The following animations are used in Professor Devaney's MA 226 course either as classroom demos, computer lab assignments, animated homework assignments, or exam questions. All of these movies are in QuickTime format.

Note: You will need a QuickTime viewer to see these animations. Click here to download a QuickTime player for a Mac or PC.

1. Phase Line Animations
   In the following exercises, you should sketch the graph of x(t) (x as a function of time t). You see a point moving along the x-axis, and you draw the graph of x(t).
   1. Phase Line Motion Exercise #1
   2. Phase Line Motion Exercise #2
   3. Phase Line Motion Exercise #3
2. Phase Line Motion Classroom Demo
   This is a demo of what motion along the phase line represents in terms of the solutions to the differential equation.
3. Phase Line Motion Quiz
   This is a good way to check to see if you can go from a graph of x(t) to motion along the x-axis.
4. Sink-source-node Demo
   This animation illustrates the differences among the various types of equilibria on the phase line.
5. One-Dimensional Bifurcation Homework
   An exercise in sketching the bifurcation diagram given a one-parameter family of graphs.
6. Saddle-Node Bifurcation Demo
   This is an animation of the bifurcation that we analyzed in class (not the modified logistic, but the other quadratic one-parameter family).
7. Motion of the Pendulum Homework/Demo
   This is a good exercise that relates solution curves in the phase plane with x(t)- and y(t)-graphs. In this case, the system represents the damped, nonlinear pendulum. Professor Devaney uses the variable y where I might use v. The derivation of the system can be found on pp. 439-443 of our text and the effects of damping are illustrated on pp. 456-457. However, you should be able to do this exercise just by thinking about what the variables mean. There is also a nice demo of the damped, nonlinear pendulum in IDE. It is called Pendulums. (Make sure that you select the damped, nonlinear option. I would also turn off the acceleration graph since acceleration is not one of the functions in our system. While you are at it, you might want to play with some of the other versions of the pendulum. The forced, damped pendulum is a blast.)
8. These two problems are similar to the phase line problems in #1 above except that the motion is now motion in the xy-plane. You should sketch two graphs, the graph of x(t) and the the graph of y(t), but I would like the two graphs to be on the same set of axes. The horizontal axis is the t-axis, and the vertical axis is either x or y.
   1. Motion in the Plane Homework #1
   2. Motion in the Plane Homework #2

Last revision: February 22, 2000