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11
Nonlinear Dynamics,
Instabilities, and Chaos
INTRODUCTION
A new area of condensed-matter physics has been defined as a
coherent subfield in the past 10 years. When solid or fluid systems are
forced away from equilibrium, they often become strongly nonlinear
and as a result exhibit instabilities leading to chaotic or nonperiodic
time evolution. In other cases, nonequilibrium systems develop pat-
terned but nonperiodic spatial structures. From a general point of view,
researchers in this field are concerned with the problem of predictabil-
ity in disordered physical systems. Under what conditions can we hope
to predict the future behavior of a system that is evolving chaotically,
or the spatial structure of one that is highly fractured? Is any predict-
ability preserved when a system behaves in an apparently irregular
way? The process of obtaining answers to these questions should have
a significant impact on fields of science extending far beyond con-
densed-matter physics.
One focus of experimental activity in this field has been the problem
of understanding fluid instabilities (pattern changes) and turbulence,
subjects of both fundamental interest and practical importance because
of the simplicity of fluids and their ubiquity in nature. However,
chaotic dynamics are increasingly being found in other areas of
condensed-matter physics, such as superconducting devices and elec-
tronic conduction in semiconductors. Phenomena exhibiting complex
215
OCR for page 216
216 A DECADE OF CONDENSED-MATTER PHYSICS
spatial patterns include the aggregation of smoke particles and the
dendritic growth of crystals.
One of the defining characteristics of this subfield is a strong
interface with the area of mathematics known as dynamical systems
theory, which provides a language and set of concepts for understand-
ing chaotic dynamics as a consequence of nonlinear models with only
a few degrees of freedom. This approach to understanding irregular
behavior is different from the conventional viewpoint of statistical
mechanics. It represents an expansion of the theoretical tools available
to condensed-matter physics. The eventual impact of these new
mathematical tools will probably be greater than that of the specific
physical systems to which they are currently being applied. Although
the discussion in this chapter emphasizes nonlinear dynamics in the
context of instabilities and chaotic motion in fluids and other con-
densed-matter systems, a much broader approach could be taken, in
which the methods of nonlinear dynamics are applied to a wide range
of interdisciplinary problems in plasma physics, astrophysics, and even
accelerator technology.
Physicists working on nonlinear problems sometimes publish in
engineering or mathematical journals, and new journals devoted spe-
cifically to nonlinear phenomena have been started. The interdiscipli-
nary nature of nonlinear dynamics creates both opportunities for
collaborations and special problems in structuring and funding re-
search. These problems are different from those of the more estab-
lished subfields of condensed-matter physics.
MAJOR ADVANCES
A New Paradigm
We now know that chaotic motion in physical systems that dissipate
energy can arise from nonlinear effects alone: that is, apparently
random behavior arises from the internal dynamics of the system rather
than from irregular external influences. The first extensive discussion
of chaotic motion of this type in a mathematical model was due to
Lorenz in 1963. Since that time, a new mathematical language and way
of thinking about nonlinear dynamics has been developed by mathe-
maticians and (more recently) physicists.
The central concept is that of a state or phase space in which the
state of a system is represented by a point, and its evolution appears as
an extended trajectory. The trajectories form geometrical shapes or
attractors in state space (see the discussion below in the section on
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NONLINEAR D YNAMICS, INSTABl LI TIES, AND CHA OS 2 17
Dynamical Systems Analysis of Experiments) whose forms provide a
way of characterizing the actual behavior of the physical system. For
example, closed loops in state space describe periodic oscillations;
attractors having the topology of a torus describe more complex oscil-
lations termed quasi-periodic; and forms known as strange attractors
describe chaotic motion. A strange attractor is a set on which trajec-
tories wander erratically. They are generally extremely complex; a
strange attractor often has the form of an infinitely folded sheet of
infinite extent.
The description and explanation of the dynamics of chaotic systems
is facilitated by a geometrical analysis of the shapes formed by the
trajectories in state space. This dynamical systems analysis has been
shown to be useful in unifying many diverse physical phenomena,
because only a limited number of fundamentally different forms or
types of attractors seem to be important experimentally. The develop-
ment of tools and language for quantifying the properties of attractors
has given physicists a powerful new way of thinking about the
dynamics of nonlinear systems.
