# Chaos theory

Chaos theory is the study of complex systems that, at first glance, appear to follow no orderly laws of mathematics or science. Chaos theory is one of the most fascinating and promising developments in late-twentieth-century mathematics and science. It provides a way of making sense out of phenomena such as weather patterns that seem to be totally without organization or order.

## Cause-and-effect and chaos

Scientists have traditionally had a rather strict cause-and-effect view of the natural world. English physicist Isaac Newton once said that if he could know the position and motion of every particle in the universe at any one moment, he could predict the future of the universe into the infinite future. He believed that all those particles follow strict physical laws. Since he knew (or so he thought) what those laws were, all he had to do was to apply them to the particles at any one point in time.

On the other hand, scientists have always realized that some events in nature appear to be just too complex to analyze by the laws of science. One of the best examples is weather patterns. Even though scientists know a great deal about the elements that make up weather, they have a very difficult time predicting what weather patterns will be. The term chaos has often been used to describe systems that are just too "messy" to understand by scientific analysis.

## Origins of chaos theory

The rise of modern chaos theory can be traced to a few particularly striking and interesting discoveries. One of these events occurred in the 1890s when French mathematician Henri Poincaré was working on the problem of the interactions of three planets with one another. The problem should have been fairly straightforward, Poincaré thought, since the gravitational laws involved were well known. The results of his calculations were so unexpected, however, that he gave up his work. He described those results as "so bizarre that I cannot bear to contemplate them."

Dutch engineer B. van der Pol encountered a similar problem in working with electrical circuits. He started out with systems that could easily be described by well-known mathematical equations. But the circuits he actually produced gave off unexpected and irregular noises for which he could not account.

## Words to Know

Attractor: An element in a chaotic system that appears to be responsible for helping the system to settle down.

Cause-and-effect: The view that humans can understand why certain events (effects) take place.

Chaos: Some behavior that appears to be so complex as to be incapable of analysis by humans.

Chaos theory: Mathematical and scientific efforts to provide cause- and-effect explanations for chaotic behavior.

Generator: Elements in a system that appear to be responsible for chaotic behavior in the system.

Law: A statement in science that summarizes how some aspect of nature is likely to behave. Laws have survived many experimental tests and are believed to be highly dependable.

Then, in 1961, American meteorologist Edward Lorenz found yet another example of chaotic behavior. Lorenz developed a system for predicting the weather based on 12 equations. The equations represented the factors we know to affect weather patterns, including atmospheric pressure, temperature, and humidity. What Lorenz found was that by making very small changes in the initial numbers used in these equations, he could produce wildly different results.

## Generators and attractors

Scientists and mathematicians now view chaotic behavior in a different way. Instead of believing that such behavior is too complex ever to understand, they have come to conclude that certain patterns exist within chaos that can be discovered and analyzed. For example, certain characteristics of a system appear to be able to generate chaotic behavior. Such characteristics are known as generators because they cause the chaotic behavior. Very small differences in a generator can lead to very large differences in a system at a later point in time.

Researchers have also found that chaotic behavior sometimes has a tendency to settle down to some form of predictable behavior. When this happens, elements within the system appear to bring various aspects of the chaos together into a more understandable pattern. Those elements are given the name attractors because they appear to attract the parts of a chaotic system to themselves.

## Applications

In theory, studies of chaos have a great many possible applications. After all, much of what goes on in the world around us seems more like chaos than a neat orderly expression of physical laws. The weather may be the best everyday example of that point. Although we know a great deal about all the elements of which weather patterns are made, we still have relatively modest success in predicting how those elements will come together to produce a specific weather pattern. Studies of chaos theory may improve these efforts.

Animal behavior also appears to be chaotic. Population experts would like very much to know how groups of organisms are likely to change over time. And, again, we know many of the elements that determine those changes, including food supplies, effects of disease, and crowding. Still, predictions of population changes—whether of white deer in the wilds of Vermont or the population of your hometown—tend to be quite inaccurate. Again, chaos theory may provide a way of making more sense out of such apparently random behavior.