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Chaos Theory

Since its inception, science relied on predictability and order. The true beauty of science was its uncanny ability to find patterns and regularity in seemingly random systems. For centuries the human mind as easily grasped and mastered the concepts of linearity. Physics illustrated the magnificent order to which the natural world obeyed. If there is a God he is indeed mathematical. Until the 19th-century Physics explained the processes of the natural world successfully, for the most part. There were still many facets of the universe that were an enigma to physicists. Mathematicians could indeed illustrate patterns in nature but there were many aspects of Mother Nature that remained a mystery to Physicists and Mathematicians alike. Mathematics is an integral part of physics. It provides order and a guide to thinking; it shows the relationship between many physical phenomenons. The error in mathematics until that point was linearity. “Clouds are not spheres, mountains are not cones, the bark is not smooth, nor does lightning travel in a straight line.” – Benoit Mandlebrot. Was it not beyond reason that a process, which is dictated by that regularity, could master a world that shows almost no predictability whatsoever?

New science and a new kind of mathematics were developed that could show the universe’s idiosyncrasies. This new amalgam of mathematics and physics takes the order of linearity and shows how it relates to the unpredictability of the world around us. It is called Chaos Theory.

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The secular definition of chaos can be misleading when the word is used in a scientific context. As defined by Webster’s dictionary chaos is total disorder. That may lead one to believe that chaos theory is indeed the study of total disorder, which it truly is not. In 1986 at a prestigious conference on Chaos another definition for chaos was introduced. It is stochastic behaviour occurring in a deterministic set. This definition of chaos was hesitantly brought forth. The scientists, mathematicians and intellectuals present were hesitant to define a concept they did not truly understand yet. They left the scientific community with a rather cryptic and oxymoronic definition of chaos. Deterministic sets behave by precise unbreakable law. Stochastic behaviour is the opposite of deterministic it has no finite laws, it is totally dependant upon chance. The dissected definition of chaos is lawless behaviour that is ruled entirely by law. (Stewart 16-17)

The principles of Chaos Theory are complex and abstract. Perhaps the simplest and most essential ideas behind chaos theory are embodied in the aphorism known as the Butterfly Effect. The butterfly effect states that the flapping of a butterfly’s wings in Hong Kong can change the weather in New York. It means that a minuscule change in the initial conditions of a system, in this case, the weather, is magnified greatly in the end conditions of that same system. The ultra-sensitivity to the initial conditions of a system was not a new and striking discovery. In fact, it was shown in ancient folklore;

“For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, the battle was lost;
For the want of a battle, the kingdom was lost!”

The smallest variation in the initial conditions of a system can result in huge differences in concluding events. There was no nail, and because of this seemingly insignificant detail in the initial condition, the kingdom was lost. Another example of the butterfly effect is two pieces of wood floating on a river. Place those two logs at nearly the same point on the river and let them go. It is absolutely impossible to predict where those logs will be later downstream. When those logs are set on the water a slight breeze, a fish that swims underneath one of them, or even a single droplet of additional water in the initial stage can totally change the end result until no resemblance between the two is seen. (Briggs, Peat 49) There is a definite correlation between that small butterfly and a storm in New York, as well as the two logs. Chaos Theory states that within the unpredictability that makes those changes there is indeed a specific order. Chaos works in order and within all order there is chaos.

The butterfly effect as well as the two logs depend solely on iteration. Iteration is the feedback that continually reabsorbs its predecessors. Iteration is a very common process, which can appear in fields as diverse as artificial intelligence or the cycling replacement of cells in the human body. (Briggs Peat 66) Iteration provides a sort of self-reference. For example, the word “time” is defined with words such as “period” or “instant”. Look up the definition of those words and it will eventually lead back to the word “time”. (Briggs Peat 68)

MIT meteorologist Edward Lorenz has the distinction of being the first person to show how iteration creates chaos. In 1960 he was solving non-linear equations on his computer that would show a model for the earth’s atmosphere. He repeated a certain forecast to check his data and when he substituted the numbers in the second time he rounded off the figures to three decimal places instead of the six he received initially. He plugged in these numbers and left the computer. He returned to a surprise. The forecast before him was not a double check on his previous information, it was a totally new forecast altogether! That three decimal place difference between the two sets of numbers had been magnified greatly in the process of solving those equations. (Briggs Peat 68-69)
Just as the butterfly effect embodies the principles of Chaos Theory, a single image has become an emblem for the early pioneers of chaos. The Lorenz attractor (Figure 1) is a magical image that resembles an owl’s mask or a butterfly’s wings. (Gleick 29) Fig. 1

