The first problem on this quiz is an exercise in evaluating the logistic map,
which looks like this.
Going back to the problem statement, we see we're supposed to use
r = 2.5 and x_0 = 0.5
By my calculator, I get 0.625 for that.
To get x_2, we plug that back in, like so.
When I plug that into my calculator, I get approximately 0.586.
By the way, what I'm really getting, because my calculator has seven
significant figures to the right of the decimal place is this
And I'm rounding to three significant figures.
To get the third iterate, we plug that back in; and when I do that I get
So the answer I get is x_3 = 0.606.
Now this gets pretty painful, so you can imagine why it might be nice to have
a computer program to do this.
Here's a simple Matlab program that does that.
I'll post this on the supplementary materials as well.
As is requested in the quiz problem, it takes three inputs:
the first initial condition x_0, the parameter value r,
and the number of iterates n.
And it spits out n iterates of the logistic map from that x_0 using that r.
So our task is to compute the tenth iterate,
starting from x = 0.2, with r = 2.6.
And I'm going to use my Matlab program to do that.
And here's the tenth iterate, right there: 0.6157.
Here's the answer.
For the third problem, you'll need to go to the app that we've written
for you to use to explore the logistic map.
You can find that app under the supplementary materials section
of the Complexity Explorer homepage for this course.
Scroll down to the current unit, which is unit 1, and the current segment,
which is segment 1.2, and there's the app.
The task was to plot 50 iterates of the logistic map with r = 2
from an initial condition x_0 = 0.2.
Remember, you have to hit the Restart Simulation button to get that to work.
The question was, "Does the orbit reach a fixed point?"
and that sure looks like it reaches a fixed point.
So the answer is definitely yes.
In question four, we're supposed to raise r from 2 (where it was in question 3),
up to 2.7 and repeat the same experiment: 50 iterates from x_0 = 0.2.
Let's try that.
The shape of the graph is a little bit different, but the orbits do
converge to a fixed point.
So the answer is again yes.
The last question asks us, "If the orbits in questions 3 and 4 both reached a fixed
point, (which is true), is that fixed point at the same value of x?"
So let's look back at the app.
Here's r = 2.7; it looks like the fixed point is above this 0.6 line on the plot.
There's r = 2.0; again, still a fixed point, but the fixed point is at x = 0.5.
So the answer is no, because those fixed points were at different values of x.