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MA 471-671

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Exam #1, 1998

1. True/False.

- a.
**F(x) = x**^{2}-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 + x**^{3} has a weakly
attracting fixed point at **x**_{0}=0.
- d.
**F(x)=x**^{2} + c has no cycles or
fixed points if **c<-2**.
- e. Suppose
**F(x)=4x(1-x)**. Then
**F**^{3}(x) has 8 fixed points.

2. Quickies. Answers only -- no partial credit.
- a. The function
**F(x) = A sin x** has an
indifferent fixed point at _________

- b. The family of functions
**F**_{A}(x) = -x^{2} + 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) = -x**^{3}. 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)x**^{2} + (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(s**_{0}s_{1}s_{2}...) = (0000.....) if
s_{3} = 0, or **F(s**_{0}s_{1}s_{2}...) =
(111....) otherwise.

8. Use algebra to find exactly where the family of
functions **F**_{c}(x) = cx + x^{2
} 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**.