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The Baker's Transformation. A dynamical system is a repeated process. It begins almost like a card trick: pick a number, any number, between 0 and 1. This number is called the "initial value". For this example, the dynamics part of the card trick is a series of instructions: double that number. If your resulting number is less than 1, leave this new number alone. If on the other hand your resulting number is greater than 1, subtract 1. Either way you end up with a number between 0 and 1 again, and this new number is called "the first iterate".
For example, if your initial value is 1/3, then the first iterate is 2/3. If your initial value is 7/10, then the first iterate is 4/10 or 2/5.
What makes this a dynamical system is repetition--that is, iteration. We plug the first iterate back in to the process to get a second iterate, and the second iterate in to get a third iterate, and so on. The sequences of iterates that correspond to the values above are
and
The first sequence repeats itself indefinitely (this is called a "periodic sequence" or "periodic orbit") and the second sequence has a repeating tail (called an "eventually periodic orbit"). But what delights dynamicists most about this system is that, unlike a card trick, there is no way to figure out in the long run what happens to a random initial value. For example, if I chose an initial value of .33333333 instead of 1/3, I get a very different sequence. The sequences will start close together, but the 30th through 35th iterates of the new choice (rounded for the purposes of display) are:
This sub-sequence does not bounce between two numbers, and is not particularly close to 1/3 or 2/3. This unpredictability, the hallmark of chaos, is known as "sensitive dependence on initial conditions". Indeed, this unpredictability is what scientists usually mean by the term "chaos".
Instead of looking at what happens to individual points, we could consider what happens to the whole collection of initial values--that is, to the line segment [0,1]. First the line segment is doubled in length (multiplying by 2), then it is cut in half and the half that sticks out is placed back on top of the first half (subtracting 1). Because this process is similar to what happens when we knead dough, this system is commonly known as the "baker's transformation".
This stretching of the line segment means that any small sub-segment eventually expands to cover the whole line segment: if a baker puts a small bit of salt in the batter, kneading the batter will mix that salt evenly through the dough. "Mixing" (which truly is the mathematical term for this consequence) is the what mathematicians usually mean by "chaos". The baker's transformation is chaotic in both the mathematical (mixing) sense, and in the scientific (sensitive dependence on initial conditions) sense.
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