New Experimental Methods
New experimental methods were necessary to allow the ideas of
nonlinear dynamics to be tested. Most importantly, laboratory com-
puter techniques have been essential for experimental studies of
complex dynamics. Automated data acquisition was required to obtain
the large quantity of data needed for analysis. Computer methods have
been needed not only to acquire data but also to analyze it. For
example, space and time Fourier transform methods, which reveal the
frequency content of the fluctuations, have been used extensively.
Numerical methods of measuring the shapes and properties of strange
attractors have been devised. Computer-enhanced shadowgraph im-
ages reveal flow patterns too feeble to be observed by ordinary
photographic or visual techniques (an example is shown in Figure
11.11. These methods are still being refined, and are gradually narrow-
ing the gap between the abstract mathematical ideas of nonlinear
dynamics and the behavior of physical systems.
Routes to Chaos
How is a chaotic fluid flow reached as it is forced more and more
strongly? The stress on the system is characterized by a dimensionless
control parameter (Reynolds number or Rayleigh number, for example,
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218 A DECADE OF CONDENSED-MATTER PHYSICS
''~7~ niU.i3~3-Set ayt_-
FIGURE 11.1 Computer-enhanced shadowgraph images of convective flow in a
container of circular cross section. The concentric flow state (a) was rendered unstable
by reducing the Rayleigh number from 1.2 times the critical value Rt to about 1.1 times
R.`.. Part of the resulting evolution of the flow field is given by bed. (Courtesy of
G. Ahlers.)
depending on the type of flow). Until about 1970, an endless sequence
of instabilities was expected, each of which changes the spatial
structure (pattern) of the flow or adds a new frequency of oscillation.
In this picture, turbulent flow was essentially a complicated
superposition of motion at many frequencies simultaneously. No
qualitative change separated laminar and turbulent motion in this view.
In 1971 a suggestion was made that turbulent flows could actually be
modeled by strange attractors having only a few degrees of freedom.
The chaotic dynamics of such models are dramatically different from a
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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 219
quasi-periodic flow with several frequencies and also quite different
from chaos produced by many interacting variables. Considerable
effort has been devoted to the testing of this hypothesis.
Over the next few years (1974-1982) many groups explored and
characterized the various sequences of instabilities leading to chaotic
fluid motion that are obtained as the control parameter is varied. A
limited number of well-defined routes to chaos were identified:
1. Period doubling, in which the basic period of oscillation doubles
repeatedly at a sequence of thresholds forming a geometric series. This
phenomenon is illustrated in Figure 1 1.2.
2. Quasi-periodicity, in which chaos is produced by several inter-
acting oscillations at incommensurate frequencies.
time.
3. Intermittency, in which laminar and chaotic phases alternate in
These basic routes to chaotic motion were found both theoretically
in simple models and experimentally in a limited class of fluid systems
where the spatial pattern of the flow is fixed and the transition occurs
continuously. We use the term chaos (or sometimes weak turbulence)
to describe the resulting noisy time-dependent motion. The flow is not
necessarily turbulent in the ordinary engineering sense because it may
not contain eddies of many different spatial scales. Many of the same
routes have also been observed in pen junction oscillators, semicon-
ductor transport phenomena, superconducting Josephson junctions,
chemical reactions, and elsewhere. These routes to chaos have univer-
sal properties that transcend the characteristics of particular systems.
The list given above is by no means exclusive; other routes occur in
model systems and may be identified experimentally.
Dynamical Systems Theory of the Routes to Turbulence
The basic sequences of instabilities or transitions found experimen-
tally are now fairly well understood theoretically. A set of theoretical
methods known as renormalization-group techniques (developed orig-
inally to describe continuous phase transitions in equilibrium systems)
were crucial in understanding the universal properties of the basic
routes to turbulence. The applicability of these methods in the new
context depends on the fact that the internal structure of strange
attractors often becomes identical (more precisely, self-similar) on all
length scales as the transition to chaos is approached.
