Chaos Theory :
An Introduction
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Chaos Theory :
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Chaos Theory describes dynamic systems such as turbulence and oscillations, weather and biological systems, and the stock make. It is the result of studies begun by Edward Lorenz in the 1960's. Lorenz was a meteorologist who used computers to simulate weather systems, using non-linear equations. He discovered inadvertently that small changes (as little as 1/1000) in the initial conditions produced dramatic changes in the overall system. This has been called the butterfly effect - if a butterfly flaps its wings in Brazil, hurricanes result in North America. This is why weather is so unpredictable. Lorenz also discovered that there was an orderliness to this chaos. If computer simulations were run long enough, spiral patterns emerged. These could be seen as graphic displays called strange attractors. The Lorenz Attractor looks like a 3 dimensional owl face. Although the system is unpredictable because the overall effect is dependent on the initial conditions, the system is also orderly in that it is confined to predetermined parameters. Therefore there is order in chaos, order that is not prescribed from without, but from within. Benoit Mandelbrot began to investigate the images that arose from nonlinear equations. He based his work on previous investigations by Gaston Julia from the 1920's. Julia theorized that iterations of a rational function stayed within confines even as the number of iterations increased to infinity. Mandelbrot found that plotting those iterations resulted in images called fractals. Like strange attractors, these images were unpredictable. Fractal geometry, unlike Euclidian geometry, can describe chaotic systems. One of the properties of fractals is that they are self-similar (Devaney,1995). That is, the images can be broken into smaller pieces that resemble the larger piece. Sierpinski's Triangle is an example. The large triangle is composed of four smaller triangles which in turn is composed of four smaller triangles, and so-on and so-on. The result is an image that looks very irregular. Clouds, coastlines, trees, protein surfaces, etc. all display this property of self-similarity and can be described using fractals. |
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| author: Kelleen Farrell | 1280 Roxie Court | Placerville, CA 95667 |
| copyright (c) 1998 | 530-295-0680 | 209-257-5537 |
