There’s more? Yes, another way of visualizing recursion, this time as reflection off the y=x (45° angle) line.

Here is yet another way of watching what happens when we plug a function’s output back into its input. When you start at a y-value and make it the new x-value, this is the same as moving horizontally to where y=x, which is the 45° diagonal line. Then you move up (or down) to the new value of the function. Each time you click on the button, you draw a dotted line from the old value to the new value.

0

0 1

Slider for setting starting value of x

f(x) = ∙x∙(1-x)

0 4

Slider for setting a

To iterate the function, click here.

- 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.

For many (but not all) values of a the attracting value is the intersection of the graph with the diagonal. Using this visualization helps explain why the orbits all tend to zero when the value of a is less than one: the graph of f(x) falls below the diagonal, and only intersects it at the origin. Similarly for the tent map: until its maximum is 1/2, the entire function is below the diagonal, so zero is the only attractor. But after that, the attractor jumps up in a discontinuous leap: the diagonal intersects the second half (the descending part) of the function. (See the Universality page.)

Sometimes the intersection becomes a repelling point, and the values move away from it. Figuring out how and when this change happens is part of understanding the bifurcation pattern.

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