Particularly noteworthy theoretical developments include prediction
of the properties of the period doubling, quasi-periodicity, and
OCR for page 220
220 A DECADE OF CONDENSED-MATTER PHYSICS
R
Rc
3.47
3.52
3.62
3.65
0 50
100 150 200
T(sec)
FIGURE 11.2 Sequence of period doublings leading to chaos in Rayleigh-B(nard
convection, as a function of the Rayleigh number R (proportional to the temperature
difference across the fluid layer). The temperature variation is periodic' and the period
doubles repeatedly as R is increased. For example, at RIRt. = 3.62, a complete cycle
includes four large peaks. (Courtesy of A. Libchaber.)
intermittence routes near the onset of chaos and the achievement of an
understanding of the effect of external noise on these phenomena.
These efforts have revealed new constants of nature and unexpected
mathematics. The discovery that the routes to chaos have universal
properties is a particularly exciting development and accounts in part
for the speed with which nonlinear dynamics (applied especially but
not exclusively to fluid motion) has been accepted as a part of
condensed-matter physics.
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NONLINEAR D YNAMICS, INSTABILITIES, AND CHAOS 22 1
Dynamical Systems Analysis of Experiments
The various routes to turbulence do not directly expose the proper-
ties of the strange attractors that characterize chaotic systems. In fact,
the goal of understanding strange behavior has not yet been achieved,
in part because it has only recently become possible to observe the
attractors directly. Whereas earlier work emphasized the use of
frequency spectra, one can now use geometrical methods to study the
forms of strange attractors in phase space and to measure their
properties. An example of the shape of trajectories in phase space near
the onset of chaotic fluid flow is shown in Figure li.3. When the
FIGURE 11.3 Trajectories in phase space (left) and Poincare sections (right) for a
rotating fluid below and above the onset of chaos. The flow is quasi-periodic in the lower
panels and chaotic in the middle and upper panels. (Courtesy of H. L. Swinney.)
OCR for page 222
222 A DECADE OF CONDENSED-MATTER PHYSICS
trajectories are shown in the two-dimensional phase space, they are
smooth in the laminar state (lower left) and convoluted in the chaotic
state (upper left). By observing the orbits in three dimensions and then
allowing them to intersect a plane, the Poincare maps on the right are
obtained. The lower one (a closed loop) is the intersection of the torus
with a plane. The upper one, where the flow is chaotic, is irregular
because the torus has been replaced by a strange attractor. The middle
case is just beyond the onset of the chaotic regime.
One of the most important properties of a strange attractor is its
dimensionality, which can be fractional. This surprising property arises
because it often consists of an infinite number of closely spaced
surfaces, yielding a dimension greater than that of a single surface but
lower than that of a solid object, i.e., a dimension between two and
three. A second important property is the rate at which nearby
trajectories diverge from each other along certain directions within the
attractor. This property of exponential divergence of nearby trajecto-
ries or sensitive dependence on initial conditions is responsible for the
chaotic behavior.
Preliminary measurements of these two properties in experiments
have provided solid evidence for the basic correctness of the dynamical
systems approach to understanding the onset of chaos in a limited class
of fluid-flow transitions. We now know that chaotic motion in a system
having 1093 degrees of freedom (i.e., a fluid) can be effectively
described in some cases by models having only a very few degrees of
freedom (variables).
Nonlinear Stability Theory
The dynamical systems approach described in the last few sections
is particularly useful for systems in which the spatial patterns of the
flow are fixed (for example, convection in a layer whose lateral
dimensions are small enough). However, in many situations (including
those of greatest practical importance) this is not the case: time
dependence is accompanied by large variations in the spatial pattern of
the flow. One widely applicable theoretical approach that has proven
successful in describing pattern changes is known as stability theory.
In this method, one examines the linear stability of the static solutions
of the nonlinear equations to infinitesimal perturbations. A consider-
able amount has been learned about various types of secondary
instabilities that modify the primary flow. Encouraging progress has
also been made in understanding the elects of the system boundaries
on the stability of the motion.
