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