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/
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