The final exam is cumulative, spanning all the material we covered during the semester. Specifically, this is Chapters 1 - 12 and 14 - 16. We covered only a portion of the material in certain chapters, while some material (such as topological conjugacy) was covered in greater, more rigorous detail than in the text, so be sure to refer to your class notes. It is highly recommended that you go over homework problems, the midterm exams, the computer projects and your class notes.

I have also listed some sample problems from Chapters 14 - 16 below. For problems from other Chapters, see the previous exam review sheets. In addition, some sample final exam questions are provided here (PDF File).

**Exam Review:** We will review for the final exam on Thursday, May 6th from 10:00 - 11:30 am
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/or prove important theorems.

- Basic Theory: iteration, the nth iterate of a function, web diagrams (graphical analysis), the Butterfly Effect
- Important Pioneers in the Field: Lorenz, Mandelbrot, Poincare, Smale, Yorke, Libchaber, Fatou and Julia, Douady and Hubbard (know who they are and what their contributions were)
- Types of Orbits: fixed points, periodic points, eventually fixed and periodic points, asymptotic orbits, infinite orbits, chaotic orbits
- Stability Types of Fixed and Periodic Points: (for both the real and complex cases), attracting, repelling, super-attracting, neutral, weakly attracting, weakly repelling
- Important Examples of Dynamical Systems: the quadratic map
**Q_c(x) = x^2 + c**or**z^2 + c**, the doubling function, the logistic map**F_lambda = lambda x(1-x)**, the shift map, piecewise linear functions - 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)
- Orbit Diagrams: how they are created, what they reveal, period-doubling bifurcations, period-n windows, self-similarity, theory of renormalization, Feigenbaum's constant
- Topological Conjugacy: definition, homeomorphism, commutative diagrams, properties that are preserved under conjugacy (ie. periodic points, basin of attraction, chaos, etc.)
- 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
- 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)
- 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
- 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
- The Importance of the Critical Orbit: Schwarzian derivative, basin of attraction, immediate basin of attraction, the consequence of having negative Schwarzian derivative everywhere
- Fractals: definition, topological dimension, fractal dimension, self-similarity, key examples (Cantor set, Koch snowflake curve, Sierpinski triangle)
- Complex Functions: properties of complex numbers (arithmetic, modulus, argument, polar form, square root), dynamics of complex linear functions, the complex derivative
- Complex Dynamics: periodic points and their stability types, the Julia set, the Fatou set, the filled Julia set, the Mandelbrot set
- Other Important Concepts/Theorems: Intermediate Value Theorem, Mean Value Theorem, Chain Rule, Fixed Point Theorem, Attracting and Repelling Fixed Point Theorems, open and closed sets, totally disconnected set, Fact 1 in the proof of the Period 3 Theorem

**Some Practice Problems:**

**Chapter 14: ** 11, 14

**Chapter 15: ** 2c, 2e, 3c, 3e, 5a, 5d, 5e, 5f, 8d, 8f, 10

**Chapter 16: ** 1, 5a, 5b, 8