This homework involves writing a program
to construct a bifurcation diagram for the logistic map.
This program will need to take several input arguments:
initial condition; a range of r values, including a step size;
the number of initial iterates; and a number of transients to discard.
The program should then output a plot of the logistic map bifurcation diagram
for a range of r.
We'll then check this program by constructing a bifurcation plot
in the given range with the given step size.
Throughout the rest of this homework,
you'll not only learn how to explore this algorithm and its parameters,
but you'll also explore the logistic map bifurcation diagram in more detail.
Here's such a program I decided to write in Matlab.
Notice the bifurcation diagram takes in an initial condition;
an r-min, which is the left range of the r values;
an r-max, the right end of the r values;
an r-step; an n, which is the number of iterates per r value;
and l, the number of transients to remove.
Notice that this program is simply a wrapper
or a for loop around the logistic map program
that you wrote in unit 1.
For convenience, that function is given right here.
At each step of the loop,
we simply plot the trajectory
from the point where the transient is being removed
to the n value against that r value.
With our program developed,
our first test was to generate a bifurcation diagram
starting at r = 2.4 to r = 4 with a step size of 0.01.
This is precisely the plot you see on the lower part of your screen.
We then needed to compare this plot to the plot seen in the homework.
As you can see, if you squint your eyes, these two figures look fairly similar.
However, near bifurcation points, for example,
our plot seems to be a lot fuzzier.
This is actually the answer to the first question.
The reason for this fuzziness near bifurcation points
is that transients get a lot longer near bifurcation values.
For example, consider the time series plot for r = 2.99.
See how long the transient behavior is?
This extended transient behavior causes the fuzzing in the bifurcation plot.
This is the answer to question b.
If transient really is the cause of this fuzziness in the bifurcation diagram,
then simply deleting more of the transient for each r value
should solve this problem.
In this figure, instead of plotting from x₅ to x₁₀₀₀,
as we did in the first figure, we plot from x₅₀₀ to x₁₀₀₀.
As you can see, the fuzziness has been eliminated
This answers question 2c.
Question 3 asks us to generate a bifurcation diagram
starting at 2.8 to 3.6
This is the bifurcation diagram shown on the screen.
It then asks, what is the last n-cycle
that is clearly different visibly from the chaotic region on your plot?
So obviously, the fixed points, the period 2,
and the period 4 are clearly visible.
However, if we zoom into this region,
you can see that period 8 is also very visible.
But period 16 is not so visible.
This is actually the answer to question 3a, n = 8.
For our knowledge about bifurcation diagrams,
we know in this period doubling cascade
there is a period 16 that lies directly after this period 8.
However, it is not clearly visible in this plot.
So what algorithmic parameter could we increase to make it visible?
My decision is the period 16 bifurcation, as well as the higher-order bifurcations,
exist in between these window slices.
In this case, if we increase the step size,
which specifies how many slices our plot has,
we can see these higher-order periods.
To test this hypothesis,
I created a bifurcation diagram with r-step equal to 10⁻⁵,
and keeping all other parameters the same.
As you can see, not only are the period 8 visible,
but now the period 16 are also clearly visible.
So, r-step is the answer to question 3b.
As an aside, I would also like to note that a combination of increasing r-step
as well as zooming in to different ranges of x values
can aid in seeing small features of the bifurcation diagram.
For example, this top fork of the bifurcation diagram
is somewhat squished in this version.
If we not only increase r-step, as we just did,
but then zoom in to this red region,
we can make this top fork much more visible.
This top fork, which is somewhat invisible in the first picture,
is now clearly visible.
Also, notice how similar this picture looks to a standard bifurcation diagram.
This is the self-similarity or fractal nature
that we have talked about throughout this unit.