TITLE: What causes bifurcation?
QUESTION [10 upvotes]: Suppose let's say we have a recursive function,
$$x_{n+1}=rx_{n}(1-x_{n})$$
From what I understand, the $x_{n+1}$ VS r graph starts to split (bifurcate) after a particular value of r (apparently it's around 3). Does that mean after they split up, we will get two values of the function for the same value of r? Am I getting it right? 
How's it possible? (I do understand that it loses its ordered nature and falls into chaos) Could someone illustrate it with an example? Help would be appreciated!  (I'm a high school student, so explaining it in simpler terms would be even more appreciated). Thank you!

REPLY [2 votes]: First of all, congratulations for you work and your observations.
Since @Milo Brandt gave very nice explanations, I shall not repeat any of them.
You can "simplify" the quadratic recurrence equation
$$x_{n+1}=r\,x_{n}(1-x_{n})$$ letting
$$x_n=\frac{r-2 z_n}{2 r}\implies z_{n+1}=\frac r 4+\frac 1r z_n^2$$ and you enter in the domain of quadratic maps which, at leat for me, is not the simplest problem.
For example, the simplest
$$c_{n+1}=c_n^2-k$$ with $c_0=0$ used to generate the Mandelbrot Set.
I shall not enter into more details but I suggest you have a look at this paper which, even if too complex for you at the time being, gives very good explanations about the bifurcation diagram. There are a few plots which are very illustrative.