Dynamical Systems

MATH 374, Final Exam

Monday, Dec. 12th, 8:30 - 11:30 am

The final exam is cumulative, spanning all the material we covered during the semester. Specifically, this is Chapters 1 - 12 and 15 - 17. We covered only a portion of the material in Chapters 16 and 17 so be sure to refer to your class notes. It is highly recommended that you go over homework problems, the midterm exams and your class notes. I have also listed some sample problems from Chapters 12, 15 - 17 below. For problems from other Chapters, see the previous review sheets.

Exam Review: We will review for the final exam on Friday, Dec. 9th from 2:00 - 3:30 pm in Swords 328. Please come prepared 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 prove important theorems.

1. Basic Theory: iteration, the nth iterate of a function, web diagrams (graphical iteration)
2. Types of Orbits: fixed points, periodic points, eventually fixed and periodic points, asymptotic orbits, infinite orbits, chaotic orbits
3. Stability Types of Fixed and Periodic Points: attracting, repelling, super-attracting, neutral, weakly attracting, weakly repelling (both real and complex case), the basin of attraction, the immediate basin of attraction
4. Important Examples of Dynamical Systems: the quadratic map Q_c(x) = x^2 + c or z^2 + c, the logistic map F_k = kx(1-x), the shift map, the doubling function, piecewise linear functions
5. Bifurcation Theory: saddle-node bifurcation, period-doubling bifurcation, bifurcation diagram, how to find bifurcation values and describe the type of bifurcation, the bifurcations of Q_c(x)
6. Orbit diagrams: how they are created, what they reveal, period-doubling bifurcations, period-n windows, general theory of renormalization, Feigenbaum's constant
7. Topological Conjugacy: definition, homeomorphism, commutative diagrams, properties that are preserved under conjugacy (ie. periodic points, basin of attraction, chaos, etc.)
8. 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.), definition of continuity
9. 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)
10. Chaos: definition, dense sets, dense orbits, topological transitivity, sensitive dependence on initial conditions (the Butterfly effect), what is preserved under topological conjugacy and what is not, examples of chaotic dynamical systems
11. 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
12. The Schwarzian Derivative, the role of the critical point, the consequence of having negative Schwarzian derivative everywhere
13. Complex Functions: properties of complex numbers (arithmetic, modulus, argument, square root), dynamics of complex linear functions, the complex derivative
14. Complex Dynamics: periodic points and their stability types, the Julia set, the Fatou set, the filled Julia set, the Mandelbrot set, the connection between the orbit of 0 and the topology of the Julia set (either connected or totally disconnected)
15. Other Important Theorems and Concepts: Intermediate Value Theorem, Mean Value Theorem, Chain Rule, Fixed Point Theorem, Attracting Fixed Point Theorem, open and closed sets

Some Practice Problems: (answers will be provided or discussed at the review session)

Chapter 12:   1e, 3, 4, 6
Chapter 15:   2c, 2e, 3c, 3e, 5b, 6, 8d, 8f, 10
Chapter 16:   2, 6b, 6c, 8
Chapter 17:   1, 2, 3