Non-Linear Equations and Chaotic Systems

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

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/