an Article from**
SCIENTIFIC AMERICAN**

DECEMBER, 1986 VOL. 254 NO. 12, 46-57.

Chaos

*There is order in chaos:
randomness has an underlying geometric form. Chaos imposes
fundamental limits on prediction, but it also suggests causal
relationships where none were previously suspected*

by James P. Crutchfield, J. Doyne Farmer, Norman H. Packard, and Robert S. Shaw

The great power of science lies in the ability to relate cause and effect. On the basis of the laws of gravitation, for example, eclipses can be predicted thousands of years in advance. There are other natural phenomena that are not as predictable. Although the movements of the atmosphere obey the laws of physics just as much as the movements of the planets do, weather forecasts are still stated in terms of probabilities. The weather, the flow of a mountain stream, the roll of the dice all have unpredictable aspects. Since there is no clear relation between cause and effect, such phenomena are said to have random elements. Yet until recently there was little reason to doubt that precise predictability could in principle be achieved. It was assumed that it was only necessary to gather and process a sufficient amount of information.

Such a viewpoint has been altered by a striking discovery: simple deterministic systems with only a few elements can generate random behavior. The randomness is fundamental; gathering more information does not make it go away. Randomness generated in this way has come to be called chaos.

A seeming paradox is that chaos is deterministic, generated by fixed rules that do not themselves involve any elements of chance. In principle the future is completely determined by the past, but in practice small uncertainties are amplified, so that even though the behavior is predictable in the short term, it is unpredictable in the long term. There is order in chaos: underlying chaotic behavior there are elegant geometric forms that create randomness in the same way as a card dealer shuffles a deck of cards or a blender mixes cake batter.

The discovery of chaos has created a new paradigm in scientific modeling. On one hand, it implies new fundamental limits on the ability to make predictions. On the other hand, the determinism inherent in chaos implies that many random phenomena are more predictable than had been thought. Random-looking information gathered in the past—and shelved because it was assumed to be too complicated—can now be explained in terms of simple laws. Chaos allows order to be found in such diverse systems as the atmosphere, dripping faucets, and the heart. The result is a revolution that is affecting many different branches of science.

What are the origins of random behavior? Brownian motion provides a classic example of randomness. A speck of dust observed through a microscope is seen to move in a continuous and erratic jiggle. This is owing to the bombardment of the dust particle by the surrounding water molecules in thermal motion. Because the water molecules are unseen and exist in great number, the detailed motion of the dust particle is thoroughly unpredictable. Here the web of causal influences among the subunits can become so tangled that the resulting pattern of behavior becomes quite random.

The chaos to be discussed here requires no large number of subunits or unseen influences. The existence of random behavior in very simple systems motivates a reexamination of the sources of randomness even in large systems such as weather.

What makes the motion of the atmosphere so much harder to
anticipate than the motion of the solar
system? Both are made up of many
parts, and both are governed by Newton's second
law, *F = m a*, which can
be viewed as a simple prescription for
predicting the future. If the forces *F
*acting on a given mass *m* are known,
then so is the acceleration *a.* It then
follows from the rules of calculus that
if the position and velocity of an object
can be measured at a given instant,
they are determined forever. This is
such a powerful idea that the 18th-century
French mathematician Pierre Simon de Laplace
once boasted that given the position and velocity of every
particle in the universe, he could predict
the future for the rest of time. Although there
are several obvious practical difficulties to achieving Laplace's
goal, for more than 100 years there
seemed to be no reason for his not being right,
at least in principle. The literal application of
Laplace's dictum to human behavior led to the
philosophical conclusion that human behavior
as completely predetermined: free
will did not exist.

**
CHAOS results from the geometric operation
of stretching. The effect is illustrated for a
painting of the French mathematician Henri Poincaré,
the originator of dynamical systems theory.
The initial image (top left) was digitized
so that a computer could perform
the stretching operation. A simple mathematical
transformation stretches the image diagonally
as though it were painted on a sheet of rubber.
Where the sheet leaves the box it is
cut and reinserted on the other side,
as is shown in panel 1.
(The number above each
panel indicates how many times the transformation
has been made.) Applying the transformation repeatedly
has the effect of scrambling the face (panels 2-4).
The net effect is a random combination of colors,
producing a homogeneous field of green (panels 10 and
18). Sometimes it happens that some of the points come
back near their initial locations,
causing a brief appearance of the original image
(panels 47-48, 239-241). The transformation shown here
is special in that the phenomenon of “Poincaré recurrence”
(as it is called in statistical mechanics) happens much more often
than usual; in a typical chaotic transformation recurrence is
exceedingly rare, occurring perhaps only once in the lifetime
of the universe. In the presence of any amount of background
fluctuations the time between recurrences is usually so long
that all information about the original image is lost.**

Twentieth-century science has seen the downfall of Laplacian determinism, for two very different reasons. The first reason is quantum mechanics. A central dogma of that theory is the Heisenberg uncertainty principle, which states that there is a fundamental limitation to the accuracy with which the position and velocity of a particle can be measured. Such uncertainty gives a good explanation for some random phenomena, such as radioactive decay. A nucleus is so small that the uncertainty principle puts a fundamental limit on the knowledge of its motion, and so it is impossible to gather enough information to predict when it will disintegrate.

