Chaos Essay, Research Paper

Introduction

Chaos is unpredictable behavior arising in a realistic system because of great sensitivity to beginning circumstances. Chaos arises in active systems. If two very close starting points diverge even a tiny bit, their future behavior is eventually unpredictable.

Every change in the system will compound with time, so, very slight changes in the starting point can lead to enormously different outcomes. Also, because of the extreme disorder, predicting the future path of the system is practically impossible. The behavior is too sensitive to the conditions, so therefore it is always changing. Chaotic behavior, although appearing random, arises from a very hard basis and it is very sensitive to any disturbances.

The system the above paragraph is referring to can be anything. A set of equations is a system, as well as weather patterns. All systems display chaotic properties. For example, weather forecasts are never totally accurate. Even if the forecast is for the week or a day, it may be totally wrong. This is due to minor disturbances in airflow. Each disturbance may be minor, but the changes will increase and add up in time. Soon, the weather will be far different than what was expected.

These same chaotic traits apply to math. Certain sets of equations, for example, can be repeated many times, creating images called fractals. Often, fractal equations consist of only an X and Y variable and a few constants. Once the equations are repeated many times, and the results are plotted on a computer screen, incredibly complex images can be produced. These pictures, called fractals, exhibit all of the chaotic traits. They are very sensitive to small changes, they are unpredictable, and they appear chaotic, even though they were created using very straightforward, non-chaotic equations. Now, that brings us to chaos theory.

Chaos theory is a critical part of science, mathematics, art and computing. It proves that the way to express an unpredictable system is in representations of the behavior of a system. So, chaos theory, which many people believe is about unpredictability, is actually about predictability in many different systems. Chaos theory arose as scientists and mathematicians started to program numbers in the computer. They tried different ways of plotting and exploring equations to get different results. After investigating, the scientists found out many new ideas and discoveries.

A common example of chaos theory is known as the Butterfly Effect. In theory, the flutter of a butterfly’s wings in China could actually effect weather patterns in New York City, thousands of miles away. This means that a very small movement can produce unpredictable and sometimes drastic results by triggering a series of events. Using mathematical rules, chaologists, scientists who specialize in the study of chaos, can create complex dynamical systems that resemble natural events like the flocking patterns of birds that land on the water, or the growth of a fern in the forest.

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History

Edward Lorenz was a meteorologist that first experimented in chaos. In 1960, he was working on weather prediction, with a set of twelve equations to model the weather. His goal was to predict the weather for a period of time. Even though it didn’t predict the weather itself, the computer program did theoretically predict what the weather might be.

One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the numbers and let it the computer run and get the result. When he came back an hour later, the sequence had changed. Instead of the same pattern as before, it diverged from the pattern and ended up completely different from the original. He soon figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.

Using all the traditional ideas of the time, his experiment should have worked. He should have gotten a sequence very close to the original sequence. When a scientist can get measurements with accuracy to three decimal places that is good enough. Since the fourth and fifth digits are impossible to measure using reasonable methods, the first three digits should have been somewhat close to the original, but Lorenz had proved this idea wrong. This work of his became known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings.

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Chaos and Fractals

One of the intriguing components of chaos theory are the complex pictures known as fractals. There is a strong link between chaos and fractals. For example, fractal geometry is a geometry that describes the chaotic systems we find in nature. Fractals are a language, a way to describe geometry. Fractal geometry is described in algorithms, which are a set of instructions on how to create the fractal. Computers translate the instructions into the patterns that we see and call fractal images.

These same chaotic traits apply to math. Certain sets of equations can be repeated many times, creating images called fractals. Let’s say these two equations consist of only an X and Y variable and a few constants. Once the equations are iterated many times, and the results are plotted on a computer screen. Soon, incredibly complex images, called fractals, can be zoomed in so much that the patterns just keep repeating. Fractals exhibit all of the chaotic traits. They are very sensitive to small changes, they are unpredictable, and therefore they are chaotic, even though they were created using very straightforward, non-chaotic equations.

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Chaos and Computers

The computer is our microscope, our telescope, and our art gallery. We cannot really explore chaos without it, and it would be almost impossible to draw a perfect fractal freehand. To produce the Mandelbrot Set on a single screen takes an estimated 6,000,000 calculations. There isn’t any sane human in this world that would be stupid enough to endure the boredom. Since computers are particularly good at patterns and repetition, they play a big part in chaos and the world.

Most people use the computer as a tool, and most computer use by scientists and chaologists are based on programming data information into the computer and instructing the computer on what output is required. Chaos theory arose as scientists and mathematicians started to program numbers into their programs and watch as lines careered around the complex plane in detailed patterns. While the scientists were experimenting with mathematicians, science and computer programming produced images that looked like nature. Some of the images produced were ferns, clouds, mountains and bacteria.

Bibliography

Devaney, Robert L. “Chaos in a Classroom” (On-line) Internet: Sun Apr 2 14:31:18

EDT 1995 Available at http://math.bu.edu/DYSYS/chaos-game/chaos-game.html

Ho, Andrew “Chaos Introduction” (On-line) Internet: November 27, 1995 Available at

http://www.students.uiuc.edu/~ag-ho/chaos/chaos.html

Chaos group “Chaos at Maryland” (On-line) Internet: December 11, 1998 Available at

http://www-chaos.umd.edu/

Trump, Dr. Matthew A. “What is Chaos? A five part-online course for everyone”

(On-line) Internet: August 14, 1998 Available at http://order.ph.utexas.edu/chaos/index.html

University Chaos Group (UCG) “Chaos Theory” (On-line) Available at

http://ikiserver.bok.ac.at/~martin/chaos.html#theory

Chaos “Chaos! Portal” (On-line) Internet: 1998 Available at

http://members.xoom.com/chaos3/first.html

Chaos Lab of Georgia Tech “Applied Chaos Lab” (On-line) Available at

http://acl2.physics.gatech.edu/aclhome.htm

Chaotic Systems Experimentation “Chaos Theory: Human Experimentation in Economics and Anthropology” (On-line) Available at

http://www.sellhigh.com/chaos.html

Prinz, Theodor “Chaos Computer Club” (On-line) Available at http://www.ccc.de/

K. Alligood, T. Sauer, J.A. Yorke “Chaos” (On-line) Available at

http://math.gmu.edu/~tsauer/chaos/intro.html

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