (Courtesy Institute for Physical Science and Technology, University of Maryland) can exhibit incredibly complicated behavior and that this behavior can be quantified is now widely appreciated and is being applied in many fields. Given the generality of mode reduction and the universality of certain as- pects of chaos, the scientific applicability of the concepts of chaotic motion will grow sig- nificantly with each step in unraveling these matters. COMPLEX CONFIGURATIONS AND PATTERN SELECTION When an extended nonlinear system is driven far from equilibrium, the many local- ized coherent structures that typically ap- pear in it can organize into an enormous range of spatial patterns, regular or random. This process is familiar in turbulent fluid flows (note the complex pattern surrouncl- ing the Red Spot in Figure I) in which tem- poral behavior is chaotic, but it also occurs in many other phenomena, ranging from me- soscaTe textures in metallurgy to markings on seashells. The resulting problem of com- plex configurations and pattern selection represents a third paradigm of nonlinearity. At present, this paradigm is being investi ~, r.'^~` Baked ~ l it. ;,;~.~..,'~:`'=a~ a - '1~ ~ _ 48 ~ .~0~ if- 5~ At.... -. . .~. ~=_~.
ORDER, CHAOS, AND PATTERNS: ASPECTS OF NONLINEARITY gated on two levels. The first level is the experimental-mathematical search for com- plicated, anisotropic configurations that go beyond the highly symmetric patterns that have been accessible via tractitional closed form, pencil-and-paper calculations. The second level is the attempt (as in various ex- perimental studies of fluid flows) to deter- mine how they arise dynamically. Nonlinear competitions can determine which particu- lar pattern emerges from the bewildering ar- ray typically explored by the chaotic interac- tion of the individual components. An increasingly tractable instance of pat- tern selection is provided by the behavior of unstable fluid interfaces, where instabilities can give rise to entrainment and to a chaotic mixing layer. There are many examples of this phenomenon. An interface separating fluids moving at different velocities is subject to shear instabilities and, through a process 0.2 -1.0 -0.2 4 ~ -~.2 -1.8 -2.0 '.,.O i ~,~,,~.""""~) .,~.. ,W, ~:~ · - -- ,-, ~ \~, ~ ,~] ................ T=1 Q _ T=2 T=3 49 known as roll-up, leads to wound-up vorti- ces along the surface. The original boundary becomes fully entangled by coherent struc- tures (vortices) in the final state. Figure 3 il- lustrates the complex patterns formed by this shear instability in a case of particular technological importance that was men- tioned earlier, namely, the vortices that occur in the wake of an aircraft. Recently, multiple- scale analytic techniques have been applied to derive approximate phase and amplitucle equations which, in some fairly simple cir- cumstances, can describe the evolution of these patterns. Another important instance of interracial instability, with potential tech- nological implications for metallurgical pro- cesses and crystal growth problems, occurs in phase transitions in supersaturated or metastable media. Here nonuniform growth of the stable phase produces fingers, known as clendrites, which compete, grow irregu Figure 3 Results of a numerical simulation of vortex sheet model for the shear layer that forms in an aircraft wake. The aircraft is flying perpen- dicular to and into the plane of the figure. The wake is shown at four positions downstream from the wing's trailing edge. Computational points are drawn on the left, and an interpolat- ing curve is drawn on the right. Initially, the vor- tex sheet is the straight line segment -1 c x c 1, y = 0, corresponding to the wing's trailing edge. Single-branched wingtip vortices form at the sheet's end points. Double-branched spirals form further inboard due to the effects of de- ployed flaps and the fuselage. The vortices' roll- up and interaction are strongly nonlinear. (Cour- tesy Robert Krasny, Courant Institute, "Compu- tation of Vortex Sheet Roll-up in the Trefftz 2. ~Plane," journal of Fluid Mechanics, in press)
larly, and produce still more complex config- urations and patterns, such as found in snowflakes. To illustrate this interracial instability in a technologically vital context, we note that the displacement of oil by water in an oil res- ervoir sometimes leads to an unstable inter- face. This Saffman-Taylor instability and the resulting viscous fingering are critical to effi- cient oil recovery; consequently geologists, petroleum engineers, theoretical physicists, applied mathematicians, computer scien- tists, and experts from other disciplines have focused intensely on this problem. The spe- cific technical issue is that almost half of the of! deposited in limestone or other porous media is typically unrecovered during ordi- nary oil extraction because it remains stuck in the pores. To recover this oil, a technique called water flooding is used, in which water is injected into the field to force out the oil. The viscous fingering phenomenon often means that nothing is recovered but the in- jected