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The Chaos Theory

Where Chaos begins, classical science ends.   Ever since physicists have
inquired into the laws of nature, the have not begun to explore irregular side
of nature, the erratic and discontinuous side, that have always puzzled
scientists.   They did not attempt to understand disorder in the atmosphere, the
turbulent sea, the oscillations of the heart and brain, and the fluctuations of
wildlife populations.   All of these things were taken for granted until in the
1970’s some American and European scientists began to investigate the randomness
of nature.
They were physicists, biologists, chemists and mathematicians but they
were all seeking one thing: connections between different kinds of irregularity.
“Physiologists found a surprising order in the chaos that develops in the human
heart, the prime cause of a sudden, unexplained death.   Ecologists explored the
rise and fall of gypsy moth populations.   Economists dug out old stock price
data and tried a new kind of analysis.   The insights that emerged led directly
into the natural world- the shapes of clouds, the paths of lightning, the
microscopic intertwining of blood vessels, the galactic clustering of stars.”
(Gleick, 1987)
The man most responsible for coming up with the Chaos Theory was
Mitchell Feigenbaum, who was one of a handful of scientists at Los Alamos, New
Mexico when he first started thinking about Chaos.   Feigenbaum was a little
known scientist from New York, with only one published work to his name.   He
was working on nothing very important, like quasi periodicity, in which he and
only he had 26 hour days instead of the usual 24.   He gave that up because he
could not bear to wake up to setting sun, which happened periodically.   He
spent most of time watching clouds from the hiking trails above the laboratory.
To him could represented a side of nature that the mainstream of physics had
passed by, a side that was fuzzy and detailed, and structured yet unpredictable.
He thought about these things quietly, without producing any work.
After he started looking, chaos seemed to be everywhere.   A flag snaps
back and forth in the wind.   A dripping faucet changes from a steady pattern to
a random one.   A rising column of smoke disappears into random swirls.   “Chaos
breaks across the lines that separate scientific disciplines.   Because it is a
science of the global nature of systems, it has brought together thinkers from
fields that have been widely separated…Chaos poses problems that defy accepted
ways of working in science.   It makes strong claims about the universal
behavior of complexity.   The first Chaos theorists, the scientists who set the
discipline in motion, shared certain sensibilities.   They had an eye for
pattern, especially pattern that appeared on different scales at the same time.
They had a taste for randomness and complexity, for jagged edges and sudden
leaps.   Believers in chaos– and they sometimes call themselves believers, or
converts, or evangelists–speculate about determinism and free will, about
evolution, about the nature of conscious intelligence.   They feel theat they
are turning back a trend in science towards reductionism, the analysis of
systems in terms of their constituent parts: quarks, chromosomes, or neutrons.
They believe that they are looking for the whole.”(Gleick, 1987)
The Chaos Theory is also called Nonlinear Dynamics, or the Complexity
theory.   They all mean the same thing though- a scientific discipline which is
based on the study of nonlinear systems. To understand the Complexity theory
people must understand the two words, nonlinear and system, to appreciate the
nature of the science.   A system can best be defined as the understanding of
the relationship between things which interact.   For example, a pile of stones
is a system which interacts based upon how they are piled. If they are piled out
of balance, the interaction results in their movement until they find a
condition under which they are in balance.   A group of stones which do not
touch one another are not a system, because there is no interaction.   A system
can be modeled. Which means another system which supposedly replicates the
behavior ofthe original system can be created. Theoretically, one can
take a second group of stones which are the same weight, shape, and density of
the first group, pile them in the same way as the first group, and predict that
they will fall into a new configuration that is the same as the first group.  Or
a mathematical representation can be made of the stones through application of
Newton’s law of gravity, to predict how future piles of the same type – and of
different types of stones – will interact.   Mathematical modeling is the key,
but not the only modeling process used for systems.
The word nonlinear has to do with understanding mathematical models used
to describe systems. Before the growth of interest in nonlinear systems, most
models were analyzed as though they were linear systems meaning that when the
mathematical formulas representing the behavior of the systems were put into a
graph form, the results looked like a straight line.   Newton used calculus as a
mathematical method for showing change in systems within the context of straight
lines. And statistics is a process of converting what is usually nonlinear data
into a linear format for analysis.
Linear systems are the classic scientific system and have been used for
hundreds of years, they are not complex, and they are easy to work with because
they are very predictable. For example, you would consider a factory a linear
system. If more inventory is added to the factory, or more employees are hired,
it would stand to reason that more pieces produced by the factory by a
significant  amount. By changing what goes into a system we should be able to
tell what comes out of it.   But as any factory manager knows, factories don’t
actually work that way.   If the amount of people, the inventory, or whatever
other variable is changed in the factory you would get widely differing results
on a day to day basis from what was predicted.    That is because a factory is a
complex nonlinear system, like most systems found in nature.
When most natural systems are modeled, their mathematical
representations do not produce straight lines on graphs, and that the system
outputs are extremely difficult to predict.  Before the chaos theory was
developed, most scientists studied nature and other random things using linear
systems.  Starting with the work of Sir Isaac Newton, physics has provided a
process for modeling nature, and the mathematical equations associated with it
have all been linear.  When a study resulted in strange answers, when a
prediction usually came true but not this one time, the failure was blamed on
experimental error or noise.
