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Hello and welcome to Unit 5.
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This is the second of two units on bifurcations.
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In this unit we'll look at bifurcations for the iterated logistic function.
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As we've seen, the iterated logistic function is capable of aperiodic behavior and chaos.
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This behavior will be reflected in the bifurcation diagram.
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And so the bifurcation diagram for these iterated functions
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will be more interesting and certainly much more complex
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than the bifurcation diagrams for differential equations
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that were the topic of the previous unit.
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I'll begin by reviewing the iterated function form of the logistic equation
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and that will lead us into its bifurcation diagram.
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So let's get started.
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So here's the logistic equation, or I'll call it now the logistic map,
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to distinguish it from the logistic differential equation.
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So this is an iterated function.
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Here's the function: rx(1-x);
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r is a parameter going from zero to 4 that will vary.
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And we'll iterate the function to produce a time series.
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We can also write that process this way: this says that the next value of x
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is a function of the current value of x: r times the current value times one minus the current value.
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And one other thing just to remind you of, is that this equation,
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the logistic equation,
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is a very simple formula. It's just a parabola, a second-order polynomial.
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So if you expand that out, that's what it looks like;
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not a very complicated function at all.
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And we can make a simple plot as well.
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It's just a downward-opening parabola; so about as simple a function as one can imagine.
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And, as we've seen though, when you iterate it,
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we can get aperiodic behavior, chaos, periodic behavior, and so on.
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So let's begin as we did for the logistic differential equation.
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I'll consider the behavior of this dynamical system for a couple different values of r.
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We'll plot, not quite a phase line, but something similar to a phase line for each r value.
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And then we'll imagine gluing those together to produce the bifurcation diagram.
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For the logistic equation, or the logistic map,
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we used a web program to plot time series for us.
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And let's take a look at that time series plotting program again.
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And we'll try out a couple r values just to remind you how this works.
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Here's the program that makes time series plots for the logistic equation.
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Let's use this to make
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time series plots for three different parameter values, three different r values.
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So I'll go down here and the first value I'm going to try is 2.9.
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So I enter 2.9,
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I ask the program to make the time series plot,
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and there it is. And we see that
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the orbit is approaching a fixed point.
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As we've seen before, this is an attracting fixed point. It takes a little while to get there,
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it wiggles back and forth, but we can see it getting closer and closer to this value.
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And if I look at numbers, it's going to around .65,
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.66; something like that. So for 2.9, we have an attracting fixed point.
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Next, let's try 3.2. So I'll go back here,
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enter a different parameter value, 3.2,
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and let's make the time series plot;
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there it is. And here we see a cycle.
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It's periodic with period 2, and it looks like it's going between
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.8 and maybe a little more than .5; we can check the numbers and look.
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And indeed, it's going between .799, about .8; and .51.
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So for this r value, when r is 3.2,
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we see periodic behavior with period 2.
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Let me try one more, and I'll do 3.8.
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So I'll go up here
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and I'll enter 3.8,
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make the time series plot, and
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it's a little hard to see what's going on here.
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It's not quite periodic;
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maybe it wants to be periodic but hasn't found the period yet.
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So in order to be sure, to see what's going on, I'm going to plot more iterates.
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Instead of 40, let's plot 200 and see if it becomes periodic or not.
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So in this view we can see that the orbit is aperiodic.
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It's not settling down into any cycle
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and it seems to range from a little bit below .2,
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maybe .18 is the minimum, and about .95 or a little bit more.
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So it doesn't fill up the entire interval;
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it doesn't make it down here, it doesn't make it to the very top.
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Let me plot -
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just to see this a little more vividly -
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I'll do a thousand; this might take a second. There it is.
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So again, we can see the orbit bounces all around.
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It's not becoming regular.
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And the minimum is about 0.18 or so, and the max is probably 0.96, 0.97.
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So...
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for this r value, the orbit is aperiodic.
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For a given r value, we can summarize the behavior of the logistic map
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with a graphical device called a final state diagram.
