In this section we will begin to get into the heart of the matter. There are only two new displays involved but we will work them to death.

This idea of iteration is fundamental to the exploration of chaos. We will practice iteration in a variety of ways but for now let's focus on the results of iterating the logistic function.

In the last section we iterated this function several times over a few values of x. (Review New y by Feedback .)

In the present case we will iterate a few times over all the points in our domain at once. This in effect produces a graph of the function itself for the zeroth iteration. For the first iteration we stop at each sample value of x in the domain, calculate a y, substitute that value for x in the function and run it again. Only the final value of y is plotted above the initial x.

Think of the effect that the maneuvers described above will have on the graph as compared to the graph of the original function. The function we are dealing with returns small numbers for small and large numbers and returns large numbers for mid-domain values. It may be thought of as mapping a straight line onto an arch, as though our segment of the real number line was stretched into a new shape by the function. Now if we take the numbers on this arch as input to the function, which is what happens in iteration, we will find the arch mapped onto a new curve.

Let us try to reason out the shape to expect for the first iteration of the logistic function. It should be zero where the original function curve was zero and it should be low where the original function curve was high. It should be high where the original function curve was medium. The logistic function was low on the ends and high in the middle, passing through medium twice. The graph of the first iterate should be lower in the middle, zero on the ends and higher between the ends and the middle.

Exactly the same hand waving argument holds for successive iterations so the graph of each iterate should be flatter on the top and steeper on the sides than the previous one. This is exactly what you will find for gains less than about 3. As iteration number increases, the curve approaches a rectangular shape indicating that whatever the starting value of x greater than 0 and less than 1, y approaches a single value. For gains greater than 3 something different happens and for reasons we will explore later there are two or more possible limiting values of y. Run the Iterations Demonstration display next.

You should see on this display how iteration develops a rectangular curve of y vs. x.

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Now we will run the display with some new parameters. You should notice that the shape of any particular iterate of the function depends strongly on the value of gain but in general they begin to take on a square wave appearance as the iteration number goes up. Beyond a certain gain the square wave gets spikes on it and as gain is pushed farther the whole shape becomes ragged. Next, on the same display you will be able to repeat the experiments you just conducted, using the other two functions.

Now let's look at the iteration of the logistic function in a different way. We have seen that under iteration, y and therefore x may settle down to some value, independent of the starting x. In successive iterations the y value gets closer and closer to the limiting value. We saw this same behavior in some of the "next 'x' selection" schemes we tried earlier. Sometimes the function edges closer to the final value from one direction, sometimes it closes in on the final value by jumping back and forth over it. We call the final value a "limit".

A limit in mathematics is a number which some variable approaches as time goes on. In this program you will be able to observe that, as you iterate the chosen function, the value of y sometimes gets closer and closer to a particular value. This is the settling down behavior mentioned above. In a sense what we will do on the Attractor Demonstration display is to test for the presence of a limit or limits under the process of iteration. You can see graphically the approach to a limit as you step through the iterations.

Because with iteration the y value of a function sometimes settles down at a particular level, it seems like there is something about that value that attracts the function. Just as a marble placed in a bowl will eventually settle in the bottom, attracted by gravity to the lowest point. This idea of an "attractor" for a mathematical function or a physical phenomena, is a powerful one. We will see this concept in several applications. As we will see, a minor change in our function greatly affects the nature of an attractor. It may have a single value or multiple values.

This idea of attractors applies not only to the simple functions that we use for demonstration but also to the functions that describe complicated physical systems. For those functions the attractors are exotic shapes in space not made up of the ordinary dimensions which we may experience; dimensions of height, width and depth. Those attractors are sometimes called strange attractors. The space in which they exist is called phase space and may have many dimensions of various measure depending on the dynamics of the system being depicted. Run the Attractor Demonstration display.

One value of the type display we have here is its intuitive connection to the geometry of iteration. We see the iterations bouncing between the curve of the function and the line y=x. The nature of the attractor at any particular gain is related to the geometry of these two lines. On the Attractor Demonstration display, look at our quadratic with gains less than 1.

Now consider the situation for values of gain between 1 and 2. In this region y is greater than the corresponding x up to the point where the function curve intersects the y=x line. Look next at the quadratic function with gains less than 2.

Now let's look at gains between 2 and 3. For gains between 2 and 3 then we have argued that the attractor is the single value found at the intersection of the function curve and the y=x line. Just as was the case for gains between 1 and 2. Above a gain of 3 , we see a different sort of attractor.

As we have been exploring the behavior of the logistic function under iteration at various gain settings we have made a point of looking at the attractor as it split into 2, 4, 8 etc. values. You may have observed that the change in gain required to get the next split decreases sharply as the splits multiply. In fact each increment in gain is less than 1/4 the previous increment. This means that there is room for more than an infinite number of splits between a gain of 3 and a gain of 3.6. So what happens at values of gain greater than 3.6?

The sort of fractured attractor we described in the last display has been called chaotic. Those regions of gain that give rise to attractors like this are called regions of chaos. Truly the appearance is that of chaos. There seems to be no pattern to the way the y value jumps around. Knowing where the current y is does not allow you to easily predict where the next iteration will take you. Keep in mind though that the next y is absolutely determined by the geometry of the function, just as was the case with the simpler attractors.

At a gain of 3.58 you will see that no matter how long you iterate, certain y values never come up. There will be two broad bands where the values lie, separated by a gap. The function visits each of these bands in turn as was the case when there was a twofold attractor. In that case each part of the attractor was single valued with the function alternating periodically between them. In this case the attractor is still periodic in the sense that the function alternates between branches, but each branch holds infinitely many values.

Having explored the logistic function under iteration, it is useful to look at the other two of our examples. The last display in this section is a research display where you have a lot of flexibility to play around with the graphical iteration of the three functions we have defined. Try various gains, base iterations and numbers of iterations for each function.

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