Homework 2



Homework 2: Due Friday 1 October


This assignment is not yet complete. A few short problems will be added Sunday or Monday.

Problem 1 is based very closely on a problem from Daniel Kaplan and Leon Glass, Understanding Nonlinear dynamics, Springer-Verlag, 1995. This is an excellent text; I used it the first time I taught this class. If you want to learn more about chaos and nonlinear dynamics, this is a great place to start.

  1. The population of a species is described by the equation f(x) = ax e^(-t), where x >= 0 and a is a positive constant.
    1. Determine the fixed points.
    2. Evaluate the stability of the fixed points.
    3. For what value of a is the first period-doubling bifurcation?
    4. For what values of a will the population go extinct starting from any initial condition?
    5. Suppose you were to make a bifurcation diagram for this equation as a function of a. What do you think the bifurcation diagram would look like and why? Which features of the diagram could you predict, and which couldn't you?
    6. (optional)Using a computer, generate a bifurcation diagram as a function of a.

  2. Calculate the Lyapunov exponent for the logistic equation with:
    1. r = 2.5
    2. r = 3.0
    3. r = 3.2

  3. Determine the fractal dimension of the following shapes.
    1. The Sierpinski triangle. (Sierpinski lived from 1882-1969. Poland put him on a stamp.) Here's another view of the Sierpinski triangle.
    2. The Sierpinski carpet.
    3. The Menger sponge.
    4. The box fractal.

[Dave] [Chaos and Complex Systems] [COA]

Web page maintained by dave@hornacek.coa.edu.