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And I introduced these in Unit 3,
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but I thought it would be good to review them quickly now.
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So this was the first r value that we experimented with
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on the website; r is 2.9.
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And we said there's a fixed point here at about
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.65 or .66.
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So I can draw that, as follows. So let me draw the line
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for the phase line and I'll use this to get it straight.
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So I'll just draw that as a line.
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And the idea is that this is zero and that's 1. And there's a fixed point at 0.66
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so that's going to go, I'll go over halfway, and
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maybe a little bit more, and I'll put a single dot there.
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And this would be - make a note -
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this is r is 2.9.
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So I'm just interested in the long-term behavior;
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that's why we call these final state diagrams. And I could imagine
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I could run the program for 40 times, maybe for 400 times,
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and then I could plot the next 100 points that would appear in the orbit.
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And in this case, those 100 points would all be the same, they'll all be at the fixed point.
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So they'll just be one single point in the final state diagram.
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Let's do the same thing for r=3.2.
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Here, there's a cycle between .8 and .55 or so.
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So let me draw the line for the final state diagram.
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There it is.
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And this time I've got two final state values: .8 and .55.
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So I'll draw these; go halfway and a little bit more,
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and then maybe there. So those might be my two values; this is r=3.2.
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And again, a way to think about the procedure for this is, we might
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run this for 40 or 400 or even 4,000 times if we wanted,
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and then plot the next hundred points. And this time the next hundred points would just
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be oscillating back and forth between these two values.
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So that would appear as just two dots here on this line.
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So that would be the final state diagram
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for r=3.2. And lastly, let's...
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think about a final state diagram for r=3.8.
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This is an aperiodic value.
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And let me
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draw the line for the final state diagram.
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(I'm using this index card
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so that all the lines are the same size.)
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Okay, so now,
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let's see, I'll write r=3.8.
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And now we see all
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the dots are more or less filling up
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all the numbers between about 0.18 and 0.96.
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So we could do this plot for
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2,000 or 20,000 times, and then plot the next several hundred in the orbit.
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And this orbit would keep bouncing around between these two values.
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And so, in between those two values, the line would fill up with dots.
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So let me draw that.
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The largest value is going to be about over here. The smallest value, .18,
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maybe that's around here. And then I would just fill
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this segment up with dots.
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(Filling it up with dots.)
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Okay, so the final state diagram might look something like this.
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and I would interpret this, if I saw this as a final state diagram,
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I'd say, "Ah, okay. It's not settling down.
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The orbit is not becoming periodic. If it was periodic, there would just be two or four,
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or however many the period was,
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dots here. But the fact that this gets
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filled up, and appears solid, is an indication that the orbit is aperiodic.
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I should mention that, if I were to run this for a really long time,
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I wouldn't fill up all of the numbers between .18 and .96 or whatever these are,
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because there are an infinite number of numbers between my fingers right now on the number line.
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In fact, there's an uncountable infinity.
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And the number of points in an orbit, even an infinite orbit, would be countable.
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So it's not that
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the interval gets completely filled up in the literal sense.
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But the dots, if we were to draw them with any sort of finite thickness,
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any nonzero thickness, would appear to just end up being a solid smudge
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of points. So these are the...
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final state diagrams
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for three different r values. They're like phase lines but they're different
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because they don't have arrows on them.
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And the reason they don't have arrows is that,
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unlike for differential equations, for these iterated functions
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there can be oscillations.
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So this one for 2.9, it wiggles back and forth.
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So it overshoots, undershoots, overshoots, undershoots. So anyway,
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we're just drawing the final states; we're not putting arrows on them.
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So before I go on, I would recommend doing the next quiz,
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which is, I'll have you practice drawing some
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final state diagrams for a few different r values. It should be pretty quick and
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I'll just make sure your brain is back in the gear,
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back in the mode, of thinking about the discrete logistic equation.
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So give those a try, and then we'll put these lines together
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and make a bifurcation diagram.