So we now have a collection of final state diagrams.
They’re the examples that I did in the last video
3.2 , 2.9 and then an aperiodic value at 3.8
and then for the quiz that perhaps you just took
you did 3.4 that’s another period 2 value
3.739 this is period 5
1,2,3,4,5 1,2,3,4,5
and then another aperiodic one 3.9
for here the dots extend from about .1 to almost 1 exactly, may be .98
so it might look something like this
so we got a collection of final state diagrams
and as we did with differential equation
we’ll form a bifurcation diagram
by gluing together a collection of final state diagrams
so let me take each of these and cut it out
so I can move it around so how do you that and we’ll see what it looks like
so now I’ve got a collection of final state diagrams
and let’s put them in order and as before put them on the side as they do that
so there is 2.9 that was period 1
here’s 2.3 that’s a stable cycle of period 2
here is 3.4
3.739
3.8
and 3.9
so this is the beginning of a bifurcation diagram for the logistic equation
remember the goal of the bifurcation diagram is to see
how the dynamical systems behavior changes
as a parameter in this case r is changed
so it gives us a global view of the range of behaviors
that dynamical systems can exhibit
in this case with just these six final state diagrams
it’s not really clear quite yet what the overall pattern might be
so in order to sort of see a pattern to connect the dots so to speak
we would need to try this out from many many more r values
and make many more phase lines and stack them all in here densely
so that we can see what happens from one r value to the next
so as you’ve probably guessed I’ll use a computer to do that work for us
and I’ll show you the program and how it works in a little bit
but first let’s focus on what the results are
so let’s see put these to the side just for a moment
here’s the bifurcation diagram for the logistic equation
the lower limit here is r equals 0
and here’s r equals 4
and then this goes from 0 to 1
so when r is between 0 and 1
between my fingers here, the, can’t quite see it here but the only fixed point is
the attracting fixed point is 0
if the growth rate is less than 1 the rabbits die out
between 1 and 3 there is an attracting fixed point
and in fact we saw that let’s see we did one for 2.9 here it is
let’s see if this is going to work,
close, ok so 2.9 that’s about there and this dot that I drew in the first example
is part of this line here then as r is increased
as the growth rate gets larger and larger
the period 1 behavior splits into period 2
and we’ve seen that here’s r equals 3.2
r equals 3.2 so the two dots from the final state diagram
show up as part of this line here
so in this region where if I go up from a single point I see two lines two dark regions
that would indicate that its period 2
here is another r value a little bit larger 3.4
and still period 2, there are only 2 dots
but the periods are little bit further apart
see if I can get both of these on at the same time
so for these two different values, it’s the same qualitative behavior
attracting cycle of period 2 but the exact locations are a little bit different
all right it’s little hard to see what’s going on in here
so we’ll zoom in here in just a moment
but first just a little bit of terminology which should be familiar
from what we did with differential equations
I’d say the system undergoes a bifurcation here r equals 3
remember a bifurcation is a sudden qualitative change in the behavior 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 to 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 two phase lines we 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 we’ll look at that more closely in a bit
but period 5 1,2,3,4,5
and then we had 2 aperiodic values at 3.8
and 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
then the ones we saw for differential equations
which isn’t surprising the logistic the logistic map and things like it
exhibit chaos aperiodic behavior
so we’ll expect it to be more 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 as a series of in this case final state diagrams
so for example if I wanted to know what’s going on right around 3.7
I would just try to blot out everything except for 3.7
and then view it as a single final state diagram
sort of imagine doing that with this, thing that I’ve made
so this is I’ve moved this so that the split shows right around 3.7
and so we would say that ahaa this looks like an aperiodic region
lots and lots of dots so it must be aperiodic
going from between this value and this value
if I want to know what’s going on at 3.2
I could move this until I’m seeing 3.2
and then I would see just these 2 dots here or small line segments
and that would mean that this is periodic with period 2
you can imagine another way to view this as r increases
you see period 2 behavior and the two values are getting further apart
they’re moving this way as I let r get larger
and then a little passed 3.4
that’s where is it there it is, there’s a bifurcation,
so now it’s period 4 1,2,3,4
like small change I go then 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 2, 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 2 so 4 goes to 8, 8 goes to 16 and so on
then we have regions of chaos here this is aperiodic but with a gap in the middle
this is it’s very narrow but this is the 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 equals 4 we have orbits that go from 0 to 1
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 kind of 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