MA 471-671

Exam #1, 1993

1. True/False.

• a. F(x) = x² has a fixed point at x= -1.
• b. Let (000....) and (010101.....) be two sequences in the sequence space Sigma. Then the distance between these two points is 1.
• c. The Cantor Middle-Thirds Set is dense in the interval [0,1].
• d. The function F(x) = -1 has no fixed points.
• e. The function G defined on the sequence space Sigma by G(s_0 s_1 s_2 .....) = (000....) if s_0 = 0, or G(s_0 s_1 s_2 .....) = (111....) if s_0 = 1$ is continuous at all points of Sigma.

2. Quickies. Answers only -- no partial credit.

• a. The function F(x) = 3x(1-x) has an indifferent fixed point at _______.
• b. The family of functions F(x) = x² + x + C has a ___________ bifurcation at C = 0.
• c. For which values of c does the function F(x) = x² + c have no cycles or fixed points whatsoever?
• d. The shift map has how many cycles of prime period 4?
• e. The family of functions S(x) = A sin x has an attracting fixed point at the origin for which A values?

3. Use graphical analysis to give a complete orbit analysis of the function F(x) = -x². 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.

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

• a. Chaos.
• b. F is continuous at x_0.
• c. The shift map sigma on the sequence space.
• d. Q is a dense subset of the set T.

5.In an essay, describe the bifurcation that occurs at c = -3/4 for the family of functions F_c(x) = x² + c. Your essay should include enough graphs and bifurcation diagrams to explain this bifurcation fully.

6. In an essay, describe the sequence space Sigma. Define the metric on Sigma. Discuss what it means for two points to be close together in Sigma. Give examples of nearby points and points that are far apart. You will be graded on the accuracy of your statements as well as the use of English in your essay (spelling, grammar, etc.)

7. Consider the subset of Sigma that consists of all sequences that have no strings of the form 0110 in them. Is this set dense? If so, prove it. If not, prove it. And, most certainly, have fun in either case.