Bifurcation means splitting into two parts: "bi" (two), and "furca" (fork).
As some functions evolve they suddenly split into two!
First we will need a function:
rx(1−x) is a good one.
x is the input value, and r is a value we want to investigate.
We will calculate the function over and over again, each time using the result as the new x value.
Let us try r=2, and start with x=0.2:
2 × 0.2(1−0.2) = 2 × 0.2 × 0.8 = 0.32
Now with the new x value: 2 × 0.32 × 0.68 = 0.4352
And once again: 2 × 0.4352 × 0.5648 = 0.49160...
And again: 2 × 0.49160... × 0.50840... = 0.49986...
And again: 2 × 0.49986... × 0.50014... = 0.50000...
We see it is settling to 0.5
But is not always so simple. Try some other r values here:
What do you notice?
- How about when r is around 3.2 ? It jumps between two values.
- And around r = 3.5 it jumps between four values.
- And what is the story around r = 3.7 ? It seems like complete chaos.
- But then try around 3.84: it briefly settles down, then goes crazy again.
We need to investigate more!
So try here. Same idea as above, but a lower plot keeps track of the last few values achieved at each r-value. It needs your help to make the plot:
Here is a closer look at it, notice the "forks" (where the name bifurcation comes from), and also notice how it goes from order to chaos and sometimes back again:
The change between order and chaos is also seen in nature.
For example populations of animals can be steady, or show this "one year many, next year few" pattern, or be just very chaotic.
Get a tap dripping.
It can start off steady (drip, drip, drip), but as you increase the flow it becomes chaotic (drips then flowing then more drips). If you get lucky you can also get the "double drip" happening (like when r is around 3.2 above).
There is a relationship between the bifurcation diagram and the Mandelbrot set. You may like to investigate that!
And this is not the only type of bifurcation in mathematics.
And there is a whole lot more to know in this really interesting subject called Dynamical Systems.