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When analyzing iterated functions, it's often easiest to plot the itinerary or the orbit.
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I'll illustrate this with an example.
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Consider again the tripling function, f(x) = 3x.
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Let's figure out the orbit for the seed x equals two. We've done this before.
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2, to get the next value of 6. We triple 6 to get 18. We triple 18 to get 54. And then tripling 54...
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Let's see...That's one hundred sixty two (162).
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So there's our orbit. There's the itinerary. I'll plot this in a type of plot called a time series plot.
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The idea is I'll plot the iterate number on the horizontal axis and the iterate value on the vertical axis.
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So, let me first sketch axes.
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So, on this axis will be time or the iterate number, and on this axis will be the iterate value, x(t).
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So, let's plot these points.
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My initial condition, my seed is two. So my initial value t is zero. I'll plot close to the origin around two.
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The first iterate is six. That's why I'll go up here to around six.
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The next iterate, when the time is two, the iterate is eighteen. That's maybe up here.
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The time is three, iterate number three. All the way up to fifty four. Maybe that's here.
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And then, the last one in this part of the orbit, when time is four, the fourth iterate, I am all the way up at hundred sixty two.
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So, this will be my time series plot. And I could connect the dots, which makes the pattern a little bit more evident.
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Although the values in between the dots don't really have a meaning. The orbit jumps from 2 to 6; it doesn't slide from 2 to 6 going through intermediate values.
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Let's take a look at some nicer version of this plot.
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There it is. So, this is a plot that I had a computer do for me. And we could see that the orbit goes from 2 to 6, to 18, to 54, to 162.
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So I am viewing the iterate number as time, and the iterate value I'm plotting on the y-axis.
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So, the point of this...This is called a time series plot, 'cause we can view the sequence of numbers as a series that advances in time.
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Is that it let us see pretty clearly that this number is growing quickly. Of course, we can see that by looking at the numbers, but this is perhaps a more compelling geometric view.
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The time series plot is a very different sort of plot than a plot of the function itself.
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The time series plot plots an orbit. It tells us the value of the orbit at time one, the first iterate, time two, time three, and so on.
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A plot of the function- here is a plot of the function f(x)=3x- tells you how the input x is related to f(x).
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Let's do another example of a time series plot. This time we'll use the squaring function, f(x)=x^2.
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My seed, my initial condition, I'll choose 1.1. So, at t equals zero initially, the value is 1.1.
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What's the next value? Well, that's determined by the function. So we square 1.1. I'll do that on a calculator. And I see that the answer is 1.21.
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OK, what's the next value? We get the next value from the function, in this case that means squaring the number. So, we square 1.21 and I get 1.46.
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What's the next value? Well, that's determined by the function. I use 1.46 as the input, so I square it and I get 2.14.
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Let's do one more. What's the next value? Well, I've taken the previous value and I square it. So, I am just squaring again and again and again.
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I get 4.59. So, iteration is repetitious, but that is sort of the point. We are doing the same thing over and over to a starting number and seeing what happens.
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OK, so now we have a time series. 1.1, 1.21, 1.46, 2.14, 4.59. It's an orbit or an itinerary.
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Let's make a time series plot. So if we do that, here's what it looks like. Again I used a computer to make one a little nicer than I could draw by hand.
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So time is down here, on the horizontal axis. When time is zero, the time series value is a little more than one, right? We started at 1.1.
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At time one, the first iterate, it grows a little bit, 1.21.
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At time two, it's about one and a half, yeah, 1.46.
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At time three, it's a little bit larger than 2. And then at time four, it's about four and a half.
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So, the idea behind a time series plot is pretty straightforward: calculate the orbit, and then just plot the successive values left to right.
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There is a good way to see if the orbit is growing, or shrinking, or doing something else.