The source of unpredictability on a large scale must be sought elsewhere, however. Some large-scale phenomena arc predictable and others are not. The distinction has nothing to do with quantum mechanics. The trajectory of a baseball, for example, is inherently predictable; a fielder intuitively makes use of the fact every time he or she catches the ball. The trajectory of a flying balloon with the air rushing out of it, in contrast, is not predictable; the balloon lurches and turns erratically at times and places that are impossible to predict. The balloon obeys Newton's laws just as much as the baseball does; then why is its behavior so much harder to predict than that of the ball?

**
OUTLOOKS OF TWO LUMINARIES on chance and probability are contrasted.
The French mathematician Pierre Simon de Laplace proposed that the
laws of nature imply strict determinism and complete predictability,
although imperfections in observations make the introduction of
probabilistic theory necessary. The quotation from Poincaré
foreshadows the contemporary view that arbitrarily small uncertainties
in the state of a system may be amplified in time and so predictions
of the distant future cannot he made.**

**Laplace, 1776**

**
“The present state of the system of nature
is evidently a consequence of what it was in the
preceding moment, and if we conceive of an intelligence
which at a given instant comprehends all the relations of the
entities of this universe, it could state the respective
positions, motions, and general affects of all these
entities at any time in the past or future.**

**
“Physical astronomy, the branch of knowledge which
does the greatest honor to the human mind, gives us an
idea, albeit imperfect, of what such an intelligence would
be. The simplicity of the law by which the celestial
bodies move, and the relations of their masses and distances,
permit analysis to follow their motions up to a certain point;
and in order to determine the state of the system of these
great bodies in past or future centuries, it suffices for
the mathematician that their position and their
velocity be given by observation for any moment in time.
Man owes that advantage to the power of the instrument he
employs, and to the small number of relations that it embraces
in its calculations. But ignorance of the different causes
involved in the production of events, as well as their
complexity, taken together with the imperfection of
analysis, prevents our reaching the same certainty about
the vast majority of phenomena. Thus there are things that
are uncertain for us, things more or less probable, and
we seek to compensate for the impossibility of knowing
them by determining their different degrees of likelihood. So it
is that we owe to the weakness of the human mind one of the most
delicate and ingenious of mathematical theories, the science
of chance or probability.”**

**Poincaré, 1903**

**
“A very small cause which escapes our notice determines
a considerable effect that we cannot fail to see, and then
we say that the effect is due to chance. If we knew exactly
the laws of nature and the situation of the universe at the
initial moment, we could predict exactly the situation of that
same universe at a succeeding moment. But even if it were the
case that the natural laws had no longer any secret for us,
we could still only know the initial situation approximately.
If that enabled us to predict the succeeding situation with
the same approximation, that is all we require, and we
should say that the phenomenon had been predicted, that it is
governed by laws. But it is not always so; it may happen that
small differences in the initial conditions produce very great
ones in the final phenomena. A small error in the former will
produce an enormous error in the latter. Prediction becomes
impossible, and we have the fortuitous phenomenon.”**

The classic example of such a dichotomy is fluid motion. Under some circumstances the motion of a fluid is laminar—even, steady, and regular— and easily predicted from equations. Under other circumstances fluid motion is turbulent—uneven, unsteady, and irregular—and difficult to predict. The transition from laminar to turbulent behavior is familiar to anyone who has been in an airplane in calm weather and then suddenly encountered a thunderstorm. What causes the essential difference between laminar and turbulent motion?

To understand fully why that is such a riddle, imagine sitting by a mountain stream. The water swirls and splashes as though it had a mind of its own, moving first one way and then another. Nevertheless, the rocks in the stream bed are firmly fixed in place, and the tributaries enter at a nearly constant rate of flow. Where, then, does the random motion of the water come from?

The late Soviet physicist Lev D. Landau is credited with an explanation of random fluid motion that held sway for many years, namely that the motion of a turbulent fluid contains many different, independent oscillations. As the fluid is made to move faster, causing it to become more turbulent, the oscillations enter the motion one at a time. Although each separate oscillation may be simple, the complicated combined motion renders the flow impossible to predict.