water, slightly polluted by traces of oil. Clearly a full understanding of this effect and ways to control it are of great impor- tance. Recent work of a combined experimental, theoretical, and computational nature has led to a semiquantitative understanding of several specific aspects of this problem. First, laboratory experiments have estab- lished, under controlled conditions, the na- ture of the complex configurations that arise in certain parameter ranges of viscous fin- gering. Figure 4 shows an image of one such configuration in a flat, effectively two-di- mensional cylindrical cell. This branched, complex configuration is a fractal. To esti- mate the fractal dimension, imagine cover- ing the image of the viscous fingering with square cells of side ~ and calculating, for a given I, the number of cells required to cover the object entirely. As the length of the side Z goes to zero, the number of cells required grows as Ilk, where ~ is the fractal dimen- sion. Performing this calulation for the vis- cous finger in Figure 3 gives ~ = 1.70 + 0.05. : it' Aft': ;:~ : in: : ~ ~,n: ~ i: ~ : I: ~ ~' : ::: : :: ~ no: ; :: I: :::: : : : ~: hi, : ~ '~ :) I: ~ : : ~: : ' `: ~ ' Figure 4 A viscous fingering effect observed when water (black) is forced through a circular inlet in the center of a flat, cylindrical Hele Shaw cell originally filled with high-viscosity fluid. The pattern has a re- producible numerical value, measured by several methods, including the one described in the text, for the fractal dimension of 1.70 + 0.05. (Courtesy G. Dac- cord, I. Nittmann, and H. E. Stanley, Physical Review Letters, 56:336, 1986) Hence, this object possesses a fractional di- mension closer to that of a plane surface (d = 2) than to that of a straight line (~1 = I). Sec- ond, in both the viscous fingering and den- dritic growth problems, analytic studies identified an intriguing nonuniqueness to certain features of the pattern selection in the simplest models. Additional physical ef- fects, such as the inclusion of surface ten- sion, were then shown to remove at least in part this nonuniqueness. Although the re- sulting pattern selection problem has not yet been fully solved, exciting recent progress includes an analytic treatment of effects be- yond all orders in perturbation theory. Third, computational simulations have sug- gestecl a number of different models and ap- proaches to the problem. Much further re- search is required, but an accurate and prac- tical procedure for modeling realistic problems now seems possible. Fractals play an essential role in several 50
ORDER, CHAOS, AND PATTERNS: ASPECTS OF NONLINEARITY other areas of practical application of the par- adigm of complex configurations. In an ef- fort to make ceramics tougher that is, able to contain a few large flaws without failing- much interest has focused on fractal crack patterns. These arise primarily from two sources: the voids that develop during the sintering process, and the materials harder thanceramics- forexample, diamond nor- mally used to machine them. Instead of mov- ing straight along the surface of the ceramic in a planar path, the propagating crack takes a more tortuous route if it interacts with some microscopic feature of the ceramic for instance, a second material added to the pri- mary constituent to enhance its toughness. Since the crack will expend more energy in moving out of the plane than it would in propagating unimpeded, it will do less dam- age to the overall ceramic. Interestingly, the fractal dimension of the crack appears to be related to the fracture toughness of the ce- ramic. Electron micrographs of cracks put into silica-nitride ceramics, one of the new high-performance materials being consid- ered for high-temperature, high-stress ap- plications such as engine parts, were used to determine the fractal dimension of the cracks. The higher the fractal dimension, the tougher the ceramic. In certain surface processes, such as roughening, fractal patterns also are ob- served. For these surface fractals, the lower limit of the fractal dimension is 2, character- istic of a perfectly smooth surface, and the upper limit is 3, a surface so rough and con- voluted that it has become a three-dimen- sional object. The complex configuration of these fractal surfaces can be very important, particularly for processes such as chemical catalysis, where in many cases the higher the fractal dimension of the surface, the greater the catalytic effect. Many further interesting and relevant il- lustrations of complex configurations and patterns can be found in nonlinear phenom- ena from virtually all disciplines. In the bio- Togical sphere, the richness of pattern forma 5 lion is particularly evident, from tigers' stripes to human digits. Certain features of the problem of morphogenesis can already be understood from plausible nonlinear mathematical models. The development of convection rolls during the transition to tur- bulence in a fluid heated from below has been extensively studied experimentally and successfully modeled using a combina- tion of computational and analytic tech- niques. On the