Dynamical Systems

MATH 374, Exam #1

Wednesday, Oct. 5, 2:00 - 3:30 pm

The first exam covers the first six chapters of the course text and the material on the first three homework assignments. It is highly recommended that you go over homework problems and your class notes. Some material was covered in greater, more rigorous detail than in the text, so be sure to go over your class notes.

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), the Butterfly effect
  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)
  4. Important Examples of Dynamical Systems: the quadratic map Q_c(x) = x^2 + c, the doubling function, the logistic map F_k = kx(1-x)
  5. Important Theorems and Concepts: Intermediate Value Theorem, Mean Value Theorem, Fixed Point Theorem (and proof of), Attracting Fixed Point Theorem (and proof of), Chain Rule
  6. 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)

Some Practice Problems: (answers will be provided in class on Monday, Oct. 3rd)

Chapter 3:   4, 7d, 7g, 10, 15, 16, 17, 18, 19
Chapter 4:   1c, 1f, 2c, 4b, 4c, 5
Chapter 5:   1g, 1h, 1j, 2b, 2e, 3, 4e, 4g, 4h, 6, 8, 9
Chapter 6:   1d, 1j, 10, 15, 16