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