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This first question
is primarily vocabulary
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and asks you to pick
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which term does not correspond
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to the same representation
as the other terms.
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These are the two different
representations of nonlinear dynamics
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that are in question in this quiz problem.
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This type of plot is known
as the cobweb diagram correlation plot,
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or return map.
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This type of plot is referred to
as the time-domain plot,
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as the x-axis is time.
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So time-domain plot is the term
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that does not correspond
to the same representation
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as the other terms in the list.
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Question 2a asks us
to use the cobweb applet
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to add one iterate at a time
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to a cobweb diagram starting
at initial condition 0.5
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using a parameter of 2.7.
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From this, we want to analyze
what kind of dynamics comes out
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and compare this with our current
knowledge of the time series plot
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to determine the dynamics.
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As we slowly add
single iterates to this map,
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what we see in the cobweb diagram
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is the red lines slowly spiraling in
to the intersection of the line xₙ+1 = xₙ
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and the parabola
defined by the logistic map.
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As we also consider
what's happening in the time domain,
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we see that we are converging
onto a single point.
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By examining both the cobweb diagram
and the time-domain plot,
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we can see that these are
fixed point dynamics.
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Part b asks us to use a correlation plot
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to analyze the stability
of these dynamics
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by choosing several initial conditions
between 0.1 and 0.9.
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To do a basic analysis of stability,
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let's apply several different
initial conditions
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to the parameter r = 2.7:
0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8.
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So it does appear
that these fixed point dynamics
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are indeed stable for r = 2.7.
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For this problem, we set r equal to 3
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and the number of initial iterates to 5,
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and have an initial condition
of x₀ equal to 0.5.
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Let's click start animation
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and see what shape
emerges in the red line.
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So as you can see, it does appear
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that the red lines
are converging to something.
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At this point, it seems like the red lines
have stopped changing
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and have converged to a small square.
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As an aside, a small square
on the cobweb plot
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is associated with a two-cycle
in the logistic map dynamics.
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The next three questions ask to match
cobweb plots in figure 1a
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to their corresponding dynamics.
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Part a asks what type of dynamics
correspond to those seen in figure 1a
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This type of cobweb plot
is associated with chaotic dynamics.
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For part b, we are told in figure 1c
that the transient has not been removed,
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and it appears
that the trajectory has converged.
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As we saw in question 2a,
this type of cobweb plot
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is associated with a fixed point dynamic.
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For part c, we're told that figure 1d
has had the transient removed
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and asked to classify the dynamic.
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This type of cobweb plot
is associated with high-period attractors.
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This next series of questions is intended
to help you understand
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the connection between time-domain plots
and cobweb plots.
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The dynamics in figure 2a
are the same as which in figure 1.
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As you can see, the dynamics in figure 2a
are the same as in figure 1b.
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One thing that might help
you understand this or see this better
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is to notice the small square
that forms in figure 1b
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and see what it corresponds to
in figure 2a.
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The dynamics in figure 2b
are a fixed point dynamic.
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As we've seen throughout this quiz,
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this corresponds to the dynamics
seen in figure 1c.
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The dynamics seen in figure 2c
in the time-domain plot
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is a high-period orbit.
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You may want to confuse this
with a chaotic orbit,
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but look how structured it is.
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As you can see, it repeats very regularly.
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This corresponds to figure 1d.
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Finally, the dynamics in figure 2d
are chaotic.
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Notice in contrast to figure 2c,
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they're a little less regular
than a periodic orbit.
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These dynamics correspond to figure 1a.
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Finally, let's look at question 5.
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As discussed in lecture,
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any time there's an intersection
between the line xₙ+1 = xₙ
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and the parabola described
by the logistic map,
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which is yellow in this plot,
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you'll have a fixed point
of the logistic map
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As you can see, there are
two such intersections, here and here.
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Obviously, the fixed point at (0,0)
is not very interesting.
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In question 5, we are asked to solve
for the interesting fixed point
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that occurs right after 0.6.
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To solve for this fixed point,
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we simply need to solve
this system of equations.
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To do so, we set them equal to each other,
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resulting in the following equation.
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Since we are not interested
in the fixed point that occurs at (0,0),
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we can eliminate these two xₙ,
resulting in this equation.
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Solving this equation,
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we then see that the fixed point
occurs at two thirds.
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We can now answer question 5a,
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as we know the fixed point at r = 3
is at x* = 2/3.