**Chaos Theory**

__ __

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

**from Interdisciplinary English, 2^{nd}
edition**

**Ó****1998-2002**

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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 linear
equations; equations which are easily solved.
**

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 **nonlinear** equations. Unfortunately, nonlinear equations are
difficult, if not impossible, to solve.

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.