Landau's theory has been disproved, however. Random behavior occurs even in very simple systems, without any need for complication or indeterminacy. The French mathematician Henri Poincaré realized this at the turn of the century when he noted that unpredictable, “fortuitous” phenomena may occur in systems where a small change in the present causes a much larger change in the future. The notion is clear if one thinks of a rock poised at the top of a hill. A tiny push one way or another is enough to send it tumbling down widely differing paths. Although the rock is sensitive to small influences only at the top of the hill, chaotic systems are sensitive at every point in their motion.

A simple example serves to illustrate just how sensitive some physical systems can be to external influences. Imagine a game of billiards, somewhat idealized so that the balls move across the table and collide with a negligible loss of energy. With a single shot the billiard player sends the collection of balls into a protracted sequence of collisions. The player naturally wants to know the effects of the shot. For how long could a player with perfect control over his or her stroke predict the cue ball's trajectory? If the player ignored an effect even as minuscule as the gravitational attraction of an electron at the edge of the galaxy, the prediction would become wrong after one minute!

The large growth in uncertainty comes about because the balls are curved, and small differences at the point of impact are amplified with each collision. The amplification is exponential: it is compounded at every collision, like the successive reproduction of bacteria with unlimited space and food. Any effect, no mailer how small, quickly reaches macroscopic proportions. That is one of the basic properties of chaos.

It is the exponential amplification of errors due to chaotic dynamics that provides the second reason for Laplace's undoing. Quantum mechanics implies that initial measurements are always uncertain, and chaos ensures that the uncertainties will quickly overwhelm the ability to make predictions. Without chaos Laplace might have hoped that errors would remain bounded, or at least grow slowly enough to allow him to make predictions over a long period. With chaos, predictions are rapidly doomed to gross inaccuracy.

The larger framework that chaos emerges from is the so-called theory of dynamical systems. A dynamical system consists of two parts: the notions of a state (the essential information about a system) and a dynamic (a rule that describes how the state evolves with time). The evolution can be visualized in a state space, an abstract construct whose coordinates are the components of the state. In general the coordinates of the state space vary with the context; for a mechanical system they might be position and velocity, but for an ecological model they might be the populations of different species.

A good example of a dynamical system is found in the simple pendulum. All that is needed to determine its motion are two variables: position and velocity. The state is thus a point in a plane, whose coordinates are position and velocity. Newton's laws provide a rule, expressed mathematically as a differential equation, that describes how the state evolves. As the pendulum swings back and forth the state moves along an “orbit”, or path, in the plane. In the ideal case of a frictionless pendulum the orbit is a loop; failing that, the orbit spirals to a point as the pendulum comes to rest.

**
STATE SPACE is a useful concept for visualizing the behavior of
a dynamical system. It is an abstract space whose coordinates are
the degrees of freedom of the system's motion.
The motion of a pendulum (top), for example,
is completely determined by its initial
position and velocity. Its state is thus a point in a plane
whose coordinates are position and velocity (bottom).
As the pendulum swings hack and forth it follows
an “orbit”, or path, through the state space.
For an ideal, frictionless pendulum the orbit is a closed
curve (bottom left); otherwise, with friction, the
orbit spirals to a point (bottom right)**.

A dynamical system's temporal evolution may happen in either continuous time or in discrete time. The former is called a flow, the latter a mapping. A pendulum moves continuously from one state to another, and so it is described by a continuous-time flow. The number of insects born each year in a specific area and the time interval between drops from a dripping faucet are more naturally described by a discrete-time mapping.

To find how a system evolves from a given initial state one can employ the dynamic (equations of motion) to move incrementally along an orbit. This method of deducing the system's behavior requires computational effort proportional to the desired length of time to follow the orbit. For simple systems such as a frictionless pendulum the equations of motion may occasionally have a closed-form solution, which is a formula that expresses any future state in terms of the initial state. A closed-form solution provides a short cut, a simpler algorithm that needs only the initial state and the final time to predict the future without stepping through intermediate states. With such a solution the algorithmic effort required to follow the motion of the system is roughly independent of the time desired. Given the equations of planetary and lunar motion and the earth's and moon's positions and velocities, for instance, eclipses may be predicted years in advance.

Success in finding closed-form solutions for a variety of simple systems during the early development of physics led to the hope that such solutions exist for any mechanical system. Unfortunately, it is now known that this is not true in general. The unpredictable behavior of chaotic dynamical systems cannot be expressed in a closed-form solution. Consequently there are no possible short cuts to predicting their behavior.

