10. (b) Left: s = 1/2. Right: s = 1. Below: s = 2. |
(c) The maximum occurs at about 0.577. |
(d) From experimenting with the graphs, we see x_{n+1} = 1 for s about 2.598. That is, the dynamics remain bounded between 0 and 1 for 0 <= s <= 2.598. |
(e) There is a single fixed point (x = 0) for 0 <= s <= 1, and in this range the fixed point appears stable. For s > 1 there are two fixed points, and x = 0 is unstable. The nonzero fixed point appears to be stable for 1 < s < 2. |
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