So here are the solutions

to quiz 2 from Unit 5.2

The first questioned asked about

r equals 1.8

What value is the fixed point?

So, I will zoom in here.

It looks like this region has 1.8 in it.

Zoom in a little bit.

And now I'm going to move the cursor

Until the r value is about 1.8

Then I will see what the x value is.

So, here where my cursor is

r is a little more than 1.8

and I see that the x value there is 0.436

So about 0.44

So the best answer to this question was 0.44

There is a fixed point at around .44 when r is 1.8

The next question asked about bifurcation

from period two to period four.

So I'm going to go back

and I'm going to zoom in here.

So bifurcation from period 2 to period 4

that occurs around here.

So when there are two lines

if you go up from here

and the line crosses

this imaginery line intersects two lines

in the bifurcation diagram

that means it would be

a final state diagram

with two dots

Here we go up we cross through four

so it is period four.

The transition occurs here

Here.

And that looks to be about 3.44, 3.45.

So that gives us an approximate r value

where this bifurcation occurs.

The last question, question 3.

Asked what goes on around 3.906.

There is a periodic window

of what period

So let me go back one.

And this will take a little bit

of trial and error exploration.

So let me zoom in.

3.9 where is that.

That looks to be around here.

I want 3.906. Let's try zooming in.

Here. Update again.

3.906.

3.906

Here is around 3.906

You can tell because

I'm reading this mouse position here.

Let's see 1,2,3,4,5.

So I see period five behavior here.

So this is a window of period five.

So, near r equals 3.906.

There is a periodic window

of period five.