Chaos Theory

Chaos Theory

By

Ron Clemens

Per 3 English

Mr.Ortiz

4/18/2005

What exactly is chaos theory? From the understanding of many scientists such as Edward Lorenz, Ian Stewart, and Robert May the chaos theory relatively means the same thing. Each of these scientists contributed to the science of chaos theory.

First and Foremost chaos theory itself comes from the seemingly half-hazard way things seem to happen in its equations, but chaos theory is really about finding the similarities between these seemingly random events in an equation.

Edward Lorenz, a meteorologist, discovered this theory when he was working on a calculation for weather prediction on his computer. He set his computer to use 12 different equations to model the weather. The computer didn't necessarily predict the weather. It just gave a guess at where the weather might be. Using these twelve different equations he tried running the model of the weather. After the equation was done he went away from his computer. Edwards wanted to see the results of his equations again so to save time he started the equations half way. He entered the number off the printout of the previous equation and let it run. Yet when he looked at his computer again the equation was drastically different as the picture shows.



All of this happened in 1961. The ideas of the time stated that you should have come out with the same results. In this time a scientist would be called "lucky" if they can get measurements with accuracy too 3 decimal places. The ideas believe that the 4th and 5th decimal places couldn't have that dramatic an effect on anything. Edward Lorenz proved them wrong. This effect later became known as the butterfly effect. Due to its relatively same comparison as a butterfly flapping its wings.

Ian Stewart wrote on Lorenz's experiment and stated "The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does."#

Lorenz later stated the to predict the weather was impossible. This led him to discover another attribute of the chaos theory. He wanted to make a simpler version of his twelve equation system but he still wanted to keep its "sensitive dependence on initial conditions". Using these conditions he narrowed the field down to 3 equations. This equation was later described as a water wheel. A water wheel is the same thing you see on the side of houses. "At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other."#

However when he graphed this "water wheel" he discovered it always ended in a double spiral as the picture shows.



In 1963 Lorenz published a paper on this Chaos theory but since he was a meteorologist his work wasn't recognized until years later.

The easiest way to describe

chaos theory is in the flip of a coin. Theoretically there are two variables in this. The time it takes the coin to hit the ground and the speed the coin is traveling at. You should be able to control these variables right? Wrong. No matter how badly you try you can never exactly control the flip of a coin or the time it takes it to fall. That is the chaos theory.

Chaos theory also relates to the prediction of biological populations. This equation for the prediction of the biological populations would be simple right? Just the exponential growth formula right? Wrong there's a lot of valuables. Those including predators, famine, and space needed for population growth. One biologist named Robert May decided to make an equaiton to see what would happen if the population growth rate changed. this equation led to a graph where as soon as it got to three the graph split in two, showing that if a population got to high the population would eventually split into two separate groups. Increasing the growth rate a little more cause this separation

to happen more and more as well. That is until the line eventually broke into chaos.Yet in this chaos he could see little bars of white. Upon closer examination of these whit lines he found that the sequence repeated itself than went into chaos again. This is known as the equation self serving itself. Meaning that it repeats itself in and of itself.

Another way that the chaos theory relates to the modern world is in the stock market. Though it in and of itself seems completely unorthodox. It can be predicted in mathematical terms. Guess what theory you would use to describe it ... that's right folks the chaos theory.

The best way to continue to explain the chaos theory is by using the media. There was a movie called The