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Chaos at r = 3.6 The last 21 iterates of Prog#9 are plotted in figure 3 for r = 3.6. Clearly, the iterates do not appear repeating themselves within the 21 iteration steps of figure 3 and, moreover, they are scattered more or less randomly over the range (0.2, 0.95). Since the erratic pattern of iterates persists in figure 3 as we include much more than 21 iterates, the period is very long and, in fact, infinitely long. We therefore say chaos is observed at r = 3.6. This comes about because the graphical iteration plot is now a loop of multiply-connected squares. But, since the squares are never closed, they will intersect with the diagonal at as many points along the x-axis as the maximum iteration number of Prog#9. Hence, the upshot is a sequence of iterates that do not repeat themselves, no matter how long we run Prog#9. Recall that the decimal digits of (figure 8 of Lesson 8) also have period  , so that they appear as a sequence of digits chosen randomly out of (0, 1, 2,..., 9). We must point out that the erratic iteration plot of Prog#9 at r = 3.6 is sensitive to the choice of initial x0. In other words, an initial population slightly different from x0 would generate a sequence of iterates that are totally different from figure 3. This is known as the sensitive dependence upon initial conditions and it is the hallmark of chaos.
Figure 3. Successive iterates of period
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