There are a few more features of the bifurcation diagram that I'd like to explore.

So, here we are back at the starting point...

...and I'll zoom in again on roughly three to four.

And we focused in the last video on these period-doubling transitions...

...from period two to four, four to eight, and so on.

But there's some other interesting structure as well.

So, we have period-doublings, and then we have these regions here where...

...there's a more-or-less dense wall of points.

And those are aperiodic orbits, they're chaotic.

But in the midst of chaotic regions of parameter values, there are windows of periodic behavior.

And we've looked at this window of period five.

Here's the window of period three.

Let me zoom in here.

And, again we can see some windows.

There's a one-two-three-four-five-six period seven window.

I zoom in even more...

...this little slice here...

...there is yet another periodic window-- nine, that's a window of period nine.

So we have these chaotic regions of parameter values, but they're interrupted every now and again...

...by these periodic windows.

The last feature I want to make note of is that there's some structure-- these waves, or veils, or something...

...within these aperiodic regions.

And, we'll see this later, but just to give you a preview, this dark band here...

...lemme zoom in just a little...

...this band here, or this band-- you can kinda see a curve coming up here...

...those are regions where there happen to be more points.

So, um, say at this value...

...the orbit is aperiodic, so as I plot more and more points, it starts to fill up...

...this interval, but more often--- the points occur more often at some values than others.

So that's where these dark lines occur: those are unusually likely or common values.

If I plot a few more points, I might overexpose it, but that might help highlight it too, let's see.

That's a little bit too much. Let's back down to, I don't know, let's try four hundred.

So there one can see, perhaps, a little better, some of this additional structure that's in here.

So, we'll explore that a little bit in some of the later units.

But the main point for now is to note that there is a tremendous amount of structure in the bifurcation diagram...

...that we see periodic windows, if we look closely in regions that we think are pure chaotic regions, in terms of the parameter value r...

...we see that there are stable regions of periodic behavior...

...we see lots and lots of period-doubling transitions...

...and we see this sort-of wavy structure as well.

So we're getting an awful lot of pattern and regularity to go along with the chaos and unpredictability that we've seen previously.