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.