HW5  Homework week 5

Read 109-131 Skip rest of Section 3.5
Read 143-147 Lab Visit 3
Read Sect. 9.3 370-5 on the Rossler attractor

Read 135-137 through Step 2.

A1. Do steps 1 and 2 of Challenge 3.

A2.  Let f be the logistic map x -> 4x(1-x) on [0,1] and
define the change of variables  y: = ax+b
Find the conjugate map g(y) i.e. y_n+1 = g(y_n)

Note that E below is NOT a checkout problem and is to be handed in.

B. Checkout. Exercise (based on Sharkovskii's theorem).  If a continuous map on an
interval has a period 3 orbit, then what are the periods of the other
periodic orbits it must have? Instead, what if it has a period 12
orbit? Or a period 2 orbit? Express your answer concisely, not as an
infinite list.

C. Checkout. (after reading section 3.4) Assume A and B are two disjoint closed
intervals and f: R -> R is continuous. Assume
B is a subset of f(A) and A is a subset of f(B).
Must there be a period two orbit? Explain.

D. Checkout. Assume f: R -> R is continuous. Under what conditions does the
equation y = f(x) have a solution x for each y?

E. Assume f: R -> R is continuous and for the open interval A = (0, 1),

f(A) contains A. Must f have a fixed point in A? If not, give an example.

What if A = [0, 1]?

T3.4  p.113 [You must show that no 2 points have the same itinerary.]

T3.10 p. 127

T3.11, p. 128 You should assume that f is monotonic on I, on J, and on K."

T3.12 p. 128.