MA 231: Three Parameter Families Lab

Three Parameter Families of Differential Equations

This lab is due Thursday, April 5, 2007 in class.

In this lab you will investigate a three parameter family of differential equations. Your goal is to provide an understandable picture of the "parameter space" of a three parameter family of linear systems (an analogue of the trace-determinant plane). In this sense you are trying to act like a scientist or mathematician whose job it is to classify all possible outcomes of a scientific or mathematical experiment.

Here is the system:

dx/dt = ay
dy/dt = bx + cy

This system depends upon three parameters a, b, and c. Your goal is to provide an an accurate and comprehensible "picture" of the a,b,c-space, indicating the regions where this system has the various types of behavior (spiral sinks, repeated eigenvalues, zero eigenvalues, saddles, etc.). Please use different colors or shadings for the different regions. Be creative! Answer the following questions about this system in order.

1. First consider the case c = 0. Compute the eigenvalues for this special case and determine the exact a,b-values where this system has different types of behavior, i.e., spiral sinks, sources, saddles, etc. Then draw an accurate picture of the a,b-plane, indicating the regions (i.e., the points (a,b)) where the two parameter family

dx/dt = ay
dy/dt = bx

has spiral sinks, sources, saddles, etc. Display all of the different types in your picture. Use different colors or shadings for different regions. Also indicate where you find the special situations: repeated and/or zero eigenvalues, etc.

2. Now repeat question 1 for some positive c value, say c = 2.

3. Describe in words and in pictures what happens to the picture in question 2 above when you take smaller c-values, with c between 0 and 2. Then describe what happens to the picture when you choose c-values larger than 2. Be creative! Perhaps present this answer to this question as frames of a movie!

4. Now repeat question 1 for some negative c-value, say c = -2.

5. Then repeat question 3 for c-values in the interval -2 < c < 0 and c < -2.

6. Now try to draw a three dimensional version of this picture, with the c axis vertical and the a,b-plane perpendicular to this axis. Again, be creative! Highlight the special cases where your system changes its type. This is tough to visualize. Maybe you can build a three-dimensional model of this space. So again, be creative!

Important Remark: You do not need to use technology for this lab, though you are free to do so if you wish.

Have fun! And be creative! (Have I said this already?)