Dynamical Systems     MATH 374

Exam #1

Thursday, Oct. 5, in class

The first exam covers Chapters 1 - 6 of Devaney's text, the material on the first three homework assignments, and the first computer project. 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.

Exam Review session: We will review for the exam on Tuesday, Oct. 3 from 7:30 - 8:45 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.

  1. Basic Theory: iteration, the nth iterate of a function, web diagrams (graphical analysis), the Butterfly Effect

  2. Important Pioneers in the Field: Lorenz, Mandelbrot, Poincare, Smale, Yorke (know who they are and what their contributions were)

  3. Types of Orbits: fixed points, periodic points, eventually fixed and periodic points, asymptotic orbits, infinite orbits, chaotic orbits

  4. Stability Types of Fixed and Periodic Points: attracting, repelling, super-attracting, neutral (weakly attracting, weakly repelling)

  5. Important Examples of Dynamical Systems: the quadratic map Qc(x) = x2 + c, the doubling function, the logistic map Fλ = λ x(1-x)

  6. Important Theorems and Concepts: Fixed Point Theorem (and proof of), Attracting Fixed Point Theorem (and proof of), Repelling Fixed Point Theorem, Intermediate Value Theorem, Mean Value Theorem, Chain Rule

  7. Bifurcation Theory: common types of bifurcations (saddle-node, period-doubling, pitchfork), the bifurcation diagram, how to find bifurcation values and describe the type of bifurcation, the bifurcations of Qc(x)

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

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, 4b, 4c, 4d, 4f, 4h, 6, 8, 9
Chapter 6:   1d, 1j, 10, 15, 16