SEEKING ORDER IN CHAOS
This article also appears in the Oak Ridge National Laboratory
Review (Vol. 25, No. 2), a quarterly research and development
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A mountain stream, a beating heart, a smallpox epidemic, and a
column of rising smoke are all examples of dynamic phenomena that
sometimes seem to behave randomly. In actuality, such processes
exhibit a special order that scientists and engineers are only just
beginning to understand. This special order is "deterministic
chaos," or chaos, for short.
Today chaos theory is employed to monitor and control the
interaction between particulates and gases in the turbulent flow of
a fluidized bed, thus improving its performance and reducing
emissions of gaseous pollutants. In the future chaos theory may be
used to smooth airplane flight and reduce fuel consumption. Heart
pacemakers may someday be tuned to issue warnings when they detect
undesirable heartbeat patterns that signal a heart attack days or
weeks away.
As a researcher in ORNL's Engineering Technology Division, I am
particularly interested in applying the new discipline of chaos
theory to engineering. Chaos theory seems to be especially suited
for dealing with a major difficulty that has puzzled engineers for
centuries: how to describe processes that are governed by explicit
laws of cause and effect but seem to behave almost randomly in
practice.
Turbulent fluid flow, as in fluidized beds used in industry, is a
good example of such an unruly process. Even though the governing
physical laws are known, under many conditions--such as in a
mountain stream or near the wing of an airplane--the mathematics of
fluid flow becomes extremely complicated, and the actual behavior
becomes so erratic that making detailed predictions is impossible.
Attempts to understand this problem have prompted some of the
deepest questions in modern physics.
To make at least some headway in practical situations, engineers
have traditionally relied on statistical descriptions or
"correlations" of irregular phenomena. These summarize empirical
experience--that is, measured data--in some convenient form,
usually guided by general physical constraints or relationships. In
cases where only gross or average behavior patterns are important,
these descriptions have often been useful. The development of
aircraft design correlations from wind tunnel measurements is a
good example of empirical engineering.
Unfortunately, empirical engineering can be costly and inefficient.
Numerous trial-and-error experiments must be performed to generate
correlations relevant to situations of interest. Correlations are
also only valid for the parameter ranges tested; extrapolation to
untested situations is risky at best.
More recently, engineers have attempted to go beyond empiricism by
developing computer models that simulate physical laws and details
to a high degree of precision. This approach has been made possible
by the increasing availability of high-speed, large-memory
computers. Although successful in some cases, such
"number-crunching" approaches are often costly and complicated, may
contain many untestable assumptions, and frequently yield little
improvement in fundamental understanding.
Another important problem with detailed computer models is that
they are sometimes "linearized" to make them more tractable. In
other words, all physical influences are assumed to be directly
additive; that is, if a given magnitude of cause A produces X units
of effect B, then twice the previous magnitude of A produces 2X
units of B. Nonlinear relationships are not so straightforward. For
example, doubling A might cause B to increase fourfold. We now know
that nonlinearities are often essential to the behavior of many
systems and that their removal can completely change the resulting
dynamics.
Chaos theory offers a new approach for explaining and dealing with
complicated behavior. As more engineering problems are considered
in light of this theory, it is becoming increasingly apparent that
irregularity is often a key to understanding the fundamental nature
of dynamic systems. Apparent randomness can result not just from
unknown physical parameters, measurement error, and outside noise,
but also from simple, nonlinear deterministic components. In fact,
many systems appear to be dominated by deterministic randomness.
CHAOS VERSUS RANDOMNESS
Deterministic chaos should not be confused with the common notion
of total disorder. In a sense, deterministic chaos is the opposite
of disorder because it actually refers to highly structured
behavior. The word chaos was coined by early investigators because
the structure can be invisible to the casual observer, thus giving
the impression of randomness.
As the adjective deterministic implies, this type of chaos arises
in physical processes or systems that follow explicit rules of
cause and effect. Thus, in practice we can model the current state
of a deterministic system as a unique function of its previous
state. Typically, the relationship between past and future is
expressed in terms of finite difference or differential equations.
The motion of a satellite around the earth is a familiar
deterministic system. Given an initial satellite position and
velocity, the laws of Newtonian mechanics make possible
construction of a model that unambiguously predicts the satellite's
position and velocity at the next moment. By contrast, in processes
such as the decay of an atomic nucleus or the motion of an electron
in an atom, the governing laws of quantum physics allow predictions
only of the probabilities of succeeding states. In these latter
cases an unambiguous relationship no longer exists between
conditions at one instant in time and the next.
WHAT CAUSES CHAOS?
