Dynamical Systems     MATH 374

Exam #2

Thursday, April 15, in class

The second exam covers Chapters 7 - 12, the material on homework assignments 4, 5 and 6 and computer projects 2 and 3. It is highly recommended that you go over homework problems and your class notes. 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: We will review for the exam during class on Tuesday, April 13th. Please come with specific questions.

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.

1. Orbit Diagrams: how they are created, what they reveal, period-doubling bifurcations, period-n windows, self-similarity, theory of renormalization
2. Topological Conjugacy: definition, homeomorphism, commutative diagrams, properties that are preserved under conjugacy (ie. periodic points, basin of attraction, chaos, etc.)
3. 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
4. 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, the "tent map" T(x) from Ch. 7 Exercises 9 - 15
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)
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: Schwarzian derivative, basin of attraction, immediate basin of attraction, the consequence of having negative Schwarzian derivative everywhere
9. Other Important Concepts: open and closed sets, totally disconnected set, Fact 1 in the proof of the Period 3 Theorem

Some Practice Problems:

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