Free Simply Complexity by Neil Johnson

filing cabinet, the file heads toward the middle and stays there. Now let’s suppose that the systematic intern had chosen a value slightly larger than
r
= 3, such as
r
= 3.2. In this case, the resulting time-series of file locations eventually repeats itself – in particular, the following pattern emerges:
. . . 0.80 0.51 0.80 0.51 0.80 . . .
     
    In technical jargon, the time-series has become periodic and hence repeats itself after every two steps. It therefore has a period equal to two. This is very strange since there is nothing in the rule which the systematic intern uses which would suggest that the file should move between two shelves in such an ordered way. And yet the file just bounces back and forth between these two shelves like the tick-tock of a reliable clock. Amazing – but things get even stranger on our route to chaos as
r
increases toward
r
= 4.
    Suppose that the systematic intern chooses a slightly larger value of
r
, such as
r
= 3.5. The resulting time-series suddenly stops repeating itself after every two steps, and instead repeats itself after every four steps. It has a period of four. So the file will move between four shelves in an ordered way, going back to the same shelf every four steps. Yet all the systematic intern did was to change slightly the number
r
in the rule that he was using.
    Further increasing
r
toward
r
= 3.6 gives a time-series of period eight, then sixteen, then thirty-two. In fact it keeps doubling in this way until the period is so long that it looks like it never repeats itself. And this is just like the Chaos that we saw earlier for
r
= 4. In fact, Chaos can simply be seen as a periodic pattern whose period is so long, that the pattern never repeats itself. Now that really is remarkable behavior, by anybody’s standards.
    We can represent all this with a special type of diagram which shows the final shelf locations, i.e. the
S
values, for different values of
r
. Since it is easy to work out these final
S
values as long as you have a calculator lying around, we might as well go ahead and plot the final
S
values for all
r
values at the same time. The result is shown schematically in figure 3.2 and what it means isthis: pick a value for r, and find it on the horizontal line. Then look straight up vertically to read off the corresponding values of the black lines. These are the final shelf locations, i.e. the
S
values, between which the file will end up bouncing forever – these shelves, and no others. So, for example, take the case
r
= 3.2 which we considered earlier. Find this value on the horizontal line in the diagram. Then if you look straight up, you will find the two values 0.51 and 0.80 which are the two values which keep repeating themselves in the time-series:

     
    Figure 3.2 A schematic diagram showing the shelf location where the file ends up after many steps, for a range of
r
values. As the
r
value increases above 3, the number of locations that the file ends up moving between doubles rapidly. In the region of
r
= 3.6, which is shown magnified, the number of locations becomes so large that the value of
S
never seems to repeat itself. For this reason, the pattern ends up looking like a solid line. But it isn’t solid. Instead it is like an extremely fine dust containing many, many points. It is called a “fractal”. For
r
values from 3.6 to 4 (not shown) the dynamics remain chaotic, apart from occasional glimpses of periodic behavior.
     
. . . 0.80 0.51 0.80 0.51 0.80 . . .
     
    For values of
r
near 3.6, the file ends up perpetually moving between different locations with the associated value of
S
never repeating itself. As a result, there seem to be so many points on the diagram that it looks like a solid vertical line. But it isn’t – instead it is like a very fine dust of points. And here comes somethingrather peculiar: there are actually an infinite number of points, and an infinite number of gaps. So this apparent line, which is shown magnified on the