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Chaos Theory Essay, Research Paper

A General View of the Chaos Theory

Abstract

This paper shows the development and uses of the chaos theory through its initial beginnings to its modern day principles. It will discuss the early pioneers in this science along with the developments that have occurred more recently. The chaos theory will be explained and defined in such a way as to make it understandable for anybody with a basic understanding of mathematics. The chaos theory is not just a formula, but a whole new way of examining data to determine the outcome of a system. This paper will also show modern uses of this theory and the potential uses of it in the future.

Introduction

Throughout history, educational methods and theories have mirrored the beliefs of the times about the workings of the human mind. As our technology has grown, our understandings of the universe and its abstract happenings have also. In order to fully understand these new discoveries we must continually develop new formulas. Not only do these formulas and ideas help us solve complex math problems, but they also lead to new insights in both the natural world and the human body. In addition to insights, these new ideas create more problems to be solved, in a never-ending process of learning. In the beginning of this century human kind believed that they had the universe nearly figured out. Then Albert Einstein arrived to enlighten and astound people with their misunderstanding. His new theories and formulas completely revolutionized the patterns of thought. Although the chaos theory probably won’t have the similar broad effect, it has already begun to have a major impact on our lives.

Main Body

“Many people believe that chaos theory is about disorder, but order is at the very heart of chaos.” The chaos theory as we know it had its beginnings in the in the late nineteenth century. Henri Poincare, a great French theoretical scientist, won a contest created by King Oscar II of Sweden. The solution to this contest was the first known work upon the Chaos theory. The contest read:

“Given a system of arbitrarily many mass points that attract each other according to Newton’s Laws, try to find, under the assumption that no two points ever collide, a representation of the coordinates of each point as a series in a variable that is some known function of time and for all whose values the series converge uniformly.”

This problem is called the n-body problem, and Poincare solved it, but only in a few specific cases. His work upon the problem comprised several books, but through these contributions he began the ability to find insights into chaos.

“The modern study of chaotic dynamics maybe said to have begun in 1963, when meteorologist Edward Lorenz demonstrated that a simple, deterministic model of thermal convection in the Earth’s atmosphere.” Lorenz was one of the earliest pioneers of chaos theory, and a meteorologist at MIT. “In 1960, Lorenz began a project to simulate weather patterns on a computer system. Lacking much memory, the computer was unable to create complex patterns, but it was able to show interactions between major meteorological events.” Lorenz, in a later attempt to duplicate the project, accidentally entered data that was off by under one-millionth of the original. When the computer played out this scenario, the result was drastically different from the original. “This discovery created the groundwork of the chaos theory: In a system, small deviations can result in large changes. This concept is now known as the Butterfly Effect.” After viewing the results of this mistake, Lorenz was able to deduce some astounding theories.

“He demonstrated visually that there was structure in his chaotic weather model that, when plotted in three dimensions, fell onto a butterfly-shaped fractal set of points of a type now known as a strange attractor. Lorenz rediscovered chaos and proved that long-range forecasting of the weather was impossible”

After seeing this plot he attempted more experiments that would show the same results. His most famous attempt was a water wheel.

“Examining the motion of his chaotic water wheel, Lorenz was able to model its motion using a series of equations and three variables. Lorenz hypothesized that a graph of these variables would stop at a given point or that a loop would eventually be reformed and retraced.”

Contrary to Lorenz’s prediction neither happened. “It traced a strange, distinctive shape, a kind of double spiral in three dimensions, like a butterfly with its two wings. The shape signaled pure disorder, since no point or pattern of points ever recurred.”

This new pattern discovered by Lorenz was iteration. Iteration is important to the understanding of chaos. There are five types of outcomes for consecutive number iterations. “The equation may converge, or it may become intermittent. The equation may increase quickly or become periodic, but the most interesting case is chaos.” It is this last type of iteration that is most important to the chaos theory. Lorenz’s experiment, when graphed, represented a new iteration. It was named “Lorenz’s Strange Attractor” and, when plotted, was this pattern:

This butterfly shaped graph has become one of the most important in the science of chaos since its invent.

The third famous researcher of chaos, and perhaps best known because of a type of set named after him, is Benoit Mandelbrot.

“He was a mathematician, but he relied upon the formulation of shapes in his head to solve problems, rather than the traditional algebraic approaches. Although this ostracized him from his colleagues, Mandelbrot’s unique ability proved invaluable.”

While he worked at IBM, Mandelbrot was asked to figure out a problem that was occurring in data

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