At first glance, the idea that irregularity can arise from
deterministic behavior would seem to be a contradiction. In
mathematical terms, deterministic irregularity is related to a
property known as sensitivity to initial conditions. This is a
technical form of the more common notion that sometimes even very
slight changes in starting conditions can result in drastic changes
to the final outcome. A familiar example of this notion is the
saying: "For want of a nail the shoe was lost, for want of a shoe
the horse was lost, for want of a horse the rider was lost, for
want of a rider the battle was lost," and so on.
When dynamic systems exhibit sensitivity to initial conditions, our
ability to predict what will happen next decreases exponentially
over time. Even if we know the deterministic law relating the
states at time t and time t+1, we will never be able to know the
starting state at time t with infinite precision. We may be able to
predict the next state with acceptable accuracy, but as we
extrapolate farther into the future, the effect of the initial
uncertainty grows so rapidly that it soon dominates our result.
Before long we have no hope of making an accurate prediction. To a
casual observer it can appear that the system is being perturbed by
some random influence.
The most common cause of sensitivity to initial conditions is the
presence of nonlinearities in the governing relationship between
sequential states in time. As described previously, nonlinearities
are characterized by nonproportional changes in the result for a
given change in input. For example, the equation y = x2 is a
nonlinear relationship between x and y. Doubling x quadruples y.
The equation y = 2x , on the other hand, is linear. The first
equation can be linearized by assuming that, for cases of interest,
x is nearly constant at some value k and rewriting the relationship
as y = kx. In effect, this change forces the equation to be linear.
The importance of nonlinearities was almost completely unrecognized
by the scientific and engineering communities until within the past
two years or so. Before that time nonlinearities were basically
ignored because they were mathematically "messy." Most models that
contained nonlinearities were linearized so that they could be
solved in analytical form.
With the advent of modern computers, it became possible to observe
the results of nonlinearities directly without the need to resort
to analytical solutions. As a result, a few key researchers quickly
began to realize that science had been systematically excluding
(albeit unknowingly) a major reality in the behavior of the natural
world. Surprisingly, this realization has been slow to take hold in
the technical community as a whole. Scientists and engineers still
have a strong tendency to linearize models, even though they may be
solving those models using a computer. We are still strongly
influenced by a legacy of linear thinking.
CHAOTIC STRUCTURE IN THERMAL CONVECTION
To better understand how the inherent order in chaos can be
visualized, it is helpful to consider another simple example.
Anyone who has driven on a hot desert road during the day has seen
the dancing mirages produced by the air as it rises from the heated
surface. The motion of the air in this case is a form of natural
thermal convection, induced by the air's lowered density as it
warms near the road. This same basic process produces the thermals
exploited by birds and hang glider pilots. A simplified
mathematical model of this process was developed about 30 years ago
by Professor Edward Lorenz of the Massachusetts Institute of
Technology. Mathematically, the model consists of three ordinary
differential equations that describe how the global patterns of air
flow and temperature change over time. Nonlinearities are prominent
in two of the equations.
Integrating the model over time produces a simulation of the
patterns that would be observed in the layer of air. The figure on
this page is a depiction of the Lorenz variable representing the
air flow (designated by convention as X) plotted against time. The
figures show two different traces, each representing the behavior
that would occur for slightly different starting conditions. Note
the rapid divergence in similarity between the two traces after an
initial period during which they remain close together.
We could be content to just observe the variation of flow and
temperature patterns individually over time, but a much more useful
approach is to plot the flow and temperature variables against each
other as in the figure onp. 40. Recall that X represents flow; Y
and Z represent horizontal and vertical temperature variations. At
each moment in time, the Lorenz model produces values for X, Y, and
Z that represent a complete description of the dynamical state of
the convecting air layer. When continously plotted, the XYZ
combinations produce a "map," or "trajectory," of the patterns
produced over time. Such plots are termed state-space or
phase-space trajectories.
As we see in the figure on p. 40, the XYZ patterns produced by the
Lorenz model vary continually over time, never settling down to a
single point and never exactly repeating. Nevertheless, over
relatively long periods of time the Lorenz phase-space trajectory
begins to fill in a distinctive shape called the attractor.
Magnifying this structure reveals an endless level of repeating
detail that can be described in terms of a new mathematical tool
called fractal geometry. Fractal geometry is a universal
characteristic of chaotic processes involving friction, which are
of primary interest to engineers. Nonchaotic systems also have
attractors, but their attractors are not fractals.
To distinguish chaotic attractors from ordinary ones, the former
are called strange attractors. Attractors provide a unique way of
defining and comparing the structure of different chaotic
processes. In effect, it is possible to use the attractor as a kind
of dynamic "fingerprint." Not only do attractors allow us to more
conveniently picture the underlying patterns, they can be
mathematically analyzed to obtain numerical descriptions. Even
slight changes in the chaotic behavior (when the system parameters,
operating conditions, or inputs are changed, for example) are
readily detectable as changes in the attractor characteristics.
