Summary of population dynamics

Let us plot in figure 4 the iteration results of table 1 for the parameter range r = (0, 4). Although the bifurcations to period 2 and 4 iterates are evident in figure 4, the evidence for period 8 and 16 bifurcations are completely absent because of the large increment 0.1 used in table 1. Besides, one cannot really infer chaos from a splattering of only 10 iterates, which are certainly not long enough for us to say something definite about the period of iterations. For a better demonstration of period-doubling bifurcations toward chaos, it is therefore necessary to step through the parameter range by an increment much smaller than 0.1, and also display more than 10 iterates. In fact, this has been incorporated into program Prog#10 to generate a bifurcation diagram, which clearly indicates the sequence of

Period 1 period 2 period 4 period 8 … chaos.

Figure 4. Plotting the iterates of table 1

But, you still cannot see the presence of period 16 in the bifurcation diagram. However, to show the period 16 iterates, it requires a much finer graphics resolution than what is used to generate the bifurcation diagram in Prog#10. It is interesting to point out that chaos once initiated will not persist in the entire parameter range r = (3.57, 4). In fact, there are three distinct breaks at r 3.63, 3.74 and 3.83 in the bifurcation diagram of Prog#10, for which we have observed the period 6, period 5 and period 3 iterates, respectively. (Can you verify this by Prog#9?) In this way, the population dynamics drifts in and out of chaos in an unpredictable fashion for r 3.57.

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