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

**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 (

**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

**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.)

*Maintainer*: leader@rose-hulman.edu.