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Report of the
Research Bnefing Panel on
Order, Chaos, and Patterns:
Aspects of Nonlinearity

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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

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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:

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· "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

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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

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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

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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

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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

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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

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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.... -.
. .~.
~=_~.

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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)

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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

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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

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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

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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

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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.