Orbit Histograms

Computer Lab

Log on and select Logistic. From the main menu, choose "Orbit Histograms."

In our study of the logistic model, we have seen that when the value of parameter a exceeds 3.57, orbits are likely to be chaotic. Today you will spend some more time examining what a chaotic orbit looks like. This will help us to build our mathematical definition of the term chaos.

1) A chaotic orbit can never hit the same number twice. Why not?

2) As you know, chaotic orbits bounce all over the place. But do they go everywhere? Do they spend more time in certain neighborhoods than in others? That's hard to say when you're just looking at a list of data. Orbit histograms help keep track of where an orbit visits. To begin, you will look at the orbits when parameter a is set at 4.

Each neighborhood is an interval of numbers somewhere between 0 and 1000. You can make the width of that interval as wide or skinny as you wish. Start with an interval width of 100, so that there are just ten neighborhoods altogether. Pick some arbitrary seed and ask for 10000 iterations. The computer will run the recursion and display a bar graph to show how many times the orbit hits each neighborhood. That bar graph is called a histogram. Each of the tick marks on the vertical axis represents 5% of the number of iterations you requested.

3) Maintaining the same interval width and same number of iterations, try several different seeds to see where the orbit goes. Does it seem to be evenly distributed, or are certain neighborhoods more popular? Does it depend on the value of your seed? Share your data with other group members.

4) Now change the value of parameter a to 3.8 and look at histograms for several different seeds. You will see that certain neighborhoods never get any visits at all. That didn't happen when a was set at 4. What's the difference? (Web diagrams can help you answer this.)

5) Keep parameter a set at 3.8, but change the interval width to 20. Find the most popular neighborhoods. Fashion yourself a little ruler made from graph paper, so that you can hold it up to the screen and estimate the numerical value of the popular neighborhood(s). (The screen goes from 0 to 1000 across the bottom.) Then go to the bifurcation diagram and look at a window from 3.75 to 3.85. Make a new ruler that you can hold vertically to allow you to estimate numerical values. (Now the screen goes from 0 to 1000 vertically.) Find the place(s) on the bifurcation diagram that corresponds to the most popular neighborhood(s) for 3.8. Do you see anything significant on the bifurcation diagram? Try this for several different values of parameter a.

6) These orbit histograms display yet another strange characteristic of chaos: while the orbits appear to wander aimlessly, they are not random. They seem to prefer certain neighborhoods. But that is only a preference, not a restriction, because they also make occasional visits to unpopular neighborhoods. In fact, they eventually go to every neighborhood. I can make the size of my neighborhood as small as I want-width of one-millionth, or even one-trillionth-anything greater than zero will do. Then I just sit back and wait and eventually I will get a visit.

This phenomenon is known as mixing, and it is one of the three essential ingredients of the mathematical definition of chaos. The term mixing was coined because mathematicians made the analogy of dropping a teaspoon of spices (let's say cinnamon) into a bowl of dough. As the baker mixes the dough, the particles of spice will be spread throughout the mass of dough.

The other two ingredients are sensitivity, which we have already studied, and density of periodic points, which I will take a stab at explaining now. None of this is easy.

When I described mixing above, I was a little bit sloppy. I may have implied that you will see visits to every neighborhood, no matter what seed you pick to start your orbit. That's not true. There are some very special cases of seeds. For example, you might be using an a value of 4, and pick the seed 750. When you run 750 through the recursion rule:

P_n+1 = 4(750)(1 - .001(750))

you get 750 again. (Do the arithmetic to check that out.) That orbit will be forever stuck on 750, so it certainly won't visit every neighborhood. So equilibrium is one of the special cases. You could also use this wacky number for your seed:

P₁ = 5 + Ö 5 * 1000
8

With parameter a still set at 4, that P1 value will produce this result:

P₁ = 5 - Ö 5 * 1000
8

and if you iterate again, you return to your original P₁. Thus this orbit is forever caught in a two-cycle and certainly will not visit every neighborhood.

There also exists a three-cycle that works when parameter a is 4, a seventeen-cycle, any cycle length you want. The special seeds that cause these cycles are known as "periodic points." None of these cycles is attractive, of course, so you are not going to see them unless you happen to get extremely lucky. The overwhelming odds are that you will pick a seed that leads to a chaotic orbit, one that visits every neighborhood. Even if you did happen to pick one of the periodic points, if you tried to run it on the computer, there would be round-off error and the phenomenon of sensitivity would throw the orbit out of its cycle. The only way to see a cycle in this situation is to start out with the perfect seed and then do the arithmetic perfectly.

This gets back to our debate about infinite sets inside of other infinite sets. There is an infinite number of periodic points for a = 4. That infinite set is dense, too, because between any two periodic points that I find, I can always find another periodic point. Even with an infinite number of possibilities, though, my probability of finding such a point just by chance is practically zero. The size of the set of other seeds, the ones that go to chaotic orbits, is so incredibly huge, that I am virtually certain to pick one of them.

Now back to be a little more careful about my description of mixing. Since there are always periodic points lurking around, I can't say for sure that any seed picked will visit every neighborhood. We already know that a seed of 750 won't do that. Still, I could take a little interval around 750, and I would be sure to find a seed that does visit every neighborhood.

In the same way, chaotic orbits will never visit 750 exactly, because if they did they would get stuck there. Even so, they can get arbitrarily close to 750. They are guaranteed to visit 750's neighborhood. Here is a more formal definition of mixing, taken from a math textbook:

For any two open intervals I and J (which can be arbitrarily small, but must have a non-zero length) one can find initial values in I which, when iterated, will eventually lead to points in J.

Call me over when you are ready to discuss all of this.