I'm currently trying to work through the following two exercises from Robert L. Devaney's "A First Course in Chaotic Dynamical Systems" [p. 95].
In Figure 8.5, note the sudden appearance of the period-3 cycle and its window. Explain why this window opens so suddenly.
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In Figure 8.5, the period-3 window also closes abruptly. Explain why the orbit of $0$ suddenly occupies a much larger interval as the period-3 window closes.
I don't want to get into copyright issues with "Figure 8.5", so I will instead modify the questions to deal with the logistic map $x \mapsto r x(1-x)$, and this figure from Wikipedia.
What I want to know is why does the period-3 window open and close abruptly.
I'm not even sure that I understand the questions, but regardless, my reasoning is as follows.
Opening: since $3$ is an odd number the period-3 window cannot come up out of a period doubling bifurcation.
Closing: No idea.
I would appreciate if anyone explained the meaning of the questions (i.e. the words "suddenly" and "abruptly"), and also help in giving more convincing arguments than mine.
Thanks!