(sources) http://www.faqs.org/faqs/fractal

http://www.imho.com/grae/chaos/chaos.html

A: Chaos is apparently unpredictable behaviour arising in a deterministic system because of great sensitivity to initial conditions. Chaos arises in a dynamical system if two arbitrarily close starting points diverge exponentially, so that their future behaviour is eventually unpredictable.

Weather is considered chaotic since arbitrarily small variations in initial conditions can result in radically different weather later. This may limit the possibilities of long-term weather forecasting. (The canonical example is the possibility of a butterfly's sneeze affecting the weather enough to cause a hurricane weeks later.)

Devaney defines a function as chaotic if it has sensitive dependence on initial conditions, it is topologically transitive, and periodic points are
dense. In other words, it is unpredictable, indecomposable, and yet contains
regularity. All good and Yorke define chaos as a trajectory that is exponentially unstable
and neither periodic or asymptotically periodic. That is, it oscillates irregularly without settling down.

Put simply, it is the idea that it is possible to get completely random results from normal equations. Chaos theory also covers the reverse: finding the order in what appears to be completely random data.
When was chaos first discovered? The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn't predict the weather itself. However this 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 number off his printout and left to let it run.
When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. (See figure 1.) Eventually he 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.

By all conventional ideas of the time, it should have worked. He should have gotten a sequence very close to the original sequence. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Surely the fourth and fifth, impossible to measure using reasonable methods, can't have a huge effect on the outcome of the experiment. Lorenz proved this idea wrong.
This effect came to be 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.
The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)
This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behaviour of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment. Such things are impossible to avoid in even the most isolated lab. With a starting number of 2, the final result can be entirely different from the same system with a starting value of 2.000001. It is simply impossible to achieve this level of accuracy - just try and measure something to the nearest millionth of an inch!
From this idea, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory.

http://millbrook.lib.rmit.edu.au/exploring.html Exploring Chaos and Fractals

http://www.cc.duth.gr/~mboudour/nonlin.html Chaos and Complexity (by M.Bourdour)

http://ucmp1.berkeley.edu/henon.html Experimental interactive henon attractor

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