First question is whether or not the logistic map bifurcation diagram is in fact fractal
To answer this question, lets look at the bifurcation diagram
As you can see, the bifurcation diagram is in fact self-similar
By self-similar, we mean that the small-scale structure, like that seen in the red circle, is similar to the large-scale structure seen in the entire window
In fact, if we were to zoom in on the red circle, we would see a complete copy of the logistic maps bifurcation diagram
Similarly, if we were to zoom in on this region for example, or this small region, we would see another copy of the bifurcation diagram
Each one of these tiny self-similar copies has infinitely more self-similar copies contained within it
So for this reason we would say that the bifurcation diagram of the logistic map is in fact fractal
As we just saw, the logistic maps bifurcation diagram is in fact fractal, so this question is true
The next question concerns whether all periodic orbits in the logistic map have even periods
To answer this question, lets again look at the bifurcation diagram
If you look in this window up here you can see where the red arrow is pointing you see a period three orbit
Onetwothree
Since 3 is not even, the question will be false
As an aside, its interesting to note that, given any natural number, its possible to pick a parameter r which will give you an orbital logistic map of that period
As we just saw, the logistic map has a period 3 orbit, and as a period 3 orbit is not even, this statement has to be false
This next question asks us to classify the particular range of the logistic map bifurcation diagram as either a numerical effect, period-doubling cascade, or chaotic attractor
The red box in this picture outlines the interesting behavior thats being discussed in that question
This is called a period-doubling cascade, for obvious reasons
So you go from a fixed point to period 2, to period 4, et cetera, period-doubling all the way up to chaos
So we could best describe the behavior of this region as a period-doubling cascade
Its important to realize that this is not a chaotic attractor
A bifurcation diagram is in parameter space, not in state space, so this cannot be an attractor
It may be tempting to think that, because youre going from left to right, that youre quote-unquote attracting to chaos
but this is not the way that you should think about an attractor
Instead we should describe this as a period-doubling cascade
As we just showed, this particular range of the logistic map bifurcation diagram is best described as a period-doubling cascade
The next question concerns whether there are multiple period-doubling cascades in the logistic map bifurcation diagram
A period-doubling cascade actually occurs not only in this first region that we just discussed, but in each one of these blank windows of structure
So for example, in this very large one, you begin with period 3, but then you perform a period-doubling cascade back to chaos
Each one of these periodic windows will have the same period-doubling cascade occur
So there are multiple period-doubling cascades
As we just saw, there are in fact multiple period-doubling cascades in the logistic map bifurcation diagram, so this question is true
The final question asks us to explain what the dark veils are that occur in the logistic map bifurcation diagram
These beautiful dark lines that appear in the logistic map bifurcation diagram are sometimes called veils, or dark veils
Your task in this problem is to figure out what is the cause of these veils, or if theyre just a numerical effect
The easiest way for me to personally understand these dark veils is to not actually look at the bifurcation diagram, but to look at a time-domain plot in an area of the bifurcation diagram where these dark veils occur
That is, to look at a time-domain plot for an r value which has these dark veils
So what well do is well look at the time-domain plot where we have these three dark veils
Here is such a time domain plot
So what we see is that, in this chaotic region, sometimes were kind of going all over the place
Very regularly, we stay very close to some periodic orbits
In this case, we get very close to this unstable three-cycle
Then we bounce around for a little while, then we fall off
Then well get attracted back to this unstable three-cycle again
Then well maybe travel along it for a little while, then fall off
This happens over and over and over again
If we look at this straight on, then what wed see is a tight clustering of points here, a tight clustering of points here, and a tight clustering of points here
with very sparse points in between
These tight clustering of points, over time, end up causing what you see as these dark lines or veils
To reiterate, this means that unstable periodic orbits are the cause of the dark veils
As we just showed, the dark veils that occur in the logistic map bifurcation diagram correspond to unstable periodic orbits