Full Blown Chaos

I'm here! What am I learning now?

Okay, so either you got sensitivity down or you just skipped to this section. If it's the second, thanks for making this tutorial that much more difficult to write. To summarize, sensitivity is one of the conditions necessary for chaos, and is basically the concept that even minor differences in initial conditions of a system in chaos cause dramatically different results to occur in that system over the long run. It is often summarized in an interesting phrase called the Butterfly Effect.

Would you mind confusing me some more?

No problem. In the Butterfly Effect section, we learned that a weather system is chaotic; however, what thing other than sensitivity made the system chaotic. Well, there is basically one fundamental thing that made these systems chaotic: iteration. Iteration of y=-x function

Was that an attack on my mother?

Iteration is basically repeating a set of instructions over and over again beginning with some initial conditions. An example would be counting from zero. You start with the number zero, and then you carry out the instructions of adding one to the number you just got to get to the next number you are counting and repeat it over and over again. A graphical example of iteration can be seen on the left. You can see an example of it here.

What is so chaotic about counting?

Obviously, any iterative process is not automatically chaotic. There are actually three conditions an iterative process must have in order to be considered chaotic. They are: Lorenz attractor

Now, let us look at periodicity since you already know about sensitivity. It is the idea that some patterns repeat in the system after a while. The image on the right shows the Lorenz attractor, developed from plotting weather patterns and it displays this periodicity as well.

You just lost me

You will probably need a better example than a weather system to understand mixing when we get to it, and yes I slept through meteorology as well. The best such example is to look at quadratic functions. How can quadratic functions be chaotic, you ask? That's what people at Los Alamos were wondering about Mitchell Feigenbaum when he came to do research there in 1974.

Not another history lesson

Feigenbaum was supposed to be working with complex differential equations, yet he was analyzing one of the most basic types of functions, something that any high school student who takes algebra knows about. A parabola doesn't really sound that chaotic. They can be represented in the following manner:

f(x) = k*x(1-x)

Well, Feibenbaum learned that iterating with this function for with a small k value caused the function to settle at one point regardless of the starting position. As he increased the k value, he evenentually found functions that settled at the same two points, then four points, and so on, until eventually the pattern did not settle down and chaos was established. You can see an example of this chaotic system here.

You assumed that we stayed awake in algebra last semester

Using the Feibenbaum example let us look at mixing. Mixing is basically the idea that given enough runs of iteration, the system will go through all sets of intervals. Confused? Okay, let's divide all numbers between 0 and 1 into ten sections: between 0 and 0.1, etc. Let us pick the interval between .3 and .4. Now, using the chaotic function of f(x) = 4x(1-x) and having our starting value as .21, let us iterate. We get .6636 as our first response, which is not between .3 and .4, so let us continue. Plugging in .6636, we next get .8929 (I rounded for display here, but it is still to the proper decimals on my calculator), still not in the interval. The next number is .3824, which is in the interval. Go back to the example that I provided and test it out again. It will work with smaller intervals too, but it might take a while longer if you want to try it.

Last updated 8/8/99