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Iterated Maps and the Chaos Phenomenon

The basic idea of an iterated map is to take a number between 0 and 1, and then in a sequence of steps to update this number according to a fixed rule or "map". Many of the maps I will consider can be expressed in terms of standard mathematical functions, but in general all that is needed is that the map take any possible number between 0 and 1 and yield some definite number that is also between 0 and 1.

The pictures on the next two pages [150, 151] show examples of behavior obtained with four different possible choices of maps.

Cases (a) and (b) on the first page show much the same kind of complexity that we have seen in many other systems in this chapter—in both digit sequences and sizes of numbers. Case (c) shows complexity in digit sequences, but the sizes of the numbers it generates rapidly tend to 0. Case (d), however, seems essentially trivial—and shows no complexity in either digit sequences or sizes of numbers.

On the first of the next two pages all the examples start with the number 1/2—which has a simple digit sequence. But the examples on the second of the next two pages instead start with the number π/4—which has a seemingly random digit sequence.

Cases (a), (b) and (c) look very similar on both pages [150, 151], particularly in terms of sizes of numbers. But case (d) looks quite different. For on the first page it just yields 0's. But on the second page, it yields numbers whose sizes continually vary in a seemingly random way.

If one looks at digit sequences, it is rather clear why this happens. For as the picture illustrates, the so-called shift map used in case (d) simply serves to shift all digits one position to the left at each step. And this means that over the course of the evolution of the system, digits further to the right in the original number will progressively end up all the way to the left—so that insofar as these digits show randomness, this will lead to randomness in the sizes of the numbers generated.

It is important to realize, however, that in no real sense is any randomness actually being generated by the evolution of this system. Instead, it is just that randomness that was inserted in the digit sequence of the original number shows up in the results one gets.

From Stephen Wolfram: A New Kind of Science [citation]