Chapter 5, Exploration 29

The graphs below two different plots of \(y = f^2_a(x)\). The first one shows it plotted on the domain \( 0 \leq x \leq 1\) and also includes the graph of \(y = f_a(x)\) so that you can see the relationship between the two. The second one only shows the graph of \(y = f^2_a(x)\), but zooms in on the non-zero fixed point to better illustrate what happens in a period-doubling bifurcation. Use these tools to answer the following questions. Make sure the sidebar menus is minimized.

1. Describe in general terms what happens to the non-zero fixed points of \(y = f^2_a(x)\)as \(a\) increases through the bifurcation value of \(a = 3\). How many fixed points are there when \(a < 3\) and what are their stabilities? How many are there when \(a> 3\) and what are their stabilities? The first tool will be most helpful for this exercise.
2. Recall that fixed points of \(y = f^2_a(x)\) are period 2 points of \(y = f_a(x)\). Are any of the fixed points of \(y = f^2_a(x)\) actually fixed points of \(y = f_a(x)\) (i.e. not prime period 2 points)? If so, are they attracting or repelling? Are any of the fixed points of \(y = f^2_a(x)\) prime period 2 points of \(y = f_a(x)\)? If so, are they attracting or repelling?
3. In exploration 28 you showed that \(f_3(2/3) = 2/3\) and \(f'_3 (2/3) = −1\). What is \(f''_a(2/3)\)? Look at the figure first, then calculate it. How is this different from the second derivative condition in the Tangent Bifurcation Theorem? Why do you think this is important in explaining the bifurcation that you see when looking at the graph of \(y = f^2_a(x)\).