Now, with the advent of the Chaos theory and research into complex
systems theory, we know that the “noise” actually was important information
about the experiment.   When noise is added to the graph results, the results
are no longer a straight line, and are not predictable.   This noise is what was
originally referred to as the chaos in the experiment.   Since studying this
noise, this chaos, was one of the first concerns of those studying complex
systems theory, Glieck originally named the discipline Chaos Theory.
Another word that is vital to understanding the Complexity theory is
complex.   What makes us determine which system is more complex then another?
There are many discussions of this question.  In Exploring Complexity, Nobel
Laureate Ilya Prigogine explains that the complexity of the system is defined by
the complexity of the model necessary to effectively predict the behavior of the
system. The more the model must look like the actual system to predict system
results, the more complex the system is considered to be. The most complex
system example is the weather, which, as demonstrated by Edward Lorenz, can only
be effectively modeled with an exact duplicate of itself.    One example of a
simple system to model is to calculate the time it takes for a train to go from
city A to city B if it travels at a given speed.   To predict the time we need
only to know the speed that the train is traveling (in mph) and the distance (in
miles). The simple formula would be mph/m, which is a simple system.
But the pile of stones, which appears to be a simple system, is actually
very complex.   If we want to predict which stone will end up at which place in
the pile then you would have to know very detailed information about the stones,
including their weights, shapes, and starting location of each stone to make an
accurate prediction.   If there is a minor difference between the shape of one
stone in the model and the shape of the original stone, the modeled results will
be very different.   The system is very complex, thus making prediction very
The generator of unpredictability in complex systems is what Lorenz
calls “sensitivity to initial conditions” or “the butterfly effect.” The concept
means that with a complex, nonlinear system, a tiny difference in starting
position can lead to greatly varied results.   For example, in a difficult pool
shot a tiny error in aim causes a slight change in the balls path.   However,
with each ball it collides with, the ball strays farther and farther from the
intended path.   Lorenz once said that “if a butterfly is flapping its wings in
Argentina and we cannot take that action into account in our weather prediction,
then we will fail to predict a thunderstorm over our home town two weeks from
now because of this dynamic.”(Lorenz, 1987)
The general rule for complex systems is that one cannot create a model
that will accurately predict outcomes but one can create models that simulate
the processes that the system will go through to create the models.   This
realization is impacting many activities in business and other industries.   For
instance, it raises considerable questions relating to the real value of
creating organizational visions and mission statements as currently practices.
Like physics, the Chaos theory provides a foundation for the study of
all other scientific disciplines.   It is a variety of methods for incorporating
nonlinear dynamics into the study of science.   Attempts to change  the
discipline and make it a separate form of science have been strongly resisted.
The work represents a reunification of the sciences for many in the scientific
One of Lorenz’s best accomplishments supporting the Chaos Theory was the
Lorenz Attractor.   The Lorenz Attractor is based on three differential
equations, three constants, and three initial conditions. The attractor
represents the behavior of gas at any given time, and its condition at any given
time depends upon its condition at a previous time. If the initial conditions
are changed by even a tiny amount, checking the attractor at a later time will
show numbers totally different. This is because small differences will reproduce
themselves recursively until numbers are entirely unlike the original system
with the original initial conditions.   But, the plot of the attractor, or the
overall behavior of the system will be the same.
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 any secret for us, we could still know the 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 has been predicted, that it is governed by the 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…” (Poincare, 1973)
The Complexity theory has developed from mathematics, biology, and
chemistry, but mostly from physics and particularly thermodynamics, the study of
turbulence leading to the understanding of self-organizing systems and system
states (equilibrium, near equilibrium, the edge of chaos, and chaos).   “The
concept of entropy is actually the physicists application of the concept of
evolution to physical systems. The greater the entropy of a system, the more
highly evolved is the system.”( Prigogine, 1974)   The Complexity theory is also
having a major impact on quantum physics and attempts to reconcile the chaos of
quantum physics with the predictability of Newton’s universe.
With complexity theory, the distinctions between the different
disciplines of sciences are disappearing.   For example, fractal research is now
used for biological studies.   But there is a question as to whether the current
research and academic funding will support this move to interdisciplinary
Complexity is already affecting many aspects of our lives and has a
great impacts on all sciences.   It is answering previously unsolvable problems
in cosmology and quantum mechanics.   The understanding of heart arrhythmias and
brain functioning has been revolutionized by complexity research.   There have
been a number of other things developed from complexity research, such a the
SimLife, SimAnt, etc. which are a series of computer programs.   Fractal
mathematics are critical to improved information compression and encryption
schemes needed for computer networking and telecommunications. Genetic
algorithms are being applied to economic research and stock predictions.
Engineering applications range from factory scheduling to product design, with
pioneering work being done at places like DuPont and Deere & Co.
Another element of the nonlinear dynamics, Fractals, have appeared
everywhere, most recently in graphic applications like the successful Fractal
Design Painter series of products. Fractal image compression techniques are
still being researched, but promise such amazing results as 600:1 graphic
compression ratios.   The movie special effects industry would have much less
realistic clouds, rocks, and shadows without fractal graphic technology.
Though it is one of the youngest sciences, the Chaos Theory holds great
promise in the fields of meteorology, physics, mathematics, and just about
anything else you can think of.

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