Objective: We now investigate population dynamics of in the parameter range r = ( 3, 4). Recall from Lesson 9 that in the range r = ( 1, 3) the intersection of the round curve and diagonal is the stable fixed point of period 1.

As r increases beyond 3, the fixed point is now a square intersecting with the diagonal at two opposite corners. This is how the period 2 iterates are born. Similarly, the period 4 iterates arise from a loop of two connected squares, the period 8 iterates from a loop of four connected squares, etc. Since the pace of period-doubling quickens within a small parameter range, one may perceive a sudden appearance of chaos near r ~ 3.6. Here, the emergence of chaos follows the period-doubling route:

Period 1 period 2 period 4 period 8 … chaos.

Scary words: Period-doubling, chaos, bifurcation, bifurcation diagram.

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