The Strange Attractor

Because of its aesthetic qualities, the Feigenbaum diagram has acquired the nature of a symbol. Out of allegedly dry mathematics, a fundamental form arises. It describes the connection between two concepts, which have hitherto seemed quite distinct: order and chaos, differing from each other only by the values of a parameter. Indeed the two are opposite sides of the same coin. All nonlinear systems can display this typical transition. In general we speak of the Feigenbaum scenario (see Chapter 9).

Indeed the fig-tree, although we have considered it from different directions, is an entirely static picture. The development in time appears only when we see the picture build up on the screen. We will now attempt to understand the development of the attractor from a rather different point of view, using the two dimensions that we can set up in a cartesian coordinate system. The feedback parameter k will no longer appear in the graphical representation, although as before it will run continuously through the range 0 ≤ k ≤ 3. That is, we replace the independent variable k in our previous (k, p)-coordinate system by another quantity, because we want to investigate other mathematical phenomena. This trick, of playing off different parameters against each other in a coordinate system, will frequently be useful.

From the previous chapter we know that it is enough to choose k between 0 and 3. There are values between k = 1.8 and k = 3 at which we can observe the perioddoubling cascade and chaos.