Much of chaos theory focuses on the proper methods for visualizing
and describing attractors.
CHAOTIC TIME SERIES ANALYSIS
About three years ago I began collaborating with three other ORNL
colleagues in applying chaos theory to engineering systems. Our
principal interests were (and still are) to determine if particular
systems are, in fact, deterministically chaotic, to visualize and
quantify the chaotic patterns if they exist, and to use the
resulting information in practical ways such as improved design and
control. After considering various approaches, we decided that the
best approach would be to focus on the analysis of time series
measurements.
Time series measurements are sequential records of physical
variables such as velocity, temperature, or pressure that are
collected for some period of time from a process or system.
Typically, such measurements are depicted graphically as time
traces like the Lorenz X variable in the figure on p. 39. Other
familiar examples of such data are stock prices, temperature and
rainfall records, and the speed of a car's engine.
The analysis of nonchaotic time series is highly developed and
relatively routine. One common technique used by engineers is
Fourier analysis, a mathematical procedure that models variations
over time as combinations of sine waves (that is, regular waves of
constant amplitude and frequency). Fourier analysis is clearly
appropriate when the governing relationships and resulting patterns
are linear, but it is inadequate when the data being analyzed are
chaotic.
In evaluating chaotic time series measurements from engineering
systems, we have found that the most productive approach is to
construct phase-space trajectories analogous to those shown in the
figure on the opposite page. Because it is usually not possible to
measure all of the dynamic variables in a system, trajectories must
typically be constructed from a small set of selected time series
measurements. Construction is accomplished using a mathematical
technique called embedding. The embedding technique is, in fact, so
powerful that it is theoretically possible to construct a
representation of the behavior of all of the dynamical variables of
a system from measurements of a single variable. Thus, for example,
it is possible to reconstruct the contributions of Y and Z in the
Lorenz model (and thus a dynamically equivalent facsimile of the
figure at left) from measurements of the X variable alone!
The final result of the embedding procedure is a transformation of
the original time series data into a reconstructed phase-space
trajectory as shown in the figure at right. Once this
reconstruction has been accomplished, the resulting patterns can be
analyzed in detail to determine whether or not the system is
chaotic, linear, or purely random. If it is chaotic, the
distinctive measures of the attractor can be used to uniquely
define the underlying structure and compare it to other systems,
conditions, or models.
Dynamical systems often have more than three important components,
and the resulting phase-space can be difficult to visualize
directly because it requires more than the three dimensions
encountered in everyday experience. One approach to this problem is
to make projections of the trajectory in two or three dimensions.
Mathematically, however, the analytical procedures are identical,
and no serious problems occur in making quantitative descriptions
of trajectories in more than three dimensions.
SPECIFIC APPLICATIONS
We initially selected fluidized beds as a trial engineering system
for applying chaotic time series analysis. Fluidized beds are
widely used in industry to promote intimate mixing between
particulate solids and fluids (either liquids or gases). Briefly,
they consist of vertical chambers containing "beds" of solids
through which fluid flows upward. The fluid flow is sufficiently
high to suspend the solids, which swirl about the chamber in
turbulent patterns reminiscent of thermal convection. Commercially
important processes using fluidized beds include refining petroleum
and minerals and fluidized combustion of waste and high-sulfur
coal.
The turbulent flow of fluid and solids in fluidized beds is one of
the most desirable characteristics for mixing, but it is also the
source of much complexity and engineering uncertainty. Although
fluidized beds have been prominent in the chemical and petroleum
industries for several decades, their design and control is still
largely empirical. It is still common to find large
multimillion-dollar facilities initially constructed for specific
performance specifications (such as yield of a particular chemical
product or degree of sulfur removal) only to discover after startup
that expensive testing and trial-and-error equipment modifications
must be made to meet the original design goals.
Our approach in applying chaos theory to fluidized beds has been to
analyze the experimental chaotic time series of pressure and
voidage measurements taken from several different test facilities
here and at other collaborating DOE laboratories and universities.
From past experience we know that pressure and voidage fluctuations
are good indicators of the fluidization state. The results to date
clearly demonstrate the presence of deterministic chaos and suggest
the possible ranges of chaotic behavior that can be expected in
commercial fluidized beds. The figure at left illustrates three
examples of chaotic attractors constructed from fluidized-bed
pressure signals. Note that the plots shown are two-dimensional
projections of five-dimensional trajectories.
Our experience with fluidized beds demonstrates that chaotic time
series analysis is a much more discriminating method for
characterizing and monitoring fluidization state than conventional
methods such as Fourier analysis. Thus, it should be possible to
use chaotic measures to design and control fluidized beds more
effectively than in the past.
