MA 231: Three Parameter Families Lab

Three Parameter Families of Differential Equations

This lab is due Thursday, October 30, in class.

In this lab you will investigate several three parameter families of differential equations. In each case you are to provide an understadable picture of the parameter space (the analogue of the trace-determinant plane).

System #1:

dx/dt = ax + by
dy/dt = cx

In this case the parameters are a, b, and c. You should provide 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, etc.). Using different colors could help. Be sure to state the exact parameter values where you have the special situations: repeated and/or zero eigenvalues. You might also consider providing a two dimensional "movie" of this parameter space, by looking at various planes where one of the parameters is held fixed. Make sure you explain all of your pictures so that anyone can understand them (like me), not just an expert in ODEs.

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

Then repeat this procedure for the following system:

System #2:

dx/dt = 2x + az
dy/dt = by
dz/dt = cx

Although this is a three-dimensional system, you do not need any special knowledge to solve it. That is, this system can be (easily) solved using methods already developed for two-dimensional systems. In this case, you should provide some representative three-dimensional phase portraits (the x,y,z-plane for the various situations that arise as well as the pictures in parameter space.

Have fun!