The state space nonetheless provides a powerful tool for describing the behavior of chaotic systems. The usefulness of the state-space picture lies in its ability to represent behavior in geometric form. For example, a pendulum that moves with friction eventually comes to a halt, which in the state space means the orbit approaches a point. The point does not move—it is a fixed point—and since it attracts nearby orbits, it is known as an attractor. If the pendulum is given a small push, it returns to the same fixed-point attractor. Any system that comes to rest with the passage of time can be characterized by a fixed point in state space. This is an example of a very general phenomenon, where losses due to friction or viscosity, for example, cause orbits to be attracted to a smaller region of the state space with lower dimension. Any such region is called an attractor. Roughly speaking, an attractor is what the behavior of a system settles down to, or is attracted to. Some systems do not come to rest in the long term but instead cycle periodically through a sequence of states. An example is the pendulum clock, in which energy lost to friction is replaced by a mainspring or weights. The pendulum repeats the same motion over and over again. In the state space such a motion corresponds to a cycle, or periodic orbit. No matter how the pendulum is set swinging, the cycle approached in the long-term limit is same. Such attractors are therefore called limit cycles. Another familiar system with a limit-cycle attractor is the heart.

A system may have several attractors. If that is the case, different initial conditions may evolve to different attractors. The set of points that evolve to an attractor is called its basin of attraction. The pendulum clock has two such basins: small displacements of the pendulum from its rest position result in a return to rest; with large displacements, however, the clock begins to tick as the pendulum executes a stable oscillation.

The next most complicated form of attractor is a torus, which resembles the surface of a doughnut. This shape describes motion made up of two independent oscillations, sometimes called quasi-periodic motion. (Physical examples can be constructed from driven electrical oscillators.) The orbit winds around the torus in state space, one frequency determined by how fast the orbit circles the doughnut in the short direction, the other regulated by how fast the orbit circles the long way around. Attractors may also be higher-dimensional tori, since they represent the combination of more than two oscillations.

The important feature of quasi-periodic motion is that in spite of its complexity it is predictable. Even though the orbit may never exactly repeat itself, if the frequencies that make up the motion have no common divisor, the motion remains regular. Orbits that start on the torus near one another remain near one another, and long-term predictability is guaranteed.

**
ATTRACTORS are geometric forms that characterize long-term
behavior in the state space. Roughly speaking, an attractor is
what the behavior of a system settles down to, or is attracted to.
Here attractors are shown in blue and initial states states in red.
Trajectories ( green) from the initial states eventually
approach the attractors. The simplest kind of attractor is a
fixed point (top left). Such an attractor corresponds
to a pendulum subject to friction;
the pendulum always comes to the same rest position, regardless
of how it is started swinging (see right half of illustration
on preceding page). The next most complicated attractor is
a limit cycle (top middle), which forms a closed loop
in the state space. A limit cycle describes stable oscillations,
such as the motion of a pendulum clock and the beating of a heart.
Compound oscillations, or quasi-periodic behavior, correspond to
a torus attractor (top right). All three attractors are
predictable: their behavior can be forecast as accurately as desired.
Chaotic attractors, on the other hand, correspond to unpredictable
motions and have a more complicated geometric form. Three examples
of chaotic attractors are shown in the bottom row; from left to
right they are the work of Edward N. Lorenz, Otto E. Rössler and
one of the authors (Shaw), respectively. The images were prepared
by using simple systems of differential equations having a
three-dimensional state space.**

Until fairly recently, fixed points, limit cycles, and tori were the only known attractors. In 1963 Edward N. Lorenz of the Massachusetts Institute of Technology discovered a concrete example of a low-dimensional system that displayed complex behavior. Motivated by the desire to understand the unpredictability of the weather, he began with the equations of motion for fluid flow (the atmosphere can be considered a fluid), and by simplifying them he obtained a system that had just three degrees of freedom. Nevertheless, the system behaved in an apparently random fashion that could not be adequately characterized by any of the three attractors then known. The attractor he observed, which is now known as the Lorenz attractor, was the first example of a chaotic, or strange, attractor.

Employing a digital computer to simulate his simple model, Lorenz elucidated the basic mechanism responsible for the randomness he observed: microscopic perturbations are amplified to affect macroscopic behavior. Two orbits with nearby initial conditions diverge exponentially fast and so stay close together for only a short time. The situation is qualitatively different for nonchaotic attractors. For these, nearby orbits stay close to one another, small errors remain bounded and the behavior is predictable.

The key to understanding chaotic
behavior lies in understanding a simple stretching
and folding operation,
which takes place in the state space.
Exponential divergence is a local feature: because
attractors have finite
size, two orbits on a chaotic attractor
cannot diverge exponentially forever.
Consequently the attractor must fold
over onto itself. Although orbits diverge and
follow increasingly different
paths, they eventually must pass close
to one another again. The orbits on a
chaotic attractor are shuffled by this
process, much as a deck of cards is
shuffled by a dealer. The randomness
of the chaotic orbits is the result of
the shuffling process. The process of
stretching and folding happens repeatedly,
creating folds within folds ad infinitum.
A chaotic attractor is, in other
words, a fractal: an object that reveals
more detail as it is increasingly magnified
[*see illustration on page 53*].

