The Problem.
Part V. Divergence Diagrams (extensions of Feigenbaum diagrams.)

The divergence diagram for the logistic map provides a way to see periodic points that still remain when a > 4. Each shade (color) corresponds to the number of iterates before x > 1 (after which the orbit will clearly diverge to negative infinity.) Refer to the Divergence Diagram given in the background materials preceding the problem sets.

1) Find the equations of the boundary curves for the largest region in which orbits diverge after one iteration. Plot the curves and compare with the divergence diagram.

2) Find the equations of the boundary curves for the two regions in which orbits diverge after two iterations. Plot these curves along with the curves found in the first problem and compare to the divergence diagram.

3) Find the equations of the boundary curves for the four regions in which orbits diverge after three iterations. Plot these curves along with the curves found in the first two problems and compare to the divergence diagram.

4) Note that other than the largest region, the other regions each have a maximum width. Find the a-value for which the boundary curves found in problem 2 are the widest.