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Chaos theory

Chaos Theory: A Modern Revelation

How is it that a subject so entwined in nearly every aspect of every-day life, can be so overlooked? Such is the case of the recently discovered chaos theory. Upon hearing the term, one would deem it as just another proposition that scientists seem to manufacture endlessly. However, in An Introduction to Chaos Theory and Fractal Geometry, Manus J. Donahue III states that the name chaos theory is in itself actually a contradiction. He explains that it "...leads the reader to believe that mathematicians have discovered some new and definitive knowledge..." (n.p.). On the contrary, the first descriptions of chaos theory were discovered nearly thirty years ago. Now while this still may be fairly recent, at least in the realm of science discoveries, chaos theory is definitely not, by any means, any definitive knowledge. Even to this day, scientists are discovering new areas that chaos can be applied to. Chaos is not a definitive term in itself, either. Being a complex science that has applications in nearly realm of understanding, chaos theory cannot be defined in any one specific way. Jonathan Mendelson and Elana Blumenthal state the general definition of chaos, in Chaos Theory and Fractals, as the "...turmoil, turbulence, primordial abyss, and undesired randomness..." (n.p.). However, this is not the definition of chaos that scientists use. For instance, in his book, Science of Chaos, Christopher Lampton defines chaos as a system where minute variations in ordered initial conditions can have a significant enough impact on the results to make them seem random (13). Aside from the many varying definitions of Chaos theory, it can also be applied to a large number of aspects of every-day life. Some of these aspects include weather prediction, fractal geometry, and nonlinear dynamics.

The pre-existing notions of chaos began to re-emerge in the late 1960's by Edward Lorenz, an aspiring meteorologist studying at MIT. The Butterfly Effect, a concept found by mistake, became a basic principal that was used to describe chaos theory (Donahue n.p.). The name was coined by Lorenz based on the belief that if a butterfly were to flap its wings in Asia, weather patterns would be effected by it in New York (Lampton 68). It was believed that the small air currents made by the butterfly's flapping wings could, eventually, lead to huge storm systems, weather variation, and drastic unpredicted effects with the help of convection currents and other chaotic occurrences. Lorenz happened upon the principle while using a computer to analyze a sequence of data in order to predict what the weather might be. As described by Greg Rae in his article Chaos Theory: A Brief Introduction, " day, instead of starting the sequence of data over, Lorenz decided to take a shortcut and start in the middle of the sequence and proceeded to type in the data from an earlier printout" (n.p.). James Gleick furthers the description and states that Lorenz then left to take a short break, and when he returned, what he discovered from the diverging results meant not only that long-range weather forecasting was out of business, but also that a new science was emerging (17). Lorenz found that the new values that he had input seemed to coincide with the original values to an extent and then diverged wildly. He concluded that the effect that his data experienced was sensitivity to initial conditions. Eventually Lorenz used his Butterfly Effect to come to the conjecture that "Small variations in initial conditions result in huge, dynamic transformations in concluding events" (Donahue n.p.). This meant that even the minutest differences in any input in a chaotic system will result in traumatic effects, and that it would be nearly impossible to forecast the weather over a long period of time. After compiling and analyzing his unbelievable findings, Lorenz then developed three equations from the field of fluid dynamics to represent his results graphically.;

x' = 3(y - x)
y' = -xz + 26.5x - y
z' = xy - z

When he graphed these on a three-dimensional plane, the equations demonstrated the characteristics of a chaotic system exactly. The graph was both ordered and chaotic at the same time. The equations resulted in a graph that looped back and forth, but never crossed or settled down to a single point (Rae n.p.). This graph came to be known as the Lorenz Attractor, and proved that seemingly chaotic and random systems could show order in their results (Donahue).

Lorenz Attractor

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After discovering the workings of chaos in his attractor, Lorenz experimented with a new concept involving his new theory to support his findings in chaos, a waterwheel. His equations that he had previously graphed dealt with convection yet also strangely seemed to describe the pattern of a simple waterwheel. After numerous trials, he found that without even varying the speed of the water flow, he could produce an entirely chaotic system with the waterwheel, where the buckets changed speed and direction seemingly randomly (Gleik 27).