other hand, understanding the pattern formation seen in fully devel- oped, three-dimensional turbulence re- mains one of the most challenging problems of modern science. Finally, a fascinating class of discrete non- linear dynamic systems, known as cellular automata, exhibit remarkable pattern forma- tion properties and are currently being sub- jected to rigorous mathematical scrutiny. At a more speculative level, these highly dis- crete systems have suggested novel compu- tational algorithms often called lattice-gas models for solving certain continuum non- linear partial differential equations. These al- gorithms may prove especially valuable for computers based on massively parallel archi- tectures, although both their virtues and their limitations require further study. This section has focused only on those par- adigms of nonlinear science that have been most thoroughly developed and explored, but there are clear indications of many other emerging paradigms. Two are particularly exciting. The concept of adaptation refers to nonlinear dynamic systems that adapt or evolve in response to changes in their envi- ronment. Here one crucial aspect is that the nonlinear equations describing the system can themselves be modified on a slow time scale. Among the initial tentative applica- tions of this concept are models for the hu- man immune system and for autocatalytic networks of proteins. A related but some- what distinct concept is often termed con- nectionism and reflects the appealing idea that many simple structures connected to- gether can exhibit complex behavior collec
lively because of the connections. Recent specific instances of this approach include mathematical models called neural net- works. Typically only loosely patterned after true neurological systems, these models are remarkable in their promise for being able to learn behavior from experience. The con- cepts of familiar dynamic systems such as basins of attraction and coexistence of multi- ple stable patterns-have already played a crucial role in interpreting the behavior of these more complex systems. ISSUES, RECOMMENDATIONS, AND CONCLUSIONS INTERNATIONAL STANDING OF U.S. WORK Researchers in the United States have played a significant but not dominant role in the recent achievements of nonlinear sci- ence. In particular, they have made uniquely important contributions to the experimental mathematics aspect and also provided sub- stantial insights into the experimental and analytic aspects. Nonetheless, the reception of this work within the U.S. scientific com- munity has not been comparable to that seen elsewhere, especially in the Soviet Union and France. In both those countries long- standing traditions in mathematical physics and applied mathematics have helped to stimulate interest in nonlinear phenomena, and the high level of importance that many leading scientists attach to this enterprise is readily noticed in their public comments and in their contributions to the field, particu- larly in the analytic and experimental areas. Since connections with the active French groups are fairly well established, special emphasis should be placed on strengthening ties with the Soviet efforts, for the United States stands to gain considerably from in- creased interaction with Russian researchers in this area. From the Soviet perspective, the U.S. leadership in experimental mathemat- ics provides a natural quid pro quo. 52 In many other countries, work of the high- est caliber has been accomplished, and in some there has already been an institutional response, with centers focusing on nonlin- ear problems established at several universi- ties. It is worth noting that the European Sci- ence Foundation recently hosted a meeting on nonlinear science at which the creation of a major European institute on the subject was discussed. In summary, U.S. research in nonlinear science is of high quality and is widely recog- nized internationally. Ironically, recognition of this subject within the American scientific community is less developed. Indeed, the interdisciplinary character of the field ap- pears to be problematic for U. S. institutions and agencies. In particular, typical U.S. uni- versities, having departmental structures fairly rigidly defined along traditional disci- plines, appear to lack the flexibility to re- spond adequately to this subject. Students, while interested, seem worried (for good reason) about future positions. In general, while there is strong individual motivation, one hardly senses a more communal na- tional one. PERSONNEL Given the interdisciplinary character of nonlinear science, we expect that most of the successful long-term research efforts in this subject will typically result from experts in widely different fields pooling their intellec- tual resources. Accordingly, agencies and academic administrators should consider both the support of loosely coordinated re- search networks and the creation of more fo- cused centers in this area. At the same time, however, since many outstanding contribu- tions can be traced to scientists working es- sentially alone, it is vital to foster and reward high-quality individual research. In particu- lar, the needs of younger scientists eager to become involved but anxious about the lack of a disciplinary base must be confronted. In- creased support for junior faculty, postdoc