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NONLINEAR D YNAMICS. INSTABlLITlES, AND CHAOS 223
Pattern Evolution
Studies of the evolution of spatial flow patterns resulting from
instabilities are an important step in understanding the onset of chaotic
How. During the past few years, an increasing effort has been devoted
to studies of the evolution of flows, especially convective flows, by
laser Doppler mapping and other methods. Particular attention has
been directed to the effect of defects or dislocations on the evolution of
the flow patterns.
Theoretical efforts have proceeded in parallel with the experiments.
Most of the important qualitative phenomena seem to be predictable on
the basis of two-dimensional model equations that are far simpler than
the full three-dimensional hydrodynamic equations. These models are
used to describe approximately the slow spatial variations of the basic
flow structure. Numerical computations based on such model equa-
tions have been carried out successfully, and an example is shown in
Figure 11.4. One remarkable development is that near the onset of
convection? the evolution may possibly be described by the minimiza-
tion of a quantity that depends on the concentration of defects, the
amount of curvature in the pattern, and the orientation of the flow field
with respect to the boundaries. Though evolution toward a minimum is
surely not a general property of nonequilibrium systems, the usefulness
of equilibrium concepts (approach to a minimum) in this context, even
under restricted circumstances, is an important result.
Many other systems also show fascinating patterns and pattern
selection. The growth of crystals from the melt exhibits complex
dendritic or snowflake forms. The growth of soot particles shows
somewhat less regular, but fundamentally similar, patterned growth.
These spatial structures are, like the attractors of the fluid, strange
objects having fractional dimension.
Various types of random motions can similarly be described in terms
of objects with fractional dimension and can probably be understood
using the language of dynamical systems. For example, the invasive
percolation of a two-phase fluid in a porous medium, various aggrega-
tion processes, and the behavior of electrons lopping in a random
material all show dynamically produced structures of fractional dimen-
sionality.
Instabilities in Other Dissipative Systems
Instabilities and chaos occur in many other condensed-matter sys-
tems. For example, Josephson junction (superconducting) oscillators
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224 A DECADE OF CONDENSED-MATTER PHYSICS
r=50
r = 230
r -1631
r = 13450
FIGURE 11.4 Numerical simulation of convective-pattern evolution based on ~ 2-D
model, starting from random initial conditions. The simplification of the flow into
pattern of textured rolls as time evolves is at least qualitatively similar to the phenomena
observed experimentally.
show chaotic dynamics, a fact that may be important in device
applications. Cooled extrinsic germanium photoconductors exhibit a
series of progressively more complex instabilities with an increasing
applied dc electric field. Niobium triselenide shows periodic and
chaotic fluctuations in its conductivity that are related to the interac-
tion of a charge-density wave with the crystal lattice. The spatial
arrangement of adsorbed atoms on a periodic substrate and the
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NONLINEAR D YNAMICS, INS TABl LI TIES, AND CHAOS 225
electronic energy bands of a periodic crystal in a magnetic field both
show complex structures that can be understood using the methods of
nonlinear dynamics. The dynamics of chemically reacting systems are
chaotic in some cases and in fact show a close correspondence with
low-dimensional dynamic models.
Instabilities are also important in many aspects of the study of phase
transitions between various states of matter. Progress has been made in
studies of the nucleation of first-order phase transitions (i.e., the
thermally activated initiation of a phase transition from a metastable
state) and spinodal decomposition (phase transitions occurring from an
initially unstable state). These processes are highly nonlinear and have
been elucidated recently by light-scattering experiments on systems
undergoing phase separation.
Nonlinear Dynamics of Conservative Systems
The nonequilibrium behavior of condensed-matter systems usually
involves the dissipation of energy, and this means that the models of
interest have attractors onto which the orbits in phase space converge.
However, the nonlinear dynamics of conservative (nondissipative)
systems has closely related properties and also gives rise to chaotic
behavior. In this section, we briefly note that the history of this field is
much older than is generally appreciated.
