The final state diagram for the logistic map. The x-axis is the value of the parameter a. The first period-doubling sequence begins at a=3.0 with the bifurcation of the stable fixed point into a stable period 2 orbit. This period doubling sequence culmninates near a=3.6 where 'chaotic' (or ergodic) bands appear. Note the 'shadow lines' and periodic windows. These can be explained (see the text Section 11.4).