One control application that has already been developed is a
fluidization control module now being incorporated into uranium
processing facilities at the Oak Ridge Y-12 Plant. An essential
component in the Y-12 control scheme is a device that reconstructs
the fluidized bed's attractor from pressure measurements and
continually analyzes its key features. When the attractor structure
begins to deviate significantly from optimum values, a control
signal is sent to correct the bed operation, making it operate more
efficiently than was previously possible.
While continuing to work with fluidized beds, we have expanded our
scope of potential chaos applications into other areas such as
pulsed combustion, motor current signature analysis, nuclear
reactor monitoring, and internal combustion engine diagnostics.
Results from preliminary investigations in several of these areas
are very encouraging.
For example, using simplified computer models, we and collaborators
from DOE's Morgantown Energy Technology Center have demonstrated
that chaos can result from interactions among the basic heat, mass,
and momentum balances in pulsed combustion (see the figure at
right). Because these same basic balances are present in most
flames, we believe that chaos may affect many types of combustion.
It is known that fluctuations of flow, concentration, and
temperature in flames are often major factors in the production of
pollutants such as nitrogen oxides and unburned hydrocarbons. If
such fluctuations are the product of deterministic chaos, it should
be possible to use chaos theory and chaotic time series analysis to
develop totally new methods for reducing undesirable combustion
emissions from automobile engines, power plants, and waste
incinerators.
FUTURE PLANS AND POSSIBILITIES
We plan to continue exploring the use of chaos theory as a basis
for engineering diagnostics. As described previously, we will
attempt to use chaotic time series analysis on systems of interest
to detect subtle changes in their attractors that indicate the
onset of undesirable performance or imminent failure of critical
components.
Beyond providing improved diagnostics, we anticipate that chaos
theory will contribute to the development of radically new control
strategies. For example, chaos theory may be combined with
pattern-recognition techniques, parallel computers, and high-speed
algorithms for extremely rapid "on-line" decision making.
Sufficient data processing speed would allow the introduction of
real-time control changes to force the system to behave in a
particular fashion. An example of an application of this idea would
be in the development of "smart" monitors for automobile engines.
Such monitors could evaluate an attractor representing engine
performance and determine if the pattern meets certain criteria
(e.g., less than some allowable maximum for production of nitrogen
oxide, an air pollutant). If the desired performance criteria are
not met, some appropriate parameter adjustment would be made (e.g.,
increasing the spark advance). Such adjustments, or
"perturbations," could be made almost continuously to effectively
mold the engine's attractor into its optimum form.
In summary, because it is appropriate for many technical
disciplines and applications, use of chaotic time series analysis
as a basic research tool and engineering methodology will likely
expand widely over the next few years.
SUGGESTED READING
The following short list of references is recommended for those
interested in learning more about chaos: James Gleick, Chaos:
Making a New Science, Viking Press, New York. Nontechnical but very
readable review of major developments and players. Ian Stewart,
Does God Play Dice?, Basil Blackwell, Cambridge, Massachusetts.
Very readable introduction to the basic concepts and philosophy
behind chaos theory. G. L. Baker and J. P. Gollub, Chaotic
Dynamics: An Introduction, Cambridge University Press, New York.
Technical presentation of chaos at the undergraduate and first-year
graduate student level. Francis Moon, Chaotic Vibrations, John
Wiley and Sons, New York. Technical introduction to chaos for
engineers and applied scientists. J. P. Eckmann and D. Ruelle,
"Ergodic theory of chaos and strange attractors," Rev. Mod. Phys.,
Vol. 57, No. 3, Part 1, pp. 617-656. Highly technical discussion
suitable for those familiar with chaos theory but not specifically
acquainted with chaotic time series analysis. Biographical
SketchStuart Daw is a development staff member in ORNL's
Engineering Technology Division. He holds a Ph.D. degree in
chemical engineering from the University of Tennessee. He served as
a research and development engineer at E. I. du Pont de Nemours and
Company before coming to ORNL in 1979. Three years ago, he became
interested in using chaos theory to describe behavior in
fluidized-bed reactors. Since then he has collaborated with
researchers from ORNL's Engineering Physics and Mathematics
Division and Instrumentation and Controls Division to develop
chaotic time series analysis as a generic research tool. As a
result of this collaboration, Daw was a co-winner of a 1991
Technical Achievement Award from Martin Marietta Energy Systems,
Inc. He is a member of the ORNL Graduate Fellow Selection Panel,
the Board of Advisors of the Central States Section of the
Combustion Institute, and the American Flame Research Committee.
C. Stuart Daw
(keywords: chaos, deterministic chaos)
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Date Posted: 2/7/94 (ktb)