There is a very specific range for the "r" parameter.
Seemingly simple arrangements can show chaotic behavior. (After all, xt+1 = rxt (1 - xt) uses simple arithmetic on one parameter and one variable.)
It's chaotic, but it's not random. If it were random, there would be an equal probability that the result of the next iteration would be anywhere. But that's not true. Wherever it is, it will be somewhere on that plot.
There is sensitive dependence on initial conditions. Very small changes in r can result in a period doubling.
We know that systematic period doubling ends at about r=3.6, after which chaotic behavior sets in.
Its fractal. If you plotted the period doubling for "r" between 3.5 and 3.9 or "r" between 3.51 and 3.59 or "r" between 3.591 and 3.599, the shape would be the same.
There are patterns. Look at the curving density lines.
Chaotic behavior can be interrupted. Look at the non-chaotic region just past r = 3.8.
Is there a lot of pattern and predictability? Yes. Does it comport with our familiar sense of pattern and predictability - certainly not with mine. That's what I like about 4.669... as a symbol for understanding how programs work and what they do.
If you are not satisfied with my overly simple explanation of 4.669…, try some of the following.