**CHAOTIC ATTRACTOR has a much more complicated structure than
a predictable attractor such as a point, a limit cycle, or a torus.
Observed at large scales, a chaotic attractor
is not a smooth surface but one with folds in it.
The illustration shows the steps in
making a chaotic attractor for the simplest case: the Rössler
attractor ( bottom). First,
nearby trajectories on the object must “stretch”,
or diverge, exponentially (top), here the
distance between neighboring trajectories roughly doubles.
Second, to keep the object compact, it must “fold”
back onto itself (middle): the surface bends onto
itself so that the two ends meet. The Rössler attractor has
been observed in many systems, from fluid flows
to chemical reactions, illustrating Einstein's maxim
that nature prefers simple forms.**

Chaos mixes the orbits in state space
in precisely the same way as a baker
mixes bread dough by kneading it. One
can imagine what happens to nearby
trajectories on a chaotic attractor by
placing a drop of blue food coloring in
the dough. The kneading is a combination
of two actions: rolling out the
dough, in which the food coloring is
spread out, and folding the dough
over. At first the blob of food coloring
simply gets longer, but eventually it is
folded, and after considerable time the
blob is stretched and refolded many
times. On close inspection the dough
consists of many layers of alternating
blue and white. After only 20 steps the
initial blob has been stretched to more
than a million times its original length,
and its thickness has shrunk to the molecular
level. The blue dye is thoroughly mixed with
the dough. Chaos
works the same way, except that instead of mixing dough it mixes the
state space. Inspired by this picture of mixing. Otto E. Rössler
of the University of Tübingen created the simplest
example of a chaotic attractor in a flow
[*see illustration on preceding page*].

When observations are made on a physical system, it is impossible to specify the state of the system exactly owing to the inevitable errors in measurement. Instead the state of the system is located not at a single point but rather within a small region of state space. Although quantum uncertainty sets the ultimate size of the region, in practice different kinds of noise limit measurement precision by introducing substantially larger errors. The small region specified by a measurement is analogous to the blob of blue dye in the dough.

**
DIVERGENCE of nearby trajectories is the underlying reason
chaos leads to unpredictability. A perfect measurement would
correspond to a point in the state space, but any real measurement is
inaccurate, generating a cloud of uncertainty. The true state might
be anywhere inside the cloud. As shown here for the Lorenz attractor,
the uncertainty of the initial measurement is represented
by 10,000 red dots, initially so close together that they are
indistinguishable. As each point moves under the action of the
equations, the cloud is stretched into a long, thin thread, which then
folds over onto itself many times, until the points are spread over
the entire attractor. Prediction has now become impossible: the
final state can be anywhere on the attractor. For a predictable
attractor, in contrast, all the final states remain close together.
The numbers above the illustrations are in units of 1/200
second.**

Locating the system in a small region of state space by carrying out a measurement yields a certain amount of information about the system. The more accurate the measurement is, the more knowledge an observer gains about the system's state. Conversely, the larger the region, the more uncertain the observer. Since nearby points in nonchaotic systems stay close as they evolve in time, a measurement provides a certain amount of information that is preserved with time. This is exactly the sense in which such systems are predictable: initial measurements contain information that can be used to predict future behavior. In other words, predictable dynamical systems are not particularly sensitive to measurement errors.

The stretching and folding operation of a chaotic attractor systematically removes the initial information and replaces it with new information: the stretch makes small-scale uncertainties larger, the fold brings widely separated trajectories together and erases large-scale information. Thus chaotic attractors act as a kind of pump bringing microscopic fluctuations up to a macroscopic expression. In this light it is clear that no exact solution, no short cut to tell the future, can exist. After a brief time interval the uncertainty specified by the initial measurement covers the entire attractor and all predictive power is lost: there is simply no causal connection between past and future.

Chaotic attractors function locally as noise amplifiers. A small fluctuation due perhaps to thermal noise will cause a large deflection in the orbit position soon afterward. But there is an important sense in which chaotic attractors differ from simple noise amplifiers. Because the stretching and folding operation is assumed to be repetitive and continuous, any tiny fluctuation will eventually dominate the motion, and the qualitative behavior is independent of noise level. Hence chaotic systems cannot directly be “quieted”, by lowering the temperature, for example. Chaotic systems generate randomness on their own without the need for any external random inputs. Random behavior comes from more than just the amplification of errors and the loss of the ability to predict; it is due to the complex orbits generated by stretching and folding.

