Superstable fixed point

Try Example 10.1.1 on a hand calculator by pressing the x button over and over. You ll see that for sufficiently small Xg, the convergence to x =0 is extremely rapid. Fixed points with multiplier A = 0 are called superstable because perturbations decay like 7/ 7/g , which is much faster than the usual 7/ A"7, at an ordinary stable point. 

This is a fair comparison because the maps have the same stability properties x, is a superstable fixed point for both of them. Please notice that to obtain Figure 10.7.2b, we took the second iterate of f and increased r from / (, to / ,. This r-shifting is a basic part of the renormalization procedure. 

There is no reason to stop at For instance, we can renormalize to generate it too has a superstable fixed point if we shift r to / 2. The same reasoning as above yields... 

Superstable fixed point) Find the value of r at which the logistic map has a superstable fixed point. 

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