What happens to the sequence x, f(x), f(f(x)), etc... for the particular function a∙x∙(1-x)?

On this page we make things more graphical. First, we show what the function f(x) looks like as you change the value of a. Next, instead of clicking a button to see the orbit values one by one, we plot them also on a graph.

If you are unfamiliar with the terminology of iterated functions and orbits, you should visit the preceding pages in this series, by clicking the “Previous” link at the top of this page.

The graph on the left is the conventional graph of the logistic map: you can see that it starts and ends at zero and always has a peak in the middle. The value of a determines the height of that peak. (It is a/4.)

The graph on the right is the *sequence* of values x, f(x), f(f(x)), etc. starting (arbitrarily) with an x value of 0.15. So it is not a continuous function itself, but a series of distinct, discrete values. For low values of a the graph rapidly approaches a constant value. As a increases, the graph shows a very complicated pattern of oscillations.

Graph of f(x)Graph of orbit, starting with 0.15

0

01

0 4

Slider for setting a

- a = This has a single attractor at value 0. This should be no surprise!
- In fact, all values of a from 0 to 1 have orbits that head to zero.
- a = This has a single attractor at value 0.5.
- At a = 3 the orbit turns into a period-2 cycle. The next three values show the period-doubling behavior as a increases.
- a = The orbit values settle down to a period-2 cycle.
- a = The values fall into a period-4 cycle.
- a = The values fall into a period-8 cycle. (Takes a while to settle in.)
- From here up to the value a ≈ 3.57 the periods keep doubling, and every power of 2 occurs. After that, things are rather complicated until a reaches 1 + 2√2 ≈ 3.828, at which point a cycle of length 3 appears.
- a = The values fall into a period-3 cycle!
- After this, as a increases the periods again double, to 6, 12, 24, etc. until chaos breaks out again.
- a = The values change chaotically.

The graph on the left shows that the logistic map has both domain and range from zero to one. That makes it well-defined to keep plugging the output back in to the input: they both vary over the same values. If it looks like part of a parabola, that’s because it is. Later, on the Universality page, we will see that the exact shape of the function does not matter, as long as it goes up and then comes back down. That is very general! We use this function because it is easy to calculate.

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