In this set of demonstrations the "invariant measure" is estimated as the histogram (see diagnostics) for various values of a. In addition, to understand the relationship of the histogram to the dynamics it is useful to look at the orbits directly, and the power spectrum introduced in chapter 2. (When running the histogram or power spectrum demonstrations an initial plot will appear given by collecting data over a small number of iterations. The plot should be refined by continuing the iteration by hitting Start. The speed of the iteration process can usually be increased to near its maximum value, and the iteration runs faster if the number of iterations is not displayed.)
Notice that for a=3.58, above the period doubling accumulation point, the power spectrum shows peaks corresponding to period 2 and period 4 that remain perfectly sharp (limited by the resolution of the algorithm), whereas the higher period peaks have become broadened, and there is also a broad band background. The histogram shows 4 disjoint bands. Looking carefully at the time evolution shows that under iteration x jumps between the bands, in a completely predictable order, yielding the sharp peaks, whereas the location within the bands shows the features of chaos - apparent stochasticity, sensitive dependence on initial conditions etc. The peaks in the power spectrum remain sharp because the stochasticity never affects the predictable jumping between the bands, so that the "error" in e.g. the period 4 motion remains bounded for all time. It is a similar argument that shows that thermal fluctuations of the atomic positions leave the Bragg peaks sharp in X-ray diffraction off a crystal, although reducing the amplitude (by the Debye-Waller factor).
Changing a to 3.6 leads to a band-merging - now only the period 2 peak remains sharp, and there are two disjoint bands in the histogram. In fact the band merging transitions moving away from the accumulation point ac mirrors the sequence of period doubling transitions approaching ac from below, as we will see in more detail in chapter 17.
At a=4 the bands have merged to give chaotic dynamics over the whole of the unit interval. The shape of the histogram can be analytically calculated at this value.
Return to discussion.