**
CHAOTIC ATTRACTORS are fractals: objects that reveal more detail
as they are increasingly magnified. Chaos naturally produces fractals.
As nearby trajectories expand they must eventually fold over close
to one another for the motion to remain finite. This is repeated
again and again, generating folds within folds, ad infinitum. As
a result chaotic attractors have a beautiful microscopic structure.
Michel Hénon of the Nice Observatory in France discovered a simple
rule that stretches and folds the plane, moving each point to a
new location. Starting from a single initial point, each successive
point obtained by repeatedly applying Hénon's rule is plotted.
The resulting geometric form ( a) provides a simple example of a
chaotic attractor. The small box is magnified by a factor of 10
in b. By repeating the process (f, d) the
microscopic structure of the attractor is revealed in detail.
The bottom illustration depicts another part of the Hénon
attractor.**

It should be noted that chaotic as well as nonchaotic behavior can occur in dissipationless, energy-conserving systems. Here orbits do not relax onto an attractor but instead are confined to an energy surface. Dissipation is, however, important in many if not most real-world systems, and one can expect the concept of attractor to be generally useful.

Low-dimensional chaotic attractors open a new realm of dynamical systems theory, but the question remains of whether they are relevant to randomness observed in physical systems. The first experimental evidence supporting the hypothesis that chaotic attractors underlie random motion in fluid flow was rather indirect.The experiment was done in 1974 by Jerry P. Gollub of Haverford College and Harry L. Swinney of the University of Texas at Austin. The evidence was indirect because the investigators focused not on the attractor itself but rather on statistical properties characterizing the attractor.

The system they examined was a
Couette cell, which consists of two
concentric cylinders. The space between the
cylinders is filled with a fluid, and one or
both cylinders are rotated with a fixed angular
velocity. As the angular velocity increases,
the fluid shows progressively more complex
flow patterns, with a complicated time
dependence [*see illustration on
this page*]. Gollub and Swinney essentially
measured the velocity of the fluid at a
given spot. As they increased the rotation rate,
they observed transitions from a velocity that
is constant in time to a periodically varying
velocity and finally to an aperiodically varying
velocity. The transition to aperiodic motion was
the focus of the experiment.

The experiment was designed distinguish between two theoretical pictures that predicted different scenarios for the behavior of the fluid as the rotation rate of the fluid was varied. The Landau picture of random fluid motion predicted that an ever higher number of independent fluid oscillations should be excited as the rotation rate is increased. The associated attractor would be a high-dimensional torus. The Landau picture had been challenged by David Ruelle of the Institut des Hautes Études Scientifiques near Paris and Floris Takens of the University of Gröningen in the Netherlands. They gave mathematical arguments suggesting that the attractor associated with the Landau picture would not be likely to occur in fluid motion. Instead their results suggested that any possible high-dimensional tori might give way to a chaotic attractor, as originally postulated by Lorenz.

Gollub and Swinney found that for low rates of rotation the flow of the fluid did not change in time: the underlying attractor was a fixed point. As the rotation was increased the water began to oscillate with one independent frequency, corresponding to a limit-cycle attractor (a periodic orbit), and as the rotation was increased still further the oscillation took on two independent frequencies, corresponding to a two-dimensional torus attractor. Landau's theory predicted that as the rotation rate was further increased the pattern would continue: more distinct frequencies would gradually appear. Instead, at a critical rotation rate a continuous range of frequencies suddenly appeared. Such an observation was consistent with Lorenz' “deterministic nonperiodic flow,” lending credence to his idea that chaotic attractors underlie fluid turbulence.

**EXPERIMENTAL EVIDENCE supports
the hypothesis that chaotic attractors underlie some kinds of random
motion in fluid flow. Shown here are successive pictures
of water in a Couette cell, which consists
of two nested cylinders. The space between
the cylinders is filled with water and the
inner cylinder is rotated with a certain angular
velocity ( a). As the angular velocity
is increased, the fluid shows a progressively
more complex flow pattern (b), which
becomes irregular (c) and then chaotic
(d).**

Although the analysis of Gollub and Swinney bolstered the notion that chaotic attractors might underlie some random motion in fluid flow, their work was by no means conclusive. One would like to explicitly demonstrate the existence in experimental data of a simple chaotic attractor. Typically, however, an experiment does not record all facets of a system but only a few. Gollub and Swinney could not record, for example, the entire Couette flow but only the fluid velocity at a single point. The task of the investigator is to “reconstruct” the attractor from the limited data. Clearly that cannot always be done; if the attractor is too complicated, something will be lost. In some cases, however, it is possible to reconstruct the dynamics on the basis of limited data.

A technique introduced by us and put on a firm mathematical foundation by Takens made it possible to reconstruct a state space and look for chaotic attractors. The basic idea is that the evolution of any single component of a system is determined by the other components with which it interacts. Information about the relevant components is thus implicitly contained in the history of any single component. To reconstruct an “equivalent” state space, one simply looks at a single component and treats the measured values at fixed time delays (one second ago, two seconds ago, and so on, for example) as though they were new dimensions.

