Is Chaos Predictable?

By: Aga Freund

What Is Chaos Theory?

“Chaos theory attempts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions. A common example of this is known as the Butterfly Effect. It states that, in theory, the flutter of a butterfly's wings in China could, in fact, actually effect weather patterns in New York City, thousands of miles away. In other words, it is possible that a very small occurance can produce unpredictable and sometimes drastic results by triggering a series of increasingly significant events.” – from http://library.thinkquest.org/3120/

History of ChaosMeteorologist, Edward Lorenz, was working on weather prediction problem in 1960, using a computer set up with a set of twelve equations modeling the weather. It predicted theoretically what might be the weather and not what it was. In 1961, he went back to a specific sequence, starting in the middle and let it run. After one hour, the sequence developed otherwise. Ended up totally different than the original. Difference: computer saved to six decimal places, he did to three on the printout to save the paper.

History of Chaos Cont.

Lorenz proved that digits after the third decimal place can have a big effect on the outcome of the experiment, which became to be known as the butterfly effect. Meaning, that small events can have huge effect on changing the prediction of long-term behavior. This phenomenon is also known as sensitive dependence on initial conditions. So, he indicated that predicting the weather accurately is impossible, which led him to eventually develop chaos theory. He came down from twelve to three equations in the system (Lorenz system), of which the outcome always stayed on a curve, double spiral. He called the image he got the Lorenz attractor.

Mathematics of Chaos

Scientists and mathematicians started to play with plotting and exploring equations. It produced nature-like looking pictures (ferns, clouds, mountains, and bacteria). They acted the same as stock change, populations, and chemical reactions simultaneously. The theory had to do with lots of different intellectual domains, so they began plotting fractals. Chaotic systems have defining features and are not random.

Fractal geometry, described in algorithms, explains chaotic systems found in nature.

Defining Features of Chaotic Systems

1. Deterministic – something determines their behavior.

2. Very sensitive to initial conditions, making the system fairly unpredictable.

3. Appear disorderly or random, but are not.

Animated Fractals: http://library.thinkquest.org/3120/library.html

Real World & Chaos

Analysis of chaos indicated that the market prices, while highly random, have a trend, the amount of which differs from market to market and from time frame to time frame.

http://library.thinkquest.org/3120/realife.html

The coastline of Great Britain is infinite

http://library.thinkquest.org/3120/realife.html

Long range weather forecasting is not possible to be completely right due to effects of chaos.

ExampleGeorge T. Yurkon tried

to use eight significant digits on initial conditions, however, the butterfly turned out lopsided.

Example

Example

http://www.csuohio.edu/sciences/dept/physics/physicsweb/kaufman/yurkon/chaos.html

Some Examples of Chaos

The Cantor Set

The Sierpenski Triangle and its Area

The Mandelbrot Set

The Julia Set

http://library.thinkquest.org/2647/chaos/chaos.htm

Conclusion I think that chaos can be predicted to happen in

some cases. However, its outcome cannot be calculated precisely. The range for the outcome to occur in, though, is possible to be calculated using the Lorenz System’s three differential equations.

Questions

Resources -Websites Used

http://library.thinkquest.org/3120/

http://library.thinkquest.org/3120/history.html

http://en.wikipedia.org/wiki/Lorenz_system

http://library.thinkquest.org/3120/history.html

http://library.thinkquest.org/3120/library.html

http://library.thinkquest.org/3120/realife.html

http://www.csuohio.edu/sciences/dept/physics/physicsweb/kaufman/yurkon/chaos.html

http://library.thinkquest.org/2647/chaos/chaos.htm