Jason Putorti Chaos and the Logistic Map February 2001

Full report: html and MS Word

Applications of Chaos Theory

I found several several little tidbits around the Internet that show chaos theory applied to the world around us. The most obvious I already knew about was fractals, I have explored fractal systems for quite some time and they are truly remarkable application of mathematics. Worlds within worlds, the closer you look, the more you see, constantly changing and seemingly random patterns emerging before your eyes. Mandelbrot himself found an interesting application that I found amusing:

How long is the coastline of Britain? His mathematical colleagues were miffed, to say the least, at such an annoying waste of their time on such insignificant problems. Theof their time on such insignificant problems. They told him to look it up.  Of course, Madelbrot had a reason for his peculiar question, quite an interesting reason. Look up the coastline of Britain yourself, in some encyclopedia. Whatever figure you get, it is wrong. Quite simply, the coastline of Britain is infinite.  You protest that this is impossible. Well, consider this. Consider looking at Britain on a very large-scale map. Draw the simplest two-dimensional shape possible, a triangle, which circumscribes Britain as closely as possible. The perimeter of this shape approximates the perimeter of Britain. However, this area is of course highly inaccurate. Increasing the amount of vertices of the shape going around the coastline, and the area will become closer. The more vertices there are, the closer the circumscribing line will be able to conform to the dips and the protrusions of Britain's rugged coast. There is one problem, however. Each time the number of vertices increases, the perimeter increases. It must increase, because of the triangle inequality. Moreover, the number of vertices never reaches a maximum. There is no point at which one can say that a shape defines the coastline of Britain. After all, exactly circumscribing the coast of Britain would entail encircling every rock, every tide pool, and every pebble that happens to lie on the edge of Britain. Thus, the coastline of Britain is infiritain is infinite.” –The Chaos Experience, Thinkquest.org

Chaos theory has been used to explain nearly every aspect of human life; the famous butterfly effect details how a seemingly miniscule force could affect storms on the other side of the planet. Edward Lorenz showed how the bifurcation effect that we looked at earlier is consistent with attempts to predict the weather in any amount of time into the future. Why is the weather [forecast] right sometimes and off others? We put all the variables into the system and what happens? 1/1000 decimal place in the results dramatically diverged the results.

Another interesting fact I came across had to do with stock markets and their relation to the tree-like fractals:

While the branches get smaller and smaller, each is similar in structure to the larger branches and the tree as a whole. Similarly, in market price action, as you look at monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance. Just as with natural objects, as you move in closer and closer, you see more and more detail. Another characteristic of chaotic markets is called "sensitive dependence on initial conditions." This is what makes dynamic market systems so difficult to predict. Because we cannot accurately describe the current situation band because errors in the description are hard to f description are hard to find due to the system's overall complexity, accurate predictions become impossible. Even if we could predict tomorrow's stock market change exactly (which we can't), we would still have zero accuracy trying to predict only twenty days ahead. A number of thoughtful traders and experts have suggested that those trading with intra day data such as five-minute bar charts are trading random noise and thus wasting their time. Over time, they are doomed to failure by the costs of trading. At the same time these experts say that longer-term price action is not random. Traders can succeed trading from daily or weekly charts if they follow trends. The question naturally arises how can short-term data be random and longer-term data not be in the same market? If short-term (random) data accumulates to form long-term data, wouldn't that also have to be random? As it turns out, such a paradox can exist.” –The Chaos Experience, Thinkquest.org

Note   Chaos theory is undoubtedly a hot, if not the hottest topic in modern mathematics. Ever read Jurrasic Park? The possibilities for explanation of natural phenomena are endless including religion. Truly fascinating!