The delayed values can be viewed as new coordinates, defining a single point in a multidimensional state space. Repeating the procedure and taking delays relative to different times generates many such points. One can then use other techniques to test whether or not these points lie on a chaotic attractor. Although this representation is in many respects arbitrary, it turns out that the important properties of an attractor are preserved by it and do not depend on the details of how the reconstruction is done.

The example we shall use to illustrate the technique has the advantage of being familiar and accessible to nearly everyone. Most people are aware of the periodic pattern of drops emerging from a dripping faucet. The time between successive drops can be quite regular, and more than one insomniac has been kept awake waiting for the next drop to fall. Less familiar is the behavior of a faucet at a somewhat higher flow rate. One can often find a regime where the drops, while still falling separately, fall in a never repeating patter, like an infinitely inventive drummer. (This is an experiment easily carried out personally; the faucets without the little screens work best.) The changes between periodic and random-seeming patterns are reminiscent of the transition between laminar and turbulent fluid flow. Could a simple chaotic attractor underlie this randomness?

**
DRIPPING FAUCET is an example of a common system that can
undergo a chaotic transition. The underlying attractor is
reconstructed by plotting the time intervals between successive
drops in pairs, as is shown at the top of the illustration.
Attractors reconstructed from an actual dripping faucet
( a, c) compare favorably with attractors
generated by following variants of Hénon's rule (b, d).
(The entire Hénon attractor is shown on page 53.) Illustrations
e and f were reconstructed from high rates of water
flow and presumably represent the cross sections of hitherto
unseen chaotic attractors. Time delay coordinates were employed
in each of the plots. The horizontal coordinate is t_{n},
the time interval between drop n and drop n-1. The
vertical coordinate is the next time interval, t_{n+1}
and the third coordinate, visualized as coming out of the page,
is t_{n+2}. Each point is thus determined by a triplet
of numbers (t_{n}, t_{n+1}, t_{n+2})
that have been plotted for a set of 4,094 data samples.
Simulated noise was added to illustrations b and
d.**

The experimental study of dripping faucet was done at the University of California at Santa Cruz by one of us (Shaw) in collaboration with Peter L. Scott, Stephen C. Pope, and Philip J. Martein. The first form of the experiment consisted in allowing the drops from an ordinary faucet to fall on a microphone and measuring the time intervals between the resulting sound pulses. Typical results from a somewhat more refined experiment are shown on the preceding page. By plotting the time intervals between drops in pairs, one effectively takes a cross section of the underlying attractor. In the periodic regime, for example, the meniscus where the drops are detaching is moving in a smooth, repetitive manner, which could be represented by a limit cycle in the state space. But this smooth motion is inaccessible in the actual experiment; all that is recorded is the time intervals between the breaking off of the individual drops. This is like applying a stroboscopic light to regular motion around a loop. If the timing is right, one sees only a fixed point.

The exciting result of the experiment was that chaotic attractors were indeed found in the nonperiodic regime of the dripping faucet. It could have been the case that the randomness of the drops was due to unseen influences, such as small vibrations or air currents. If that was so, there would be no particular relation between one interval and the next, and the plot of the data taken in pairs would have shown only a featureless blob. The fact that any structure at all appears in the plots shows the randomness has a deterministic underpinning. In particular, many data sets show the horse-shoelike shape that is the signature of the simple stretching and folding process discussed above. The characteristic shape can be thought of as a “snapshot” of a fold in progress, for example, a cross section partway around the Rössler attractor shown on page 51. Other data sets seem more complicated; these may be cross sections of higher-dimensional attractors. The geometry of attractors above three dimensions is almost completely unknown at this time.

If a system is chaotic, how chaotic is it? A measure of chaos is the “entropy” of the motion, which roughly speaking is the average rate of stretching and folding, or the average rate at which information is produced. Another statistic is the “dimension” of the attractor. If a system is simple, its behavior should be described by a low-dimensional attractor in the state space, such as the examples given in this article. Several numbers may be required to specify the state of a more complicated system, and its corresponding attractor would therefore be higher-dimensional.

The technique of reconstruction, combined with measurements of entropy and dimension, makes it possible to reexamine the fluid flow originally studied by Gollub and Swinney. This was done by members of Swinney's group in collaboration with two of us (Crutchfield and Farmer). The reconstruction technique enabled us to make images of the underlying attractor. The images do not give the striking demonstration of a low-dimensional attractor that studies of other systems, such as the dripping faucet, do. Measurements of the entropy and dimension reveal, however, that irregular fluid motion near the transition in Couette flow can be described by chaotic attractors. As the rotation rate of the Couette cell increases so do the entropy and dimension of the underlying attractors.