For 200 years after Newton, mathematicians and physicists sought to
integrate Newton's equations and generally assumed that all Newton-
ian systems would eventually be found to be integrable or separable
into equations for each degree of freedom and therefore to behave
smoothly and predictably. However, Maxwell pointed out a century
ago that the motion of a gas of hard spheres could quickly (in 10-~°
second) be observably altered by a small perturbation and therefore
could not be predictable. Early in its history, the probabilistic nature of
statistical mechanics was realized not to be obviously consistent with
an integrable and predictable Newtonian dynamics. Slightly later, in
his study of the gravitational N-body problem, Poincare proved that
the energy is the only well-behaved constant of the motion for an
isolated system, thereby casting grave doubts on the notion that
Newton's equations could be integrated. Moreover, the wildly erratic
character of the orbits of a system of three bodies led him to remark,
"Determinism is a fantasy due to Laplace."
The nature of chaotic motion in conservative systems has been
elucidated over the years by many distinguished scientists, including
Birkhoff, Gibbs, and Hopf. An especially important concept is that of
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226 A DECADE OF CONDENSED-MATTER PHYSICS
mixing, in which the flow of trajectories on a surface of constant energy
in phase space mixes in the same way as the fluid particles in a stirred
martini do. We now know that there are many conservative systems
that are not only mixing but also truly chaotic in the sense that the final
state is exponentially sensitive to changes in the initial conditions.
Newtonian dynamics is thus quite rich; it contains the predictable
systems of textbook physics as well as the fully chaotic, random
systems envisioned by statistical mechanics. How can one go from one
to the other? The much celebrated Kolmogorov-Arnol'd-Moser (KAM)
theorem (1954) showed that small perturbations to integrable systems
leave most orbits nearly unchanged (with suitable qualifications) while
also creating a small set of chaotic orbits. It is this small chaotic set that
can grow, as the conditions of the KAM theorem are violated, to yield
fully chaotic motion.
The final and recent development in conservative dynamics comes
via information theory. The basic notion here is that a chaotic system
or orbit exhibits much greater variety, and therefore contains much
more information, than does an orderly integrable system or orbit. This
variety is sometimes quantified using the concept of metric entropy. If
finite accuracy measurements of an arbitrary observable of the system,
made at regular time intervals from the beginning of time to the
present, do not permit precise prediction of the next measurement, the
system has positive metric entropy, i.e., information is gained from
each measurement. Another concept recently adapted from informa-
tion theory is that of algorithmic complexity, which measures the
unpredictability of individual orbits. An orbit is said to have maximum
algorithmic complexity provided its information content cannot be
encoded in an algorithm simpler or shorter than that which simply
generates a copy of the orbit. Maximum complexity implies random-
ness or chaos, and this concept helps to eliminate the apparent paradox
of random behavior in deterministic systems.
Many connections have been established between the nonlinear
dynamics of conservative systems and the dynamics of dissipative
systems.
General Remarks
The developments described in the preceding sections are only a
small sampling of the activities in nonlinear dynamics. Systems with
instabilities leading to chaotic behavior have been emphasized. The
research has been highly international, with the United States making
major contributions but not dominating the field. Although physicists
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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 227
have added something new to the problem of understanding instabili-
ties and the transition to weak turbulence, the majority of current
research in fluid mechanics is done by the engineering community (and
is described in the report of the Panel on the Physics of Plasmas and
Fluids).
The impact of this research is likely to be substantial because (at the
mathematical methods of nonlinear dynamics are helping to solve
many previously intractable problems in physics; (b) instabilities and
chaotic motion are widespread phenomena with common features; and
(c) the growing understanding of the origins of chaotic behavior and the
limits of predictability contribute to the foundations of physics.
CURRENT FRONTIERS
The field of nonlinear dynamics, instabilities, and chaos is undergo-
ing a period of rapid expansion. A considerable number of important
problems merit attention.
