Homework 1



Homework 1: Due Friday 24 September


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.

The first three problems should require very little algebra. The last problem will require a moderate amount of algebra.

  1. Find a function for a finite-difference equation with:
    1. Four fixed points, all of which are unstable.
    2. Eleven fixed points, three of which are stable.
    3. No fixed points.
    Hint: just give the function's graph. Do no algebra!

  2. Consider the function f(x) = x^3.
    1. Plot the function. Choose several different initial conditions and show their graphical iterates.
    2. Based on your graphical work, find all fixed points and classify them.
    3. Using the algebraic criteria (i.e. by finding the slope at the fixed points), determine the stability of all fixed points.
    4. Does your algebra agree with the graphical results?

  3. Consider the function f(x) = -x + 4
    1. Plot the function. Choose several different initial conditions and show their graphical iterates.
    2. Does this function have a fixed point?
    3. What type of fixed point is it? Discuss.

  4. For the logistic map
    1. Calculate the fixed point as a function of r.
    2. Algebraically determine the r value at which the fixed point loses stability.
    3. Does the bifurcation diagram confirm your algebraic calculation?



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

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