but first just a little bit of terminology
which should be familiar
from what we did with differential equations
I would say the system undergoes a bifurcation here
at r = 3
remember a bifurcation is a sudden
qualitative change in the behaviour of a
dynamical system
as a parameter is varied continuously
so the qualitative change here
is that the fixed point here splits into two
so we go from an attractor of period 1
to an attractor of period 2
so that's a bifurcation
and it's called a period doubling bifurcation
because the period doubles
here we see we have a bifurcation
from period 2 to period 4
so that's another period doubling bifurcation
ok, let's zoom in on the bifurcation diagram
let's look at just this portion
let's look at what's going on from 3 to 4
since this is where a lot of the interesting action is
so here, I've zoomed in,
and this is a bifurcation diagram from 3 to 4
So we see in this region, from 3 to about a little more than 3.4,
the behavior is period 2
Here are the final state diagrams we drew previously.
There's 3.2, and there's 3.4,
and they line up pretty well.
Let's see if I can get a few more on here.
Here's 3.739, and that corresponds to
this funny region here
(this light region)
and we'll look at that more closely in a bit.
But period 5 (1, 2, 3, 4, 5)
and then we had two 8 periodic values,
at 3.8 ... at around there,
that looks pretty good
and then 3.9 which is right around there.
So, the bifurcation diagram for the logistic map
looks quite different than the one we saw
for differential equations,
which isn't surprising,
the logistic map, and things like it,
exhibit chaos, aperiodic behavior,
so we'd expect it to be a richer bifurcation diagram,
and have more features to look at.
But remember, the thing about bifurcation diagrams,
to interpret them,
remember that they began their life (in this case)
as a series of final state diagrams
so for example, if I wanted to know
what's going on right about 3.7,
I would just try to blot out everything except for 3.7,
and then view it as a single final state diagram.
I sort of imagined doing that with this thing that I've made.
So ... I've moved this so that the slit shows right around 3.7,
and so we would say that this looks like an aperiodic region,
lots and lots of dots,
so it must be aperiodic,
going between this value and this value.
If I wanted to know what' going on at 3.2,
I could move this until I'm seeing 3.2,
and then I would see just these two dots here,
(or small line segments)
and that would mean that this is periodic with period 2.
And imagine another way to view this,
as r increases, we see period 2 behavior,
and the two values are getting farther apart.
They're moving this way, as I let r get larger.
And then a little past 3.4,
(where is it? there it is - there's a bifurcation)
so now it's period 4 (1, 2, 3, 4).
A small change in r (that's moving this)
leads to a qualitative change
in the behavior of the dynamical system.
In this case we go from a cycle of period 2 to a cycle of period 4
and then as I increase r further still,
there's a region of period 8 (1, 2, 3, 4, 5, 6, 7, 8),
each period splits into two, so 4 goes to 8,
8 goes to 16, and so on,
then we have regions of chaos.
Here, this is aperiodic, with gap in the middle.
This is very narrow but this is a period 5 value
we saw before.
More aperiodic regions.... here's a period 3 gap.
1, 2, 3 ... I think we investigated that,
maybe back in unit 2,
and then finally up at r = 4, we have
orbits that go from zero to one,
and so it would fill this entire interval.
Ok, so this is the bifurcation diagram for the logistic equation.
We'll spend lots more time exploring this,
but first I would recommend doing the quiz.
It should be quick, and it will just
check your understanding of this lecture,
and then we will look at an online program
that will let you do much, much more exploring
with the bifurcation diagram for the logistic equation.