Bifurcation Sequences
As new systems are investigated, it is likely that new distinct
bifurcation sequences will be observed. Furthermore, the universality
of the various bifurcation sequences still needs to be explored in some
cases. For example, how universal is the breakup of a two-torus in real
experiments? Much work remains to be done before the different types
of transition not only those that are mathematically possible but also
those that occur physically are classified. Are there universal phe-
nomena associated with abrupt nonequilibrium transitions? If so, what
are they? These phenomena are reminiscent of, but more complicated
than, those associated with first-order phase transitions.
Little experimental work has been done on systems with two or more
control parameters (codimension-two bifurcations). Since theorists
have begun to classify the types of bifurcations that can occur in such
systems, we anticipate that this will be a fruitful area for further
experimental work.
The problem of connecting the universal phenomena near the chaotic
transition with the nonuniversal behavior farther away is a challenge
for the future. We do not yet know how to predict in advance what
bifurcation sequence will occur in a given physical situation. At
present, we can only say that if there is period doubling (for example),
it must have certain asymptotic properties.
The effects of sequences of instabilities on heat, mass, and momen
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228 A DECADE OF CONDENSED-MATTER PHYSICS
tum transport in fluids have not been adequately studied and could
have practical applications (for example, in the problem of lubrica-
tion).
Patterns
There are a large number of general questions associated with the
formation of patterns in condensed-matter systems. How are patterns
selected? Under what conditions, if any, is there a functional whose
extreme describe selected steady states? Under what circumstances
are these states sensitive to initial conditions? To boundary conditions?
Conversely, how can one understand the apparent inevitability of some
patterns, e.g., dendritic structures? How can one understand the
selection of quantized modes (for example, in Taylor-Couette flow)?
Can group-theoretical methods be useful in understanding or classify-
ing possible patterns?
What is the role of noise in pattern selection? Can one understand
pattern selection in some cases as being caused by competition
involving attraction to a large number of steady states? Is the system
driven from one state to another by external or dynamical noise? If so,
what is the source of this noise? Nonlinear theories quantifying the
strength of attraction to different attractors may eventually have to be
developed.
Pattern formation is a particularly interdisciplinary problem, and
physicists can benefit from interaction with metallurgists, geologists,
meteorologists, biologists, and chemists. All of these groups of scien-
tists deal with the production of intricate patterns and with the pro-
gression from patterned to chaotic behavior. Can we begin to see
general rules and approaches that cut across disciplines? For example,
pattern selection in hydrodynamic systems has conceptual similarities
to nucleation in the physics of phase transitions. Can these similarities
be used to understand these processes better? In general, the process
of pattern selection is likely to become much more recognized than it
has been in the past for its discriminatory power to reveal mechanisms.
Numerical Simulations
Analysis will (probably) continue to provide insight into behavior
just beyond the onset of instabilities and will play an important role in
understanding pattern selection. However, the difficulty of extending
nonlinear stability theory beyond the second instability is clear.
Accurate simulations have recently been achieved for Couette flow in
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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS
229
the regime beyond the second instability. These simulations are dif-
ficult and require about 105 spectral modes. An important problem for
the future is the simulation of the onset and growth of turbulence,
which will require even more modes and larger computers than those
now available. When properly carried out, simulations have the power
to help us understand the physics of these difficult nonlinear problems.
Numerical studies of low-dimensional models will also be important
in the future. Much of what we know about nonlinear dynamics comes
from numerical calculations, and this will continue to be the case,
especially as systems with several control parameters are investigated.
Experimental Methods
Now that the usefulness of phase-space methods has been demon-
strated, we anticipate that a great deal of attention will be given to the
development of reliable ways of measuring such quantities as the
dimensionality, Lyapunov exponents, and metric entropy of strange
attractors. One important goal of this work is to understand chaotic
behavior significantly above its onset and to begin to understand the
relationship between chaotic time dependence and spatial variations.
Experimenters are finding that increasingly precise methods are
required to answer the most interesting questions. Laser Doppler
techniques can be made more precise by using correlation techniques;
digital imaging methods will make it possible to study time dependence
and spatial structure simultaneously.
