__Chaos
Theory__

__ __

**by Loretta
F. Kasper, Ph.D.**

**from Interdisciplinary
English © 1998. pp. 70-73.**

** **

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Chaos
theory is a mathematical theory first developed in the 1970's. Chaos theory deals with the breakdown of
ordered systems into chaotic, or disordered, ones. The word, ** chaos**,
means "utter confusion or random disorder." It is relatively easy to describe the behavior of certain kinds
of events mathematically. For example
pendulums appear to follow a regular pattern of movement which can be
mathematically described by

Although
the discipline of mathematics is very good at dealing with regularity, it is
not so good at dealing with turbulence.
In the 1960's, scientists were trying to use computers to understand and
to predict the weather. The weather is
an example of an event whose behavior is ** turbulent**. Turbulent events are described by

So
that we may better understand this concept, let's contrast a linear with a
nonlinear system. In a linear system,
a small change produces a small and easily quantifiable, or measurable,
change. However, a nonlinear system is
highly sensitive to and dependent upon its ** initial
conditions**. Small or virtually
immeasurable differences in initial conditions can lead to tremendously
differing results.

Because
the behavior of weather systems is so dependent upon initial conditions, it is
virtually impossible to predict the weather.
Tiny differences in initial conditions lead to wide differences in
outcomes. That is the essence of
nonlinear dynamics, which is another name for chaos theory. So, chaos theory originally grew out of
attempts to make computer models of the weather.

Chaotic,
or turbulent, systems can be found in many other aspects of our lives. Mathematics, science, and society itself are
all in a constant state of seemingly random change. The study of the ** dynamics**
of such systems is an essential part of the effort to understand the principles
of order that form the basis of all real systems, from ecosystems to social
systems to the universe as a whole.

The
possibility of chaos in a natural system was first proposed by the French
mathematician, Henri Poincare, in the late 19th century, in his work on
planetary orbits. Many years later, in
1963, the American meteorologist, Edward Lorenz, demonstrated that thermal
convection in the Earth's atmosphere was a chaotic system. In the mid-1970's the American physicist,
Mitchell Feigenbaum, predicted that when an ordered system begins to break down
into chaos, there would be a consistent sequence of doubling transitions. Feigenbaum then went on to calculate a ** numerical constant** that governs
the doubling process. A numerical
constant is a quantity whose value, under specified conditions, does not
change. Feigenbaum showed that his
results could be applied to a wide range of chaotic systems, and the numerical
constant he calculated is called Feigenbaum's number.

On
the surface, the movement of chaotic systems appears to be totally random,
unpredictable, and without form. Yet,
cutting edge research into the mathematics of chaos has demonstrated that the
motion of a chaotic system is not completely random, and that events within
chaotic systems do operate according to a definite form and pattern, evolving
in predictably unpredictable ways

So
that we may better understand the order within the chaotic system, let's
consider a simple example. To the
outside observer, your desk appears to be a chaotic mess with papers and books
scattered all over in a seemingly haphazard fashion. It appears that you will be unable to find anything amid all the
clutter. Yet when asked to produce a
specific piece of paper, you plow through the chaos to the exact spot and find
it easily. You are able to do this time
and again, until one day, when someone decides to "help" you clean up
your messy desk and straightens everything out. Now the order and the pattern of your chaotic system has been
changed and as a result you cannot find anything.

We
now know that the natural world, like your desk, is characterized by an orderly
form of chaos. Because of the great
calculating power of the computer, scientists are now able to discover and to
analyze patterns in events that were previously believed to be purely
random. Nevertheless, although we now
know that all natural events do follow a certain pattern, the outcome of these
events is still unpredictable.

Mathematicians,
like the rest of us, have observed such events and the changes that result from
them. These scientists have tried to
infuse order into the chaos by plotting equations that might allow them to
predict the outcomes of certain small events.
These equations involve ** the
principle of iteration**.
Iteration means taking the answer to an equation and feeding it back
into the equation, over and over.

These
iterative calculations are very long and boring. They are virtually impossible to complete without the aid of a
computer. As you learned in the
previous unit, computers are very good at performing mechanical, repetitive
calculations. So, mathematicians feed
data into the computer and then instruct the computer about the required
output. Then they watch as the computer
solves the equation and produces patterns of numbers. Throughout this process, the scientists play with different
equations, substituting different sets of numbers to see what happens.

Through
their work with computers, scientists have discovered that although chaotic
systems appear to be random, they are not.
Rather, chaotic systems are deterministic, which means that these
systems are ruled by a determining equation.
However because chaotic systems are also very sensitive to the initial
conditions, the slightest change in the starting point can lead to very
different outcomes. The system and its
outcome are determined by minor variations in starting conditions, and when
these conditions cannot be stabilized, the system and its outcome are both
fairly unpredictable. Finally, while
chaos theory states that order exists within ** random** complex systems, it also asserts that seemingly
simple systems can produce highly complex and unpredictable behavior.

An
example of a seemingly simple system which can produce complex behavior is the ** trajectory** of a pool ball after
it is hit. In theory we should be able
to

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