Part of the problem with explaining chaos theory is that the word "chaos" has different meanings. The normal everyday meaning of the word is "a condition or place of great disorder or confusion"*, which sounds similar to the meaning of randomness: "having no specific pattern"*. However, when we use the word chaos in a mathematical or scientific sense, it means something very different.
I find that the most common mistake beginners make is associating the everyday meaning of chaos with the mathematical meaning. Most academic institutions will refer to chaos by a different name such as "non-linear dynamics" (don't bother trying to figure out why they would call it that if you don't already know). However, most people are more familiar with the term chaos math and I think it sounds more interesting so I will use it throughout this page.
For starters, a chaotic system is not a random system. A chaotic system sometimes SEEMS random if you do not recognize that it is chaotic. For example, a roulette wheel is a chaotic system (trust me, it's not random, I'll explain why later). If you look at a fairly normal system such as a ball bouncing straight up and down against the ground, and you want to find out how high this ball will be after a certain amount of time, you would want to find out at what height the ball is dropped from, the strength of gravity, etc. and then you could easily plug these numbers into equations and then get the right answer. This is what happens in theory. In practice, since you will have to actually make measurements, what might happen is that your measurement might be a millimeter too high. If you did this, what would happen to the value that you would calculate with the slightly incorrect value? You would get an answer that was extremely close to what the ball actually did. A slight error in measurement is barely even noticeable.
Now lets consider a chaotic system, such as a ball on a roulette wheel. You might say, "hold on now, isn't a roulette table random?!", but it really isn't. Doesn't a bouncing roulette ball obey the same laws of physics as the ball bouncing against the ground? Because its behaviour is governed by laws that dictate exactly where the ball will land, the roulette wheel is not random. Another way to say that this system is not random, is to say that it is deterministic. If you wanted to find out where on the roulette table a ball would land you would need to find how high the ball was dropped from, how fast the table is spinning and the dimensions of the roulette table. Then, in theory you should be able to plug these numbers into equations and get the right answer. Again, in theory, you should be able to predict how the ball bounces straight up and down against the ground as easily as you can predict how a roulette ball will bounce on a roulette wheel. Of course if you actually tried to predict where the roulette ball would land, you'd get the wrong answer (otherwise gambling would much too easy).
Why are the two different? Like I said, in theory, you should get the exact answer if you plug the numbers into equations and you measured everything exactly right. However in practice, if you make a small mistake, with the bouncing ball you'd find that the answer you got from the equations would be off by only a little. With the roulette ball, however, that small inaccuracy will mean that the ball will take a COMPLETELY different path from the one you predicted and could land on the other side of the table.
If you can understand this point, then you understand the basic concept of chaos.
When a system like a roulette wheel can be thrown dramatically off-course with a very small change, we say that it is "sensitively dependant on initial conditions".
A very popular saying that is related to chaos theory is "a butterly that flaps its wings can cause a hurricane on the other side of the world". The saying is often used in many slightly different ways, but the meaning is the same: a very small change can have huge consequences. Althought this saying sounds exagerated. It is literally true. The weather is "sensitively dependant on initial conditions". It is a chaotic system like the roulette wheel discussed previously.
Modern weather prediction works by collecting a large amount of data about the current weather and then using a computer to determine what the weather will be like in the future. The computer is basiclly taking the measurements and putting them into equations so that it can calculate the expected behaviour very quickly. Measuring instruments that are used to measure atmospheric conditions are remarkably accurate, and governments spend billions of dollars on weather prediction satellites and computers. Did you ever wonder why we're still not able to get weather forecasts for much more than a week in the future? The reason of course is that its impossible to measure or calculate the weather with perfect accuracy and since weather is a chaotic system, a tiny inaccuracy will cause your prediction to be completely different from it will really become. This tiny inaccuracy could be cause by a number of things including inaccuracy of the measuring instruments, approximation of measurements, or even a butterfly that flapped its wings unexpectedly. I hope you can see why chaotic sytems are impossible to fully predict. Even if the government got more accurate instruments and more powerful computers, it would still not be good enough. Measurements must be done at an infinite precision to avoid the effects of chaos. This is of course impossible in reality.
Although long-term weather prediction is impossible, just as predicting the final location of a roulette ball is impossible, predictions on chaotic systems can approximate the actual system for a short time before they veer off-course. That is why we are able to have relatively accurate weather forecasts for a few days in the future. In the next section on calculator chaos, you can a chaotic system actually veering off-course.
Still confused about what chaos is? Hopefully this will help. Chaos doesn't just relate to complicated systems and abstract theories - you can see it in numbers. Try iterating this function: 2x2 - 1, with a starting value for x between 0 and 1 (but not 0 or 1). If you're not familiar with the concept of iterative functions, you can caluculate an iteration of an iterative function by starting off with an initial value for x, plugging it into the equation and then using the result of the function as your value of x in the next iteration. You can do iterations of an iterative function over and over again. You can do this on a calculator, write a computer program or use a spreadsheet. I chose the value 0.75. If you look at a graph of the function, it might look a little irregular, but not particularly interesting. The interesting part is that if you chose a value like 0.74999, which is very close to 0.75, and graph it, you'll see that it is similar at first, but then becomes totally different. This ties in with how the initial state of a system can totally change what the final result will be.
The creator of the word fractal, Benoit Mandelbrot, explains, "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible - and how appropriate for our needs! - that, in addition to "fragmented" (as in fraction or refraction), fractus should also mean "irregular," both meanings being preserved in fragment."
A fractal is usually a rough geometric shape. When examining just a small part of a fractal, you notice that it often looks similar to or exactly like the whole fractal. Fractals can be based on mathematical models, but they can also occur in real life. Examples of natural fractals are clouds, coastlines, lightning and mountains. For some other examples of fractals, look in the pictures section.
Fractals are also said to have a fractal dimension or Hausdorff-Besicovitch dimension, they both mean the same thing though. A lot of people are confused by this fractal dimension because it means that an object isn't necessarily one-dimensional or two-dimensional. Objects could have a dimension of 1.5, 2.81 or 3. The actual computation of a fractal dimension is a little complicated, but it can be thought of like this: an object like a straight line is one dimensional, an object like a square is two dimensional, an object that is a very curved and crinkly line (as many fractals are) is better at filling space than a smooth line, but not as good as a square, so the dimension of the crinkly line is somewhere in between the two. Here is a fractal with dimension 1.26, it's the Koch Snowflake.