# Homework #13

1. Consider the logistic map f(x)=ax(1-x) with a>1. Show that if x0 is not in [0,1] then the forward orbit of x0 diverges (to negative infinity). Also show that if 0<a<1 and x0 is in [0,1] then the forward orbit of x0 converges to zero.

2. Draw the phase portrait of the logistic map for a in (2,3). Compute three orbits for each of a=2,2.1,2.25,2.5,2.75,2.9,3. Comment on the apparent speed of convergence of the iterates for the various values of a.

3. Draw a staircase diagram for the logistic map for a=.5,1.5,2.5,3. Compare this with the phase portrait approach.

4. Sketch the phase portraits for the maps f(x)=-x, g(x)=x^3, and h(x)=x+x^2 (for all x0). Repeat with a staircase diagram. Which fixed points, if any, are non-hyperbolic?

5. Use the Mean Value Theorem to prove that if p is a hyperbolic fixed point of a differentiable map and the multiplier for p is less than one in modulus then there is an open interval U about p such that if x0 is in U then the forward orbit of x0 converges to p.

Student-contributed MAPLE worksheet for #2,3,4. (Submitted by Bill Richardson.)