Superstable cycles

Actually, all the dips in Figure 10.5.2 should drop down to A = - >, because a superstable cycle is guaranteed to occur somewhere near the middle of each dip, and such cycles have A = -< , by Example 10.5,1. This part of the spike is toonar-row to be resolved in Figure 10.5.2. 

Feigenbaum phrased his analysis in terms of the superstable cycles, so let s get some practice with them. 

Numerics of superstable cycles) Let denote the value of r at which the logistic map has a superstable cycle of period 2". 

A superstable cycle) Consider the logistic map with r= 3.7389149. Plot the cobweb diagram, starting from x = y (the maximum of the map). You should find a superstable cycle. What is its period ... 

Superstable cycles to the rescue) The critical slowing down encountered in the previous problem is avoided if we compute instead of r ,. Here denotes the value of r at which the sine map has a superstable cycle of period 2". ... 

Unfamiliar later cycles) The final superstable cycles of periods 5,6,4,6, 5, 6 in the logistic map occur at approximately the following values of r ... 

Suppose that f has a stable p-cycle containing the point. Show that the Liapunov exponent A < 0. If the cycle is superstable, show that A = —oo. ... 

First we introduce some notation. Let /(x, r) denote a unimodal map that undergoes a period-doubling route to chaos as r increases, and suppose that is the maximum of f. Let denote the value of r at which a 2 -cycle is born, and let R denote the value of r at which the 2 -cycle is superstable. 

The renormalization theory is based on the self-similarity of the figtree—the twigs look like the earlier branches, except they are scaled down in both the x and r directions. This structure reflects the endless repetition of the same dynamical processes a 2"-cycle is born, then becomes superstable, and then loses stability in a period-doubling bifurcation. 

Here gf(x) is a universal function with superstable 2 -cycle. The case where we start with (at the onset of chaos) is the most interesting and important,... 

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