The first question on this quiz was about
the number of state variables

that the logistic map has.

Here's the logistic map.

The logistic map has a single state
variable x, so the answer is one.

The Hénon map is a two dimension map;
it has two state variables x and y.

That is, the Hénon map is a box of
mathematics that takes two values,

x_n and y_n, and returns
two values, x_(n+1) and y_(n+1).

So the state of the system is actually
a vector that looks like this.

The logistic map, in contrast, is a box of
mathematics that takes the value for the

state variable x and tells you what the
value of the state variable

is at the next time click.

That's a 1D map, Hénon is a 2D map.

So the answer to the second problem is 2.

For the third problem,
we need to use the app.

The point of this problem was to explore
different values of the initial condition

for the same value of the parameter r.

So let's play with this.

That's x = 0.2.

Here's x = 0.1.

Here's x = 0.3.

0.5, 0.6, 0.8

The pattern here, as you're seeing,
is that the transient is different

but the fixed point's the same.

And that was the question
this problem was after.

"Do all these initial conditions limit
to the same fixed point?"

And the answer is yes.

Now, is that dynamics an
attracting fixed point?

That fixed point is certainly an
attracting fixed point because it attracts

trajectories from initial conditions
around it, so the answer to question four

is also yes.

Question five is getting at the same
point a stable fixed point is the same

thing as an attracting fixed point.

An unstable fixed point repels
trajectories that start near it,

so the answer to question
five is it's stable.

To answer question six, we're going to
have to do a bit more work with the app.

Now this notation, as you'll recall, means
that x is in the set that begins at 0

and goes all the way to 1.

It's equivalent to saying this.

When you have a parenthesis instead of a
square bracket, like this, what that means

is that the set goes all the way up to
zero but does not actually get there.

And you would rewrite that
with an inequality like this.

So looking at this question, the answers
are asking us to explore whether the

basin of attraction of the fixed point
includes the entire unit interval,

including its end points, or not.

Now, we just did an exploration with x
equals 0.1, 0.2, 0.3, and up to 0.8,

so what we need to do in order to figure
out which of these answers is correct

is do a bit more work at
the ends of that range.

Let's take a look at that funny
answer for a second first.

This is the union of two sets, one of 
which runs from 0 to π/10,

and it's a closed set,
includes its end points.

With another set that begins at
π/10 and runs up to 0.99,

I can rewrite this set using
the following inequality.

Note these less than or equal signs here:
not less than, but less than or equals.

Okay, so let's get to work.

First let's see if the initial condition
x_0 = 0 is in the basin of attraction.

Aha, it's not.

It's in the basin of attraction of a
different fixed point than

this question was asking about.

Dynamical systems can have multiple
simultaneous attractors in different parts

of their state space,
and we'll get back to that.

So that tells us something.

Let's try something just a tiny bit higher
than zero and see where it goes.

There's x = 0.000001, and it
seems to go to the fixed point,

so that tells me that zero is
not in the basin of attraction.

And that rules out the second answer,
because this one says that

the basin of attraction includes zero.

Now let's look at 1.

Here's x_0 = 1.

That's clearly not in the basin of
attraction of the fixed point.

Let's try something a tiny bit smaller
and see where that goes.

That goes to the fixed point,
so this says that one is not

in the basin of attraction
of that fixed point.

Let's go back to the possible answers.

We've now ruled out this
answer and this answer.

So it's looking like maybe
this is the right one.

But we need to check on this.

The experiment that I just did, which used
x_0 = 0.999999, is not in this interval,

so now we're prepared
to finish answering this question.

This first answer is not the right one,
because the fixed point

actually was an attractor.

The second answer is not correct
because it includes one,

and we did an experiment that
showed one was not in the basin.

The third answer is not true
because we're able to establish

that one and zero were
both not in the basin.

And this one here we were able to rule
out because I tried 0.999999,

which is a little bit to the right of
this, and that was in the basin.

So the correct answer is this one:
x_0 is in the open interval (0, 1).

Question seven is about transient shape.

And we're supposed to plot the first 50
iterates starting from the same initial

condition with two different r values.

Let's do that.

Here's r = 2, r = 2.7.

One of those transients has an
oscillatory kind of convergence;

the other one is a one-sided convergence.

And the question was
about the transient shape.

And they do not have the same shape,
so the correct answer is no.

The last question is about transient speed
and what we're supposed to do

is look at r = 2.7 and r = 2.8.

Here's r = 2.7; take a close look at that.

Here's r = 2.8.

Looks to me like it's converging kind
of about here, still wiggling and

wiggling and then it converges about here.

r = 2.7, on the other hand,
wiggles to about here.

So it converges faster.

So the transients are not the same length,
and the r = 2.7 transient is quicker.

So the correct answer is faster than.

By the way, this is a typo right here,
it should be 2.8. We'll fix that.