Dynamical Systems     MATH 374

Exam #2

Tuesday, November 21, in class

The second exam covers Chapters 7 - 12, the material on homework assignments 4, 5, and 6 and the second computer project. It is highly recommended that you go over homework problems and your class notes/handouts. Some material (such as topological conjugacy) was covered in greater, more rigorous detail than in the text, so be sure to go over your class notes.

Exam Review session: We will review for the exam on Monday, Nov. 20 from 7:00 - 8:30 pm in Smith Labs 155. Please come prepared with specific questions.

Note: No calculators are allowed for this exam so be prepared to answer questions without your personal calculator.

The following topics, definitions and theorems are important material for the exam. You may be asked to define some terms precisely as well as state and/or prove important theorems, so be sure to know this material thoroughly.

1. Topological Conjugacy: definition, homeomorphism, commutative diagrams, properties that are preserved under conjugacy (e.g., periodic points, basin of attraction, topological transitivity, etc.)
2. Symbolic Dynamics: Sequence Space (Sigma_2), the metric (distance function) d(s,t) on Sigma_2, the Proximity Theorem, the shift map (sigma), properties of the shift map (continuous, contains a dense orbit, periodic points, conjugate to Q_c for c < -2, etc.), continuity of functions on Sigma_2
3. Important Examples of Dynamical Systems: the quadratic function Q_c(x) = x^2 + c, the logistic map F_k = kx(1-x), the doubling function, piecewise linear functions
4. Orbit Diagrams: how they are created, what they reveal, period-doubling bifurcations, period-n windows, self-similarity
5. The Quadratic Map for c < -2: Cantor sets (definition, general construction of, uncountable sets), the middle-thirds Cantor set, ternary expansions, the itinerary map S (the conjugacy between Q_c and sigma --- see the class handout)
6. Chaos: definition, dense sets, dense orbits, topological transitivity, sensitive dependence on initial conditions, what is preserved under topological conjugacy and what is not, examples of chaotic dynamical systems
7. Sarkovskii's Theorem: statement of, Sarkovskii's ordering of the natural numbers, converse of, the period 3 Theorem, piecewise linear functions as examples of the converse to Sarkovskii's theorem
8. The Importance of the Critical Orbit: the Schwarzian derivative, basin of attraction, immediate basin of attraction, the consequence of having negative Schwarzian derivative everywhere (Main Theorem)
9. Other Important Concepts: open and closed sets, totally disconnected set, countable and uncountable sets

Some Practice Problems: (answers will be emailed to the class)

Chapter 7:   2, 4, 7, 15
Chapter 8:   11, 12, 13
Chapter 9:   4, 6, 18c, 18g
Chapter 10:   2, 12, 13, 17, 21
Chapter 11:   6, 7
Chapter 12:   1e, 3, 4, 6