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How does chaos show up? Lest one may think that chaos has suddenly turned up at r = 3.6, we step through the parameter range r = (3.5, 3.6) with smaller increments to observe the emergence of period 4, 8, 16, and Table 2. Final iterates for r = (3.54, 3.57)
Summing up, as we raise r slightly above 3, say, r = 3.001, the fixed point is no longer a point where the round curve and diagonal intersect with each other, but a small square around that intersection point. This is how period 2 iterates are being born. Now, at r = 3.45 the fixed point is made up of a loop of 2 connected squares and thereby giving rise to the period 4 iterates. As we further raise r, the number of connected squares for the fixed point jumps to 4 and 8, and hence the period 8 and 16 iterates. As summarized in table 3, the events of period-doubling from period 1 to Table 3. Summary of period-doubling bifurcations to chaos
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, as summarized in table 2. Note that the erratic iterates first observed at r = 3.57 are not really chaotic in that they follow more or less the period 16 pattern of r = 3.564, but with the periodicity being slightly destroyed.