We anticipate that cryogenic methods will continue to be important
in this field. For example, dilute solutions of 3He in 4He are already
useful in the study of double-diffusive convection (where both concen-
tration and temperature are important), and there is some evidence that
at low temperatures these solutions may convect as classical single-
component fluids with novel properties.
Transition from Weak to Fully Developed Turbulence
The onset of chaos in large hydrodynamic systems is not well
understood. It is important to study the loss of spatial correlation and
the evolution of time dependence with increasing Reynolds (or
Rayleigh) number. Such studies will require multiple probes, digital
imaging techniques, and ever bigger computers.
Theory and experiment on fluid instabilities have emphasized the
onset of turbulence. There has also been much work (especially in the
engineering community) on strong turbulence. However' it is not
OCR for page 230
230 A DECADE OF CONDENSED-MATTER PHYSICS
known how a weakly turbulent flow at low Reynolds number evolves
into fully developed turbulence at high Reynolds number. Is the growth
in the dimension of the strange attractor slow or fast? Continuous or
discontinuous? How many independent degrees of freedom are re-
quired to describe fully developed turbulence (102 or 10'°~? At what
point does the dynamical systems approach cease to be helpful? These
exciting questions will certainly be addressed in the next few years.
Eventually, it will be important to understand the asymptotic prop-
erties of real turbulence better. Are statistical averages unique at high
Reynolds number? Theoreticians will have to learn how to extract
information about averages from manifolds of solutions. Combinations
of computer simulations and analytical boundary-layer approximations
will, it is hoped, bring us closer to an understanding of turbulent fluid
flow and turbulent convection in particular. Perhaps theories based on
optima or bounded quantities will be fruitful.
It will be important to understand the relationship between classical
turbulence and quantum turbulence (in superfluid helium). The latter is
thought to consist of a tangled mass of quantized vortex lines. Since the
circulation about a line is strictly quantized, perhaps quantum turbu-
lence may ultimately be easier to understand than classical homoge-
neous turbulence. The resulting knowledge is likely to be of engineer-
ing importance in the cooling of superconducting devices.
Conservative Systems
Despite the deep contribution to science made by the study of
integrable systems, these are' loosely speaking, no more common than
integers on the line. The richness and variety of Newtonian mechanics
has only begun to be sampled. The full physical significance of
many-body problems that are integrable (or KAM near-integrableJ and
that, therefore, do not obey the laws of thermodynamics is ripe for
development. The possibility that all Newtonian systems having null
metric entropy are analytically solvable (although they may be ergodic
and mixing) has only begun to be investigated. The full and rigorous
development of the statistical mechanics of conservative and dissipa-
tive chaotic systems, including the approach to equilibrium, remains an
uncompleted task. The transition from complete integrability to full
chaos involves a divided phase space exhibiting chaos surrounding
integrable islands of stability. Do these islands decrease in size but
never disappear as the energy of the system increases'? Does their
presence have physical consequences for the decay of correlations, as
recent numerical experiments indicate?
What is the ultimate significance of the fact that initial errors in
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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 231
measurement or computation grow exponentially in a chaotic system?
To observe the smallest details of a chaotic orbit requires unlimited
precision; to predict these details analytically requires the manipula-
tion of infinite algorithms, which, strictly speaking, cannot even be
defined. One thus begins to suspect that the study of chaos will provide
an especially deep physical meaning to Godel's theorem and the
limitation it implies for human logical systems, including science.
Although a final resolution to these matters lies in the future, algorith-
mic complexity theory has already established that most real numbers
in the continuum are, humanly speaking, undefinable.
The problem of long-time prediction for chaotic systems is especially
interesting. Using numerical as well as analytic arguments, numerous
scientists have suggested that long-time prediction is impossible. Time
complexity theory is currently being used to ask how quickly a solution
algorithm can predict the future of a dynamical system, i.e., how much
time is required to compute the behavior, in order to obtain insight into
this question of predictability.
