CHAOS THEORY

I never imagined that I would ever one day sit down and attempt to write an essay paper on Chaos Theory. As a matter of fact, I'd never even heard of Chaos Theory until we viewed a film that gave us a brief introduction to it during a class meeting one evening. I was impressed by the presentation and was literally "drawn in " by the science of Chaos.............. The event had much the same effect on me as that of the motion of a strange attractor....... I've always been very interested in science but I've never really been that good at mathematics.........and, as we all know, there is a mathematical correlation that runs between the two subjects. So , here I am.....I guess you could say that I'm initially a frustrated scientist.........traveling along a seemingly "deterministic" path of life, when suddenly my path is interrupted by a slight change in my initial condition, which has the effect of altering my direction. This experience in my life is a very simplistic scenario, it's true, and it describes an event that could very possibly lead to chaos. By the time we've arrived at the end of my paper, I'm certain that my path will have been forever changed through chaos.

Man has always been fascinated by the heavens. He's always sought out simple rules (explanations) or laws that govern the universe. Much of his attraction to the heavens, I'm sure, had to do with an appreciation for the sheer natural, physical beauty of the universe...............the brilliance of the millions of stars and planets...........sparkling within the seemingly endless blackness of space. The other part fueling this fascination for the heavens, no doubt, had to do with the curious nature of man. Man one night long ago, looked out into space and began to ponder the question of......What makes it all work? "How many are there.......".........How vast is it?.......What is the order?.......and "Why?" were perhaps just a few examples from a long lists of questions that were raised and pondered for years and years until some explanation was articulated that proved acceptable to the authors of the questions. Early Greek astronomers (circa 400B.C) worked out a mechanical system to explain the motions of the planets, This system was built on a geocentric (earth-centered) premise and taught that the planets, the sun, the moon and the stars all revolved around the Earth. Much later on (circa 1700) man had moved away from the geocentric theory and by 1720 most scientist and philosophers agreed that the sun was truly the center of the universe. At any rate.......it's always been important (perhaps it's curiosity or perhaps it's ego) for man to control things and be able to say that........."this is exactly the way it is today, right now at this moment........and that is the way it will be tomorrow." Isaac Newton theorized that the universe was a systematic organization of poetry in motion........ With clockwork precision.......The Sun, occupying the center space of the universe, with all the other planets and celestial bodies revolving in deterministic orbits around the center. Newton hypothesized that if two bodies attract each other with a force that was proportional to the square of the distance between them......the resulting motion of one body relative to the other could be shown in a mathematical curve called a conic section. This model or curve would embrace and further advance the supposition that a planet's path, moving under the effects of the Sun's gravity, would be elliptical and the the period of this motion would depend only on the average distance from the Sun. Armed with this information, Newton felt pretty good about his ability to make predictions based on this understanding of the universe. And for the most part, this understanding was correct. In Mathematical terms, Newton was able to demonstrate that the 'two-body' problem was integrable.......meaning that you could derive at a complete, practical solution to the problem by using relatively simple mathematical equations. Kepler's law of equal areas: the ratio of the square of the period and the cube of the semimajor axis (one-half the larger axis of an ellipse) is the same for all planets.

As Earth revolves about Planet

the Sun, equal areas are

swept out in equal periods

of time........

The Lorenz attractors look rather like a mask with two else, but twisted so that the left and the right-hand sides bend in different directions. How can this lead to chaos? The answer is geometrical, and simple.......Trajectories wind around the two else of the mask, where both else merge together. Points that start off close together get stretched apart as they circulate around the attractor so they 'lose contact' , and can follow independent trajectories.......making the sequences of lefts and rights unpredictable in the long term. Stretching points apart and 're-enjecting' them back into small regions, is typical of all strange attractors. (Hall, pg 52)

This paper was written as an assignment for Ian Walton's Math G -Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.