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Period 2 iterates for r = (3.1, 3.4) The last 10 iterates of r = 3.1 plotted in figure 1 alternate between the two values of x = 0.55801 and 0.76457. Hence, it is period 2, similar to the trajectory of a valley ball hit to reach two alternating heights (figure 2 of Lesson 8). To see how this has come about, let us return to the graphical iteration plot. To separate off the beginning from the ending iterates, Prog#9 plots the first 3/4 of iterates in white and the remaining 1/4 in blue. At r = 3 we have observed that the graphical iteration plot eventually winds down to a point x = 0.666... , at which the round curve and diagonal intersect with each other. On the other hand, for a larger parameter r = 3.1 the graphical iteration plot is a square (figuratively speaking, though not an exact square) around the intersection of the round curve and diagonal. That is, the square is now a fixed point, in contrast to a point at r = 3. We find that the blue square of Prog#9 intersects with the diagonal at two opposite corners at x = 0.55801 and 0.76457, and this is how the period 2 iterates have emerged at r = 3.1.
Figure 1. Successive iterates of period 2 for r = 3.1 In retrospect, we have witnessed the stable population of period 1 splitting into alternating population levels of period 2, as r is raised from 3 to 3.1. We may think of this splitting as forking out into two values from one, and hence is called the bifurcation. Needless to say, it is important to know the precise parameter value, at which a bifurcation takes place. (Project a - Birth of the period-2 population). In any event, it is not possible to determine the bifurcation parameter value by numerical experimentations alone, as we have attempted to do so here. Suffice it to say that other analysis methods are beyond the scope of the present lesson book.
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