Reading: pg. 640-653
1. Draw a staircase diagram for a=3.1 (where the period 2 cycle is attracting) for the logistic map. Then fix x0=.5 and give the time series (that is, plot x(n) as a function of n). Indicate the period two orbit on the time series. Then do a times series for a=3.4, a=1+sqrt(6), and a>1+sqrt(6), showing the period two behaviour in the first two cases and period four behaviour in the last case.
2. Let g(x)=f^4(x) (the four-fold composition of f(x), f(f(f(f(x))))). What is the degree of g(x)? Plot g(x) for x in [0,1] when a is just less than, equal to, and just greater than 1+sqrt(6). Overlay the line y=x in each case and indicate the bifurcation. Describe the roots of g(x) (e.g., zero and (a-1)/a are the fixed points of f(x), etc.); you need not find them explicitly, but account for all of them.
3. Let q(x)=f^3(x) be the three-fold composition of f(x). Show graphically that the logistic map undergoes a tangent bifurcation at a=3.83 (approximately) which results in the birth of a period three cycle by plotting y=q(x) and y=x for values of a near 3.83.
4. Show that Sf<0 for x in (0,1) for the logistic map (where Sf is the Schwarzian derivative of f).
5. Show that the property of having a negative Schwarzian derivative is preserved under composition, that is, show that if Sf<0 and Sg<0 then S(f(g))<0 if this is well-defined. Then show that if Sf<0 then S(f^n)<0 also.
Student-contributed MAPLE worksheet. (Submitted by Bill Richardson.)