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