Lorenz then tried to model the chaos of a gaseous system, like the earth’s atmosphere. He used his knowledge in the physics field of fluid dynamics to simplify three equations to invent the following three-dimensional system of equations:

dx/dt=delta*(y-x)
dy/dt=r*x-y-x*z
dz/dt=x*y-b*z

Where delta is an inconsequential constant for which Lorenz used a value of ten. The variable r is the difference in temperature between the top and the bottom of the gaseous system. The variable b is the width to height ratio of the box, which contains the gaseous system; Lorenz used 8:3. When a gas is heated from below it tends to organize itself into a cylindrical form. Hot fluid rises to the top, loses heat and falls to the bottom otherwise known as convection. As the temperature increases, the cylinder becomes wavy and then become wild and chaotic. (Gleick 25) The resulting x in the equation is the rate of rotation of the cylinder, y is the difference in temperature at opposite sides of the cylinder, and the variable z represents the difference of the gaseous system from a line, which represents temperature. When Lorenz plotted these three equations no geometrical shape or curve appeared, but the weaving object known as the Lorenz Attractor. The system never repeats itself, so the diagram never intersects. It loops around and around forever. The motion of the attractor is theoretical but it accurately conveys the action of the real system.

The dimensions seen in everyday life are rather straightforward and comforting; zero, one, two, or three. Chaos theory speculates that the world may not be all that cut and dry. Consider the dimension of a ball of string. From a great distance, the ball is a point and had no dimension. From a few feet away it looks normal and has three dimensions. From a minute distance, a single thread is seen as a weaving line with one dimension. At an even lesser distance the line turns into columns of definite thickness, it has three dimensions. Closer still the thread is lost to individual hairs the ball is again one-dimensional. (Briggs Peat 94) The twisting and turning of the ball of yarn very closely resemble the contortion of the Lorenz attractor. Both figures have a non-integral dimension, the defining trait of a fractal dimension. The irregularity and detail of these two objects illustrate fractal geometry. (Briggs Peat 95)

Fractal Geometry was developed by Benoit Mandlebrot, a polish mathematician who has influenced the work of Gaston Julia. During World War I Julia started sketching fractal shapes, which were unexplainable through the methods of Euclidean geometry. (Gleick 221) Fractals are defined by infinite detail; infinite length, no slope, a fractional dimension, self-similarity, and they can be generated by iteration. (Briggs Peat 95) An example of a fractal shape is the Koch Curve, or the Koch snowflake (Figure 2). (Gleick 93)

Fig. 2

It starts off as an equilateral triangle, adding to each side another triangle in the middle. This process is repeated to infinity. The length of the boundary created by this fractal is infinite yet the area of the curve is less than the circumscribed circle around the original triangle. (Gleick 99) “An infinitely long line surrounds a finite area.” – James Gleick

The two concepts of fractals and attractors are intimately linked. Through fractal geometry, it is found that attackers are indeed fractal curves. Wherever there is chaos there must also be its visual representation, fractal geometry. This suggests a connection between every chaotic process. The formation of branches of the human lung and the motion of a fast-flowing river can now be seen as nearly identical. Both chaotic processes emerge from a fractal order. Fractals are another amazing contradiction in Chaos Theory. Fractals are both complex and simple. They are complex because of their infinite detail and structure and as unique as the human fingerprint, no two fractals are the same, yet they are simply because they are formed by the successive applications of simple iteration. (Briggs Peat 95-97)

Benoit Mandlebrot was an intellectual renaissance man. He was a very gifted man with an amazing brain and an ego to match. He was once invited to speak at Harvard University. He entered Harvard’s Littauer Center only to find the diagram he was going to use already on the blackboard. He jokingly asked the hosting professor how his information arrived before he did. It turns out that the diagram on the board was eight years of cotton prices. Mandlebrot diagram was that of income distribution in an economy. Two unrelated topics, which showed the same trends. (Gleick 83-84) This is an example of self-similarity. It manifests itself in many other ways. Fractals are self-similar; in that case, at higher and higher magnification the fractal image resembles the original. (Figure 3)
Fig. 3

The stock market is indeed chaotic and also self-similar. It is truly random but shows an orderly trend. It is highly dependent upon initial conditions, but because it is nearly impossible to describe those initial conditions it is impossible to predict the action of the market. Short term trading is random and futile. Long term trading however is not random at all. (Gleick 85) A deterministic order comes from chaos over time.

Chaos Theory has made quite an impact on the modern world. Even in its infancy, it has been a powerful tool in shaping popular thought of the natural world. Once dismissed as a theoretical science with no practical application, chaos theory has blossomed into an intricate and beautiful pattern, much like the fractals it deals with. Chaos theory is a complex combination of math and physics, but with its mastery comes a new era in the human understanding of the world around us.
A. A violent order is disorder: and
B. A great disorder is an order.

These two things are one
Wallace Stevens
“Connoisseur of Chaos”

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