MA 471-671

Exam #1, 1998

1. True/False.

• a. F(x) = x²-2 has a fixed point at x = -2.
• b. 1/6 is a point that lies in the middle thirds Cantor set.
• c. F(x) = x + x³ has a weakly attracting fixed point at x₀=0.
• d. F(x)=x² + c has no cycles or fixed points if c<-2.
• e. Suppose F(x)=4x(1-x). Then F³(x) has 8 fixed points.

• a. The function F(x) = A sin x has an indifferent fixed point at _________

• b. The family of functions F_A(x) = -x² + A has a saddle node bifurcation at which A-values?

• c. Find all values of the parameter A for which the family of functions F_A(x) = Ax undergoes a bifurcation.

• d. The doubling function has a cycle of prime period 2 at__________

• e. Find the distance in the sequence space Sigma between the points (1111.....) and (001 001 001.....).

3. Find the total length of all of the subintervals in the interval [0,1] that are removed during the construction of the Cantor Middle-Thirds set. That is, find the length of the complement of the Cantor set in [0,1]. What can you conclude about the ``length'' of the Cantor set itself. Show all work.

4. Use graphical analysis to give a complete orbit analysis of the function F(x) = -x³. List all fixed points and cycles and tell if they are attracting or repelling. List all points whose orbits tend to cycles or fixed points and all points whose orbits tend to + or - infinity.

5. First show that 1 lies on a three cycle for the function F(x) = -(3/2)x² + (5/2)x + 1. Then determine if this cycle is attracting or repelling or neutral.

6. Definitions. Give the precise definitions of each of the following

• a. Conjugacy.

• b. The sequence space Sigma.

• c. Saddle-node bifurcation.

• d. Repelling fixed point.

• e. The shift map.

7. Is the function F: Sigma --> Sigma given below continuous? If so, prove it. If not, prove it. F(s₀s₁s₂...) = (0000.....) if s₃ = 0, or F(s₀s₁s₂...) = (111....) otherwise.

8. Use algebra to find exactly where the family of functions F_c(x) = cx + x² has a cycle of period 2. (Note that this is not the logistic function.) Where does this family undergo period doubling bifurcation(s). ?

9. Is the following subset of Sigma dense? If so prove it, If not, prove it. H = all sequences whose tenth entry is 1.