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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 Unit2

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