Finally, there is the little explored question of whether chaos
remains a meaningful concept in quantum mechanics. At present, there
is a growing amount of convincing evidence indicating that the quantum
mechanics of finite bounded systems contains no chaotic time depen-
dence. This leads to the deep suspicion that the quantum description of
a classical chaotic system does not tend to the proper classical limit;
much worse, it may be that quantum mechanics was tacitly constructed
on the notion of intergrability. If so, the inclusion of chaos may require
fundamental changes in the foundations of quantum mechanics.
Nonequilibrium Systems
There are a number of important fundamental questions having to do
with the fact that systems undergoing instabilities are generally away
from thermodynamic equilibrium. How does one characterize and
classify the steady states that are obtained under constant external
conditions away from thermodynamic equilibrium? (It is generally
assumed that such steady states have simpler behavior than arbitrary
nonequilibrium states.) Are there general techniques for classifying the
spatial and temporal patterns that emerge in such systems? What are
the essential similarities and differences between externally driven
nonequilibrium steady states, such as cellular convection or oscillatory
chemical reactions, and freely equilibrating systems such as those
undergoing nucleation or spinodal decomposition (i . e ., phase
changes)?
There is still no really fundamental understanding of the mechanisms
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232 A DECADE OF CONDENSED-MATTER PHYSICS
that control solidification patterns in, for example, snowflakes,
quenched alloys, or directionally solidified eutectic mixtures. Some
progress has been made in developing a stability-based theory of free
dendritic growth in pure substances and dilute solutions, and it seems
possible that a firm mathematical basis is being established for work
along these lines. The problems of predicting periodicities of cellular
solidification fronts or lamellar eutectics appear to have much in
common with pattern-selection problems in Rayleigh-Benard convec-
tion or chemical-reaction diffusion systems. Despite a long history of
metallurgical investigation, the solidification problem is relatively
underexplored experimentally. There is a great need for precise
measurements of morphological response to varying growth conditions
in simple, carefully characterized model systems.
In the case of phase transitions involving nucleation or spinodal
decomposition, the major opportunity now is to use neutrons, synchro-
tron radiation, and other modern methods to make real-time observa-
tions of nonequilibrium processes in alloy solids, polymers, and similar
materials. How do states of frozen equilibrium, like glasses and spin
(magnetic) glasses, fit into the picture? Can they be understood via
dynamical systems methods?
New Directions
It is important to recognize that this field is evolving rapidly and will
find application to many other systems. Instabilities in semiconductors
are important for device applications. The nonlinear current-voltage
characteristics of a great variety of materials can lead to dynamical
instabilities. Chaos is known to occur in liquid-crystal instabilities. The
properties of magnetic fluids, liquid metals, and dielectric fluids are
also likely to result in important nonlinear phenomena. The theoretical
methods of nonlinear dynamics have already found application in
models of systems with competing length scales (commensurate and
incommensurate phases) in condensed-matter systems. Although sys-
tems studied in the past have been at equilibrium, nonequilibrium
systems with competing periodicities are also important and are now
being investigated for the first time. Finally, there are many astrophys-
ical problems (for example, pulsating stars and geomagnetic reversals)
for which the methods of nonlinear dynamics may prove useful.
There are a number of important biological applications of nonlinear
dynamics, especially to neural networks. The idea that memories can
be represented by attractors in a dynamical system is an appealing
concept that may be capable of explaining some of the important
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NONLINEAR DYNAMICS. INSTABILITIES, AND CHAOS 233
properties of memory, such as the ability to recall a great deal of
information starting with a small part of it. Another neurological
application of nonlinear dynamics is the explanation of certain types of
neural excitations (e.g., hallucinations) in terms of symmetry-breaking
instabilities in the neural network. Research on networks is closely
related to advances in computer science, where efforts are being made
to apply nonlinear dynamics to the behavior of computing machines.
Phenomena in which fractal structures occur in real space rather
than in phase space are being actively studied. Examples include
diffusion-limited aggregation and the behavior of a two-phase fluid in a
porous network. These structures have patterns on all length scales
and can be understood by methods that exploit this scale invariance.
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Appendixes
Representative terms from entire chapter:
phase space
