Period 4 iterates at r = 3.5

The last 10 iterates of r = 3.5 repeat themselves after four iterations, and the period 4 iterates shown in figure 2 are quite analogous to the period 6 decimal digits of ³/₇ = . For r = 3.5 the blue iteration plot of Prog#9 settles down to a loop of doubly connected squares, and the opposite corners intersect with the diagonal at four alternating locations x = 0.38282, 0.82694, 0.50088, and 0.87500, listed in table 1. That is, the loop of doubly connected squares is now the fixed point. However, by stepping through the parameter range r = (3.4, 3.5) with the increment 0.01, the first appearance of period 4 iterates occurs at r = 3.45. (Project b - Birth of the period-4 population) In our present discussion, we shall take r = 3.45 as the bifurcation value for period 4 iterates, although a refined value can be found by an increment less than 0.01.

Figure 2. Successive iterates of period 4 for r = 3.5

[prev] [ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 ] [next]

back to the lesson list