ORDER, CHAOS, AND PATTERNS: ASPECTS OF NONLINEARITY toral fellows, and advanced predoctoral stu- dents working on nonlinear problems from an interdisciplinary perspective is clearly necessary. But it is also necessary to ensure the continued input of experts from the tra- ditional disciplines so that studies of nonlin- ear phenomena address significant and rele- vant problems. Several changes in standarct university curricula should be contemplated to bring the excitement of this field to still younger students and to train a cacire of potential re- searchers. With closed-form analyses of in- teresting nonlinear phenomena infrequent and inadequate, an increased comprehen- sion of the schemes of analysis and calcula- tion is required and the general level of mathematical and computer literacy of all natural-science students should be raised. Coursework in differential equations should inclucle more modern dynamic-systems ideas; calculus should more regularly be fol- lowed by deeper courses in analysis; me- chanics courses should stress the limitations of perturbation theory and the omnipres- ence of nonintegrability. A course in numeri- cal methods that leads to intuitive algorithm development based on deep understanding could prepare a researcher to perform mean- ingful experimental mathematics. Greater exposure should be given to topics such as modern asymptotic and multiple-scale methods, phase and amplitude equations derived from fluicts, specific examples of sol- vable soliton equations, and methods of nu- merical analysis. Fluids and continuum me- chanics should be given higher profiles in physics curricula, and introductory courses in the qualitative phenomenology of chaos and solitons and other nonlinear waves should be generally available. Further, sum- mer institutes focused on specific aspects of nonlinear science should be supported. FAC ILITIE S With respect to facilities, one of the major administrative opportunities is the creation 53 of research centers of excellence, either in in- stitutions with preexisting efforts or in re- sponse to new proposals. Crucial to this ap- proach is the provision of block or umbrella funding for the interdisciplinary research, rather than balkanization of the research by dividing support among specific disciplines. Again, however, we stress that grants sup- porting fundamental research by outstand- ing individuals in this area should be avail- able. These grants should have one or more natural homes within the organizational structures of the federal funding agencies, and special care should be taken that they are not endangered by their interdisciplinary content. On a much grander scale, perhaps one of the proposed National Science Foun- dation science and technology centers couIc! be devotect to this subject; given its interdis- ciplinary nature and broad applicability, this may be an attractive prospect. The central role of computation in nonTin- ear science clearly suggests that increased access to supercomputers at the National Science Foundation centers, the National Center for Atmospheric Research, the Na- tional Aeronautics and Space Administra- tion, the Department of Energy laboratories, and elsewhere is vital for continued progress. in particular, interagency coopera- tion in enhancing supercomputer access is essential. But apart from supercomputer ac- cess, individual researchers must be given high-powered scientific work stations with interactive graphics capabilities and a more truly interactive environment. in this matter theorists doing experimental mathematics really do neect to be regarcled as experimen- talists and supported accordingly with the appropriate hardware. Although funding agency awareness of this situation has grown dramatically over the past five years, still greater support is needed. CONCLUSIONS As a consequence of its fundamental intel- lectual appeal and potential technological
applications, nonlinear science is currently experiencing a phase of very rapid growth. During this critical period, science adminis- trators in government, education, anct in- dustry can play essential roles in further stimulating and guiding this growth. In par- ticular, they can marshal the resources nec- essary to respond to the challenging re- search opportunities. in any effort to guide this research, however, it is imperative that nonlinear science be recognized for what it 54 is: an inherently interctisciplinary effort not suited to confinement within any single con- ventional ctiscipline or department. Hence the administrative structure of research in this area is likely to remain more fragile, and in greater need of attention, than traditional subjects with their natural constituencies. Accompanying this fragility, however, is a remarkable breadth of application and the potential to influence both our basic under- standing of the world and our daily life.
