CS/CogS 3150: Nonlinear Systems

Midterm

 

These review questions are to help you get ready for the midterm. Be sure to review your notes and the books also . . .

  1. What is a dynamical system? What are linear and non-linear dynamical systems? What properties of non-linear systems make them difficult to analyse?

  2. What is the logistics equation? Why is it an important example to study? How is Feigenbaum's Constant related to it? What is a `period doubling bifurcation cascade'? What universal properties does the logistics equation reflect? What are the properties of the system for C > 4? What about for other values of C?

  3. What is phase space? Breifly discuss, with examples.

  4. Briefly discuss some differences between continuous and discrete dynamical systems. Give some examples of each.

  5. Briefly discuss the oscillator/pendulum/satellite examples. What are fixed points? What are attractors?

  6. What are fractals? Why are fractals interesting in the context of nonlinear systems? Give examples.

  7. What are the Cantor set, the Mandelbrot set, the Koch snowflake, the Sierpinski triangle/carpet/sponge? What is a fractal dimension? How is it calculated? Give an example, and calculate its fractal dimension.

  8. What is a fixed point? What is a stable set? What is an attractor? What is a strange attractor? What is a basin of attraction? What is a Julia set? What is a limit cycle?

  9. What are chaotic systems? Why are they resistant to numerical computation? Why are they worth studying?

  10. What are some of Poincare's important contributions to the study of dynamical systems? What is a Poincare section? Who else do you consider important in the study of nonlinear systems? Why?

  11. What other interesting/important topics did we discuss? Expand on at least one.

  12. Anything else?