Examples
There are many chaotic maps. Famous iterated functions include the Mandelbrot set and Iterated function systems.
Ernst Schröder, in 1870, worked out special cases of the logistic map, such as the chaotic case f(x) = 4x(1 − x), so that Ψ(x) = arcsin2(√x), hence f n(x) = sin2(2n arcsin(√x)).
A nonchaotic case he also illustrated with his method, f(x) = 2x(1 − x), yielded Ψ(x) = −½ ln(1−2x), and hence f n(x) = −½((1−2x)2n−1).
If f is the action of a group element on a set, then the iterated function corresponds to a free group.
Most functions do not have explicit general closed-form expressions for the nth iterate. The table below lists some that do. Note that all these expressions "work" for non-integer and negative n, as well as positive integer n.
| (see note) |
where: |
| (see note) |
where: |
where: |
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Note: these two special cases of ax2 + bx + c are the only cases that have a closed-form solution. Choosing b = 2 and b = 4, respectively, reduces them to the nonchaotic and chaotic special cases above.
Some of these examples are related among themselves by simple conjugacies. A few further examples, essentially amounting to simple conjugacies of Schröder's examples can be found in ref.
Read more about this topic: Iterated Function
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