To begin let us examine maps for some higher iteration numbers. First looking at iteration number 1000. You will notice that the map is quite smooth for low values of gain and it has some distinct breaks in it. The first of these is at g=1. We have already noted that for g less than 1, high powers of g are small and i1000 involves g raised to absurdly high powers (see the equations in the Phase-Control Map section) so that y remains extremely close to 0 for all g<1. Above g=1, up to about g=3, i1000 increases in a smooth curve. Something happens at g=3 and something else at about g=3.45.

White rectangles will show up on the screen to direct your attention to some area of the display. On the next display we have enclosed an area where there are sudden changes in direction in the plotted line. We will examine this area again as we add more iterations to the map. Keep in mind that we are iterating our function a great many times before we plot any values so if the attractor is single valued the function should settle down on it by the time any point is plotted. Further iterations will not change the value of the function in that case. Run the Quadratic Iteration i1000 now if you like, otherwise just look at the image.

Next we add iteration number 1001 to the picture. Notice that i1001 lies directly on top of i1000 all the way up to g=3. Only when the two iterations result in different y values do you see different lines. At about g=3.57 some sort of catastrophe evidently occurs. The map becomes extremely badly behaved above that value of gain. There seems to be no pattern to the points. Run the Quadratic Iteration i1000-i1001

In that region of the map which is peppered with points it looks like disorder reigns. Because of this appearance you will hear this described as an area of chaos. Remember when we were looking at the lower numbered iterations one at a time, the map got increasingly complex near the right hand end as the iteration number increased. Still for the low numbered iterations the plotted points fell on some discernable curve. With 1000 iterations the complexity of that curve is enormous but still there is nothing random about the location of any plotted point.

In principle you could write the equation for y as a function of g for the 1000th iteration and calculate each point, even in the chaotic region. The apparent scatter of points results from the limitations of the presentation. The complicated tangle of the map in this area must be presented on a grid of points with limited resolution. Hence the chaotic appearance. We should understand chaos to mean simply that there is no easy way based on the current iteration to predict where the next iteration will fall. It should not be taken to imply disorder.

Think back to the iteration display for the logistic function. At certain values of gain, those less than 1, the attractor of the function was 0. From g=1 to g=3 a single valued attractor dependent on gain was evident. Above g=3 we began to see the attractor branching in two, then four, then more branches with the function visiting each branch in turn. Attractors like this are called periodic attractors. For some values of gain the function seemed to never settle down. That is what we are observing here. The logistic function, deeply iterated, settles at 0 for g<1.

From g=1 to g=3 where the attractor is single valued, by the time we have iterated 1000 times one more iteration will not change the value of y so i1001 lies precisely on i1000. In general, if we iterate deeply enough to settle down on the attractor, more iteration does not matter. In our example, above g=3 where the attractor has two branches even numbered iterations settle on one of them and odd numbered iterations settle on the other. What happens then when there are four branches. Let's add a couple more iterations on our next display and see. Run the Quadratic Iterations i1000-i1003 display.

On the next display we plot 256 iterations starting at i1000. You will observe that more detail is revealed on the phase-control map. In fact we now have enough detail to do some exploring of the map. Take a look at the Quadratic Iterations i1000-i1255 display.

We will zoom in on particular areas of the map on the next displays. In later parts of the program you will have control of the magnification of portions of the screen. For now we will automatically select a zoom window denoted by the white rectangles. The first step in magnification will be to look at the window outlined on this display where one of the attractor branches fragments into chaos.

Within the chaotic region you will see a miniature version of a portion of the whole map. It is similar to the whole map but not identical. This is "self-similarity" across scales. Run the Iterations i1000-i1255 - Zoom 1.

In each of the next several displays we will select a window and magnify it to full screen on the succeeding display so that the last map in the series will be a portion of the first map magnified about 1 million times. If you push the magnification much beyond this point you may find that small uncertainties in the numbers being calculated will be amplified to the point that the map shows artifacts which arise not from the mathematics but from the computer itself. With some experience you may come to recognize and ignore these but for now we will not push that far. Run the Iterations i1000-i1255 - Zoom 2, 3 and 4 displays.

Iterating a function deeply and displaying it on a phase-control map in effect reveals the shape of the attractor in that phase-control space. The interesting features of the attractor for the logistic function occur in the gain domain from 3 to 4. It is in that area that we will concentrate our next exploration. The next display in this section shows this portion of the attractor. Notice that even in the regions of chaotic behavior, the function remains within well defined bounds. We will extend our concept of an attractor to accommodate these regions. Run the Window of Order in Chaos display.

As we will see on the next display, the attractor of the logistic function in phase-control space exhibits areas where both the periodic and chaotic nature co-exists. The next display shows the region of the attractor where it is periodic with period 3. This region is bounded on both the left and right by regions of chaos. In the chaotic region to the left you can see that the density of points is not uniform, indicating some periodicity in the chaos. Then in the periodic region to the right of the bifurcations are bands of chaos with period 3. Run the Window of order Close Up display.

In the next lesson we will examine the way the attractor cascades from a single value to multiple single values to chaos.

Are there any questions?