When analyzing iterated functions, it's often easiest to plot the itinerary or the orbit.
I'll illustrate this with an example.
Consider again the tripling function, f(x) = 3x.
Let's figure out the orbit for the seed x equals two. We've done this before.
2, to get the next value of 6. We triple 6 to get 18. We triple 18 to get 54. And then tripling 54...
Let's see...That's one hundred sixty two (162).
So there's our orbit. There's the itinerary. I'll plot this in a type of plot called a time series plot.
The idea is I'll plot the iterate number on the horizontal axis and the iterate value on the vertical axis.
So, let me first sketch axes.
So, on this axis will be time or the iterate number, and on this axis will be the iterate value, x(t).
So, let's plot these points.
My initial condition, my seed is two. So my initial value t is zero. I'll plot close to the origin around two.
The first iterate is six. That's why I'll go up here to around six.
The next iterate, when the time is two, the iterate is eighteen. That's maybe up here.
The time is three, iterate number three. All the way up to fifty four. Maybe that's here.
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.
So, this will be my time series plot. And I could connect the dots, which makes the pattern a little bit more evident.
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.
Let's take a look at some nicer version of this plot.
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.
So I am viewing the iterate number as time, and the iterate value I'm plotting on the y-axis.
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.
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.
The time series plot is a very different sort of plot than a plot of the function itself.
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.
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).
Let's do another example of a time series plot. This time we'll use the squaring function, f(x)=x^2.
My seed, my initial condition, I'll choose 1.1. So, at t equals zero initially, the value is 1.1.
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.
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.
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.
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.
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.
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.
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.
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.
At time one, the first iterate, it grows a little bit, 1.21.
At time two, it's about one and a half, yeah, 1.46.
At time three, it's a little bit larger than 2. And then at time four, it's about four and a half.
So, the idea behind a time series plot is pretty straightforward: calculate the orbit, and then just plot the successive values left to right.
There is a good way to see if the orbit is growing, or shrinking, or doing something else.