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### Non-Linear Equations and Chaotic Systems

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Date: 03/21/2002 at 02:08:12
From: Andrew Johnson
Subject: Applying Non-Linear Equations to Chaotic Systems

Dear Dr. Math,

My question is whether non-linear equations are used to predict
complex or chaotic systems such as traffic or flocking birds. As such
systems display no clear pattern and often depend greatly on their
inherent instability to govern outcomes, it would seem impossible to
apply mathematics to them. But is there any way to mathematically
predict or analyze these complex systems?

Thank you,
Andrew Johnson
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Date: 03/21/2002 at 03:08:40
From: Doctor Mitteldorf
Subject: Re: Applying Non-Linear Equations to Chaotic Systems

Dear Andrew,

This is a large and a deep question. There are two sorts of answers.
One is that frequently the detailed behavior of a system may be
unpredictable, but some large-scale aspects of the behavior can be
calculated. An airplane wing  may be tilted at a variety of angles
facing into the wind. For low angles, air flow over the wing will be
smooth and predictable. It may be possible to calculate the largest
angle that permits smooth air flow. For larger angles, the air flow
becomes "turbulent," meaning that its exact pattern becomes impossible
to calculate.

A second answer is contained in the word "modeling." When you
recognize that the large-scale parameters of a system don't obey
large-scale equations, you can still use a computer to model the
system on a small scale. Let's take your first example. In a
traditional approach, you might look for equations that tell you the
traffic density at every point on a highway. Once you realize that the
system is chaotic, you may give up on this goal, and instead program a
computer with the behavior of a car and the way its driver responds to
other cars around him. Putting together many such cars in a single
computer program, you may be able to watch the traffic patterns on
your computer screen and see how they develop. Programs like this
usually have randomness built in, using the computer's internal random
number generator.

Because of this, you can run the program over and over, getting
different results each time. Sometimes the results may be VERY
different - you may get an accident and a traffic blockage one time,
and the next time you run the program, the accident doesn't happen,
and the traffic remains unblocked. Then, you may think you're starting
to learn something about real life. Running the program over and over,
you may notice that when you set the traffic density less than X, an
accident won't cause a traffic jam, but when the density is more than
X, the traffic flow around a place where there was an accident may
take hours to recover, even after the accident is cleared. If you were
a traffic engineer, this would give you an idea that you could then
test in the real world, to see if real traffic patterns do what your
computer model does.

Here's a web site where many different types of computer models are
available for you to try. You can also download a computer language
called NetLogo, which is designed to make it easy for students to try
modeling themselves.

Connected Models - Connected Mathematics
http://www.ccl.sesp.northwestern.edu/cm/models/

The game of SimCity and its cousins are also examples of how modeling
can give insight into the behavior of chaotic systems, without
actually predicting any specific outcome.

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Calculators, Computers
High School Linear Equations

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