The work of a curious fellow
Splitting hairs...

We have seen previously how as gain changes, the nature of the attractor changes. We talked about splitting of attractors, known as bifurcation. We have demonstrated geometrically how this bifurcation takes place. We have argued that a cascade of bifurcations is a precursor to chaos. Now we will begin to think of bifurcations as a characteristic of the logistic function attractor in phase-control space. We will look at bifurcations as arising from the mathematical form of iterates of the logistic function in phase-control space.

In an earlier section we wrote expressions for y as a function of g in developing phase control maps. These functions increased in complexity exponentially as the iteration number increased but by actually iterating we were able to produce graphs of these functions. On the next few displays we will build up a picture by generating these curves for the first few iterations just to show how they combine to give an early indication of where these bifurcations come from. For our case x0=.5, beginning with i2, even and odd iterations begin to segregate themselves. If you want to see the images develop you may run through the series of displays called Bifurcation Beginnings. Otherwise you may just look at the finished image below each link.

bifurcation boundary 1
bifurcation boundary 3 bifurcation boundary 4
bifurcation boundary 5< bifurcation boundary 6

As we added higher iterates you should have noticed that the bundle of odd iterates separates into two groups as does the bundle of even iterates. This separation of the phase-control maps of the iterates of the logistic function begins to show where the bifurcations will occur. As we continue to iterate, the curves approach the actual point of bifurcation and the break becomes sharper. In the limiting case of high iteration numbers the bifurcations become quite distinct. Now we seem to have two ways, not obviously related, to determine where bifurcation occurs.

Do bifurcations occur because the slope of the function and its iterates become steeper than -1 at certain values of gain, or because the maps of the iterates in phase-control space, where y is a function of g, wiggle in a certain way? Since both considerations are equally valid, this must not be an either/or situation. Perhaps one manifestation of bifurcation derives from the other or perhaps they both arise from something even more fundamental. You should not hesitate to use either way of looking at things as suits your taste, geometry or algebra.

We have previously alluded to the fact that bifurcations happen faster and faster as you increase gain past 3.0. What we will try to do next is quantify that phenomena by graphically locating the points where bifurcations occur. This technique is not precise but will serve to illustrate a point. To do this we need to exercise the magnification feature of the program. As we zoom in on the bifurcations, we will build a table of the gains at which they occur. You may use the cursor at each display to verify the tabular value.

In using the cursor to estimate the location of bifurcations in the logistic function attractor, remember that the precise location is marked by infinite iteration. Since we iterate only a finite number of times in the interest of finishing in our lifetime, you will see the iterates begin to diverge at gains less than the actual bifurcation. This early divergence then blends into the real attractor. The best precision will be obtained by placing the cursor where your eyeball says the blunt nose of the bifurcation would be if the branches were continued until they joined.

Notice the labels on the display. They tell you in each instance what the beginning and ending iteration number is. They also indicate the boundaries of the window to which we have zoomed. You will see that as we look for the higher order bifurcations we must iterate more deeply in order to get a reasonably sharp break at the bifurcation. Also notice that all branches of the attractor bifurcate at the same gain so that you may examine any branch to locate the bifurcation. We will locate the first eight bifurcations in this series of displays. Run the Cascade Bifurcation series of displays.

Cascade Bifurcation 1 occurs at 3.000000
cascade bifurcation 1
Cascade Bifurcation 2 occurs at 3.449383
cascade bifurcation 2
Cascade Bifurcation 3 occurs at 3.544054
cascade bifurcation 3
Cascade Bifurcation 4 occurs at 3.564396
cascade bifurcation 4
Cascade Bifurcation 5 occurs at 3.568756
cascade bifurcation 5
Cascade Bifurcation 6 occurs at 3.569691
cascade bifurcation 6
Cascade Bifurcation 7 occurs at 3.569891
cascade bifurcation 7
Cascade Bifurcation 8 occurs at 3.569934
cascade bifurcation 8

There is a regularity to this cascade of bifurcations we have been looking at. The rate at which the distance to the next bifurcation decreases appears to be approximately constant. We can use the table of bifurcations that we have developed to test this proposition. On the next display we will see the attractor between gain=2.95 and gain=3.6. Immediately below are the 8 bifurcation locations, the 7 differences between adjacent bifurcations and the 6 ratios of those differences. Even with the poor precision we are using, the ratios are all close to 4.7. Run the Feigenbaum's Number display.
quadratic function cascade

When we first explored the map of the logistic function attractor we found windows of order mixed in with the chaos. Within these windows there are also cascades of bifurcations. It is as though at certain values of gain, the attractor collapses from chaotic behavior to purely periodic, then cascades through bifurcations back to chaos. It turns out that there are an infinite number of these windows of order in the regions of chaos just as there are an infinite number of bifurcations in each cascade. In each window the attractor has a characteristic period.

We will examine one of the windows of order in the chaotic region to illustrate what we have called interior cascades of bifurcations. First we see the whole map above gain of 3.4 and then zoom in, magnifying a portion in the region of chaos where there is one of these windows of order. We have chosen to expand the picture around a window where the attractor is periodic with period 5. You will notice the chosen window is outlined in white as was the case when we first explored the map. Run the Another Window of Order display and its two zoom displays as well.

another window of order
another window of order z1 another window of order z2

A table of bifurcations, differences and ratios is shown with the following display. You can see from the table that these interior bifurcation cascades converge at the same rate as the original cascade. Exactly why the logistic function attractor splits with such geometric regularity is not obvious. One would suppose that the instructions for these bifurcations, as well as all of the infinitely detailed structure of the two dimensional logistic function attractor, would be contained in the logistic function itself. After all, that function contained all the intelligence we had at the beginning. All we did was iterate it repeatedly. Run the display An Interior Feigenbaum Cascade.
interior Feigenbaum cascade cascade

The observation that the logistic function bifurcation cascades occur with such regularity was first publicized by a physicist named Mitchell Feigenbaum. The number which the ratios in our tables approach is approximately 4.6692 and is known as the Feigenbaum number. Feigenbaum recognized that both the regularity and infinite variety of the logistic function attractor have profound implications. Perhaps the instructions for building complex structures like plants and animals can be expressed in some relatively simple functions iterated repeatedly.

In the next lesson we will go on to look at the attractors for the other functions.
Are there any questions?

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