We have been studying the map of the logistic function attractor in phase-control space. We should now look at the sinusoidal and Gaussian functions in the same way. To begin with we can review the three functions. Remember that being high in the middle and low on the ends was significant to our results in iterating these functions. Run the Three Functions Demonstration display to review this.

The logistic and sinusoidal functions are symmetrical about the vertical line x=0.5. The graph of the sinusoidal function on the interval 0 to 1 is quite similar to that for the logistic function on that same interval. The amplitude is different and of course the mathematics behind the sinusoidal function is quite different from the quadratic equation defining the logistic function. Based on the appearance of the graphs we should see similar results from iterating these two functions. Run the Sine Function Phase-Control Map display to see the attractor.

As we did with the quadratic function, we can go through the bifurcations and determine the ratio by which the interval between bifurcations decreases. Without showing all the bifurcations, we display the results in the Sinusoidal Feigenbaum Cascade display.

The graph of the Gaussian function differs from the other two in important ways. It is not symmetrical on our domain, it peaks at x=1 rather than x=.5, and it is not uniformly concave downward as were the others. At about x=2, the graph shifts to being concave upward. We might reasonably expect a radically different attractor from iterating this function. Run the Gaussian Function Phase-Control Map display.

One of the things that may be obvious to you is that the maps of these attractors develop more slowly than that of the logistic function. This is because the calculation of the sine and exponential function takes longer than the simple multiplication required in the logistic function. Even though the increased calculation time is hardly noticeable for a single calculation, the repeated calculations required in iteration can accumulate quite a delay.

One of the things we see in the Gaussian attractor, which reflects its recurve nature, is unbifurcation. A region of chaos sort of gathers itself together as gain increases and collapses to a periodic and then point attractor. At some gain, the point attractor explodes into chaos again.

You may see this demonstrated in the Unbifurcation in the Gaussian display.

Again we might mark the gains where bifurcations take place and tabulate the data. This is done in the Gaussian Feigenbaum Cascade display.

The fact that this same Feigenbaum ratio keeps showing up in the attractor of unrelated functions is known as "universality" . Does this mean that the instructions on where to bifurcate the attractor are not contained in the original function?

The notion of universality is at the heart of the science of chaos. What we have seen is just an easily visible tip of a large iceberg, most of which is submerged in a sea of complexity. In studying as we have the simplest sort of nonlinear equations we have glimpsed a structure that also controls real physical systems, like flow turning turbulent and metal being magnetized, at the onset of chaotic behavior. If the rules are independent of the function, then applying the rules does not require that we necessarily understand the function details.

As in the section on iterations and attractors, you now have an opportunity to explore the attractors of all three functions. Take some time to play around with the Phase-Control Map Research display.

This completes the discussion on phase-control maps. In the next lesson we will go back and pick up another thread of the chaos story, involving sets in the complex plane.

Are there any questions?