"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 (Gleick 29)." This

Lorenz Waterwheel


Together, the Lorenz Attractor and the Lorenz Waterwheel helped to establish the foundations for the new science of Chaos. But to the science community's despair, Lorenz's discoveries would not be recognized for more than a decade, due to the fact that

his papers were published in an atmospheric science journal that received hardly any recognition. After all, he was just a meteorologist. This dissertation from

Another modern area to which chaos theory can be applied to is fractal geometry. Fractal geometry was first introduced by the French mathematical physicist Benoit Mandelbrot. Mandelbrot first utilized the concept of fractal geometry when he inquired, "How Long is the Coastline of Great Britain?" in Nature magazine. In the article, he challenged the conventional method of mathematicians at the time, which was to approximate the length with a polygonal path around the country. However, Mandelbrot realized that as finer and finer resolutions of the coastline were analyzed, the total perimeter seemed to grow infinitely. He predicted that the coastline was in fact in fractal dimensions since it was entirely rough on all scales. To answer the question would require using the complex equations involved in fractal geometry. Zaodin and Iwona Abrams claim that his discovery played a key-role in the emergence of chaos theory (28). Donahue explains that fractal geometry is quite different from the standard Euclidian geometry that is taught in schools. He states, "...fractal geometry concerns nonlinear, nonintegral systems while Euclidian geometry is mainly oriented around linear, integral systems" (n.p.). This means that instead of describing lines, fractal geometry is used to describe complicated algorithms. Fractals are created by taking certain complex equations and iterating them over and over until they eventually form an intricate and intriguing image. There are two basic principals involved in the creation of fractals. First, as Donahue clarifies, fractals are self-similar (n.p.). This means that no matter how many times the image is magnified, it looks the same on all levels. Also, fractals have dimensions that include imaginary numbers, or that are non-integer. This is another aspect that makes them different from ordinary linear graphs. Some of the most basic fractals, such as the Sierpinski Triangle and Kotch Snowflake, are formed from equilateral triangles and can be performed by hand or using a simple graphing calculator. The Sierpinski Triangle is formed by repeatedly connecting the midpoints of an equilateral triangle.

Sierpinski Triangle


The Koch Snowflake is created by adding new triangles to the sides of an existing triangle.

Spellerberg 7

Koch Snowflake


These are just a few of the simplest fractals. Fractals can become very complex, like the Mandelbrot set and Julia set, yet only simple computers are required to perform the infinite iterations. Donahue believes that fractals are more appreciated for their beauty and intricate designs rather than their mathematical importance (n.p.). The general public view fractals as fascinating, yet simple computer generated pictures. Many do not recognize the intricate mathematical principals that are behind the aesthetically pleasing image.

The major figures that played a role in discovering the concept of fractal geometry are Gaston Maurice Julia and Benoit Mandelbrot. Together, they produced some of the most easily recognizable fractals. Gaston Maurice Julia, born in Algeria and injured during World War I, dedicated his life to researching the iteration of functions. Despite his hard work, much of his research and findings on the topic were forgotten until 1970, when Benoit Mandelbrot re-discovered his work. According to Mendelson and Blumenthal, Mandelbrot "...believed that fractals were found nearly everywhere in nature, at places such as coastlines, mountains, clouds, aggregates, and galaxy clusters" (n.p.). Mandelbrot was an important precursor in the furthering of chaos into other areas of math and science. Due to much of his work, fractals were even applied to real-life world applications. Two of the largest areas of science that fractals apply to are the worlds of medicine and biology. For example, fractals have been found on the surfaces of proteins within the human body (Donahue n.p.). Even the difficulty of describing and analyzing the complex irregular morphologies in tissues and cells during disease are believed able to be surmounted by the use of fractal geometry.

According to Rae, chaos also can be applied to areas outside of science. He states that chaos is used in computer generated art, and even music. The Lorenz Attractor has even been used to create variations on classical pieces. At the moment, it is being postulated that there are chaotic characteristics in the stock market. The goal of chaos theory is to explain non-linear dynamic systems. The stock market is an example of one of these types of systems, with shares running in either cycles or sets of cycles. These cycles form the base of an algorithm and by inputting the closing prices of any given share over a period of time the algorithm can determine all the possible cycles that exist by using the appropriate fractals. Once all the cycles have been found the strongest cycle can be calculated by using basic algebra. Therefore, the behavior of any given share over any span of time can be rather accurately predicted. While stock market predictions based on chaotic concepts can be made due to the regularly evolving patterns, long term predictions are practically useless.

So now that we have an acceptable definition of what chaos is, we can start to somewhat grasp this wide-encompassing concept. For reasons of simplicity, there is no need to go into depth about the technical aspects of chaos theory. The breadth of topics that the theory is applied to is so amazingly broad, that with only a general idea of the implications related to chaos, one can understand the function of chaos in modern subjects. The most general of world applications were discovered just as chaos theory was emerging.

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