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Chaos Theory

Chaos theory is about discovering the order of things from seemingly random events and information. Mathematics, physics, economics, and philosophy are used to study initial conditions of sensitivity. This sensitivity of cause is referred to as the “Butterfly Effect.”
This phrase derives from work by Edward Lorenz in 1972. He was a meteorologist working on weather prediction. He developed a set of numerical sequences and entered them into a computer to see when a pattern would develop. He started in the middle of the sequence to save time and left the computer running.

What happened after Lorenz returned to the computer later is that the pattern had changed. He later discovered the computer had changed the numbers to six decimal places in its memory while he was printing just three decimal places. Although his starting number was .506127, he had entered the first three digits or .506. These three digits should have been enough to replicate his pattern, with the last three digits being thought of as insignificant to the outcome. Because the variation is so small the diagram showing the difference in the starting points of the two curves was comparable to a butterfly’s wings flapping and the effect was called the butterfly effect.

While Edward Lorenz’s project did not predict the weather, he did develop theories of prediction. His work posed the question “Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?” The idea is that the flap of a single butterfly's wing could have an effect on the atmosphere. The phenomenon is chaos theory which can be described as sensitive dependence on initial conditions. The slightest difference in the starting point or initial conditions can dramatically affect the long-term outcome.

http://web.mit.edu/newsoffice/2008/obit-lorenz-0416.html
Edward Lorenz, father of chaos theory and butterfly effect, dies at 90

http://eapsweb.mit.edu/research/Lorenz/Can_chaos_90.pdf
Chaos: Can Chaos and Intransitivity Lead to Interannual Variability?

http://eapsweb.mit.edu/research/Lorenz/Chaos_spontaneous_greenhouse_1991.pdf
Chaos Spontaneous Climatic Variations and the Greenhouse Effect

http://eapsweb.mit.edu/research/Lorenz/Energy_and_NWP_1960.pdf
Energy and Numerical Weather Prediction

http://eapsweb.mit.edu/research/Lorenz/Climatic_math_70.pdf
Climatic Change as a Mathematical Problem

http://eapsweb.mit.edu/research/Lorenz/Investigating_predictability_1972.pdf
Investigating the Predictability of Turbulent Motion
The difference can be so small as to be unrecognized. The slightest immeasurable inaccuracy can change the results. Lorenz determined this makes it impossible to accurately predict weather. Lorenz began searching for a simpler system that had sensitive dependence on initial conditions. His initial number pattern was made up of twelve equations. He simplified this to three equations that precisely described the action of a water wheel.

On a water wheel the water drips down to each compartment from the one above. If the water moves too slowly the wheel never turns. As the water speeds up the weight of the water turns the wheel and can be continuous. The water can also move so fast as to reverse the direction of the wheel. The mathematical equations to describe the action seemed to be random. But when Lorenz began placing them on a graph he was surprised to discover the amounts created a curve or a double spiral.

Until then there had been just two types of order known. One was a fixed state where variables were constant. The other type of order was periodic behavior when there is a loop of continual repetition. These new equations followed a spiral. This graphed image of the equations is the Lorenz attractor. That year was 1963 and Lorenz published his findings in a meteorology journal. Although revolutionary, his findings were not discovered by mathematicians or physicists for years.

The simple act of a coin toss demonstrates sensitive dependence on initial conditions. The outcome of a coin toss is affected by how far it is tossed in the air or how soon the coin will land and how fast the coin is turning or flipping. Yet these variables are not humanly controllable in any way to predict heads or tails landing face up.

Benoit Mandelbrot of IBM was a mathematician who was analyzing the fluctuations in cotton prices. None of his outcomes reflected the normal distribution so he collected all pricing data available in the century and discovered something.

The same numbers that generated irregularities from the aspect of normal distribution, created exact symmetry when scaling. While the price changes were completely random, the sequence of changes was independent on scale: daily and monthly price change curves corresponded exactly. All points of change were constant over a sixty year period.

Mandelbrot later applied this thinking to measuring coastlines for maps. The more he magnified the coastline the more inlets and bays he found. Even at the microscopic level there were inlets between grains of sand. The exact measurement of the coastline would constantly change with magnification.

Helge von Koch, a mathematician, explains this concept in a graphic now called the Koch curve.
The Koch curve presents a paradox of infinite length surrounding a finite area. To circumvent this mathematicians created fractal dimensions. The fractal dimension of the Koch curve is about 1.26. While fractals are impossible to conceive, they are logical. Fractal is defined as an image that has the attribute of self-similarity. The Lorenz Attractor and Koch curve are fractal.

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Koch.html
Helge von Koch

http://www.bookrags.com/biography/niels-fabian-helge-von-koch-wom/
Helge von Koch

http://www.tgmdev.be/curvevonkoch.php
The Von Koch Curve

http://mathworld.wolfram.com/Fractal.html
Fractal definition and examples from Wolfram Math World

http://archives.math.utk.edu/topics/fractals.html
Links all about fractals.

http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm
The Koch Curve

http://www.jimloy.com/fractals/koch.htm
The Koch Curve: graphic examples.

In the 1960s chaos was not accepted as relevant by much of the scientific community. Most research was being performed by meteorologists. Eventually a scientist named Feigenbaum discovered that bifurcations come at a constant rate or scaling factor of 4.669. This was a revolutionary discovery that would help scientists examine chaotic equations. Easy equations could be used to predict the conclusions of intricate equations.

Fractal structures exist in reality and are all self similar. Coastlines, blood capillaries, tree branches, stock market data, are all fractal. By recording time periods between drips of water from a faucet, scientists found that at specific flow velocities, the drips came at irregular times. However when the data was placed on a graph, dripping followed an exact pattern. Chaos has applications in the art world as well. Digital art uses chaos and fractals with math formulas and music has been created using fractals.

Chaos theory changed the direction of science. Chaos has had a lasting impact on the world and rates with relativity and quantum mechanics as the greatest theory discoveries of the twentieth century. The characteristics of chaos can be found in ocean currents, wind, and the very flow of blood through our bodies. The theory of chaos will be applied to science and our world with new discoveries for decades to come.

http://eapsweb.mit.edu/research/Lorenz/publications.htm
Edward Norton Lorenz Publications

http://www.stsci.edu/~lbradley/seminar/butterfly.html
The Butterfly Effect

http://www.harcourtschool.com/activity/biographies/lorenz/
Edward Norton Lorenz biography.

http://www.fortunecity.com/emachines/e11/86/mastring.html
Mastering Chaos

http://library.thinkquest.org/3120/
Chaos projects by students.

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