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