In the past few years a growing number of systems have been shown to exhibit randomness due to a simple chaotic attractor. Among them are the convection pattern of fluid heated in a small box, oscillating concentration levels in a stirred-chemical reaction, the beating of chicken-heart cells, and a large number of electrical and mechanical oscillators. In addition computer models of phenomena ranging from epidemics to the electrical activity of a nerve cell to stellar oscillations have been shown to possess this simple type of randomness. There are even experiments now under way that are searching for chaos in areas as disparate as brain waves and economics.

It should be emphasized, however, that chaos theory is far from a panacea. Many degrees of freedom can also make for complicated motions that are effectively random. Even though a given system may be known to be chaotic, the fact alone does not reveal very much. A good example is molecules bouncing off one another in a gas. Although such a system is known to be chaotic, that in itself does not make prediction of its behavior easier. So many particles are involved that all that can be hoped for is a statistical description, and the essential statistical properties can be derived without taking chaos into account.

There are other uncharted questions for which the role of chaos is unknown. What of constantly changing patterns that are spatially extended, such as the dunes of the Sahara and fully developed turbulence? It is not clear whether complex spatial patterns can be usefully described by a single attractor in a single state space. Perhaps, though, experience with the simplest attractors can serve as a guide to a more advanced picture, which may involve entire assemblages of spatially mobile deterministic forms akin to chaotic attractors.

The existence of chaos affects the scientific method itself. The classic approach to verifying a theory is to make predictions and test them against experimental data. If the phenomena are chaotic, however, long-term predictions are intrinsically impossible. This has to be taken into account in judging the merits of the theory. The process of verifying a theory thus becomes a much more delicate operation, relying on statistical and geometric properties rather than on detailed prediction.

Chaos brings a new challenge to the reductionist view that a system can be understood by breaking it down and studying each piece. This view has been prevalent in science in part because there are so many systems for which the behavior of the whole is indeed the sum of its parts. Chaos demonstrates, however, that a system can have complicated behavior that emerges as a consequence of simple, nonlinear interaction of only a few components.

The problem is becoming acute in a wide range of scientific disciplines, from describing microscopic physics to modeling macroscopic behavior of biological organisms. The ability to obtain detailed knowledge of a system's structure has undergone a tremendous advance in recent years, but the ability to integrate this knowledge has been stymied by the lack of a proper conceptual framework within which to describe qualitative behavior. For example, even with a complete map of the nervous system of a simple organism, such as the nematode studied by Sidney Brenner of the University of Cambridge, the organism's behavior cannot be deduced. Similarly, the hope that physics could be complete with an increasingly detailed understanding of fundamental physical forces and constituents is unfounded. The interaction of components on one scale can lead to complex global behavior on a larger scale that in general cannot be deduced from knowledge of the individual components.

Chaos is often seen in terms of the limitations it implies, such as lack of predictability. Nature may, however, employ chaos constructively. Through amplification of small fluctuations it can provide natural systems with access to novelty. A prey escaping a predator's attack could use chaotic flight control as an element of surprise to evade capture. Biological evolution demands genetic variability; chaos provides a means of structuring random changes, thereby providing the possibility of putting variability under evolutionary control.

Even the process of intellectual progress relies on the injection of new ideas and on new ways of connecting old ideas. Innate creativity may have an underlying chaotic process that selectively amplifies small fluctuations and molds them into macroscopic coherent mental states that are experienced as thoughts. In some cases the thoughts may be decisions, or what are perceived to be the exercise of will. In this light, chaos provides a mechanism that allows for free will within a world governed by deterministic laws.

**
TRANSITION TO CHAOS is depicted schematically by means
of a bifurcation diagram: a plot of a family of attractors
( vertical axis) versus a control parameter
(horizontal axis). The diagram was generated by a
simple dynamical system that maps one number to another.
The dynamical system used here is called a circle
map, which is specified by the iterative equation
x_{n+1} = w + x_{n} + k/2
π sin(2 π x_{n} ). For each
chosen value of the control parameter k a computer
plotted the corresponding attractor. The colors encode the
probability of finding points on the attractors: red
corresponds to regions that are visited frequently,
green to regions that are visited less frequently, and
blue to regions that are rarely visited. As k is
increased from 0 to 2 (see drawing at left), the
diagram shows two paths to chaos: a quasi-periodic route
(from k = 0 to k = 1, which corresponds
to the green region above) and a “period doubling”
route (from k = 1.4 to = 2). The quasi-periodic
route is mathematically equivalent to a path that passes
through a torus attractor. In the period-doubling route, which is
based on the limit-cycle attractor, branches appear in pairs,
following the geometric series 2, 4, 8, 16, 32, and so on. The
iterates oscillate among the pairs of branches. (At a particular
value of k—1.6, for instance—the iterates visit only
two values.) Ultimately, the branch structure becomes so fine
that a continuous band structure emerges: a threshold is
reached beyond which chaos appears.**