MA 226: Families of Linear Systems Lab

Two Parameter Families of Linear Systems

This lab is due **Tuesday, April 12, 2016** in class.

In this lab you will investigate four different two-parameter families
of linear systems of differential equations. In each case the parameters
will be **a** and **b** where both **a** and **b** are real.
Your goal is to provide a
picture of the "parameter plane" for each of these
families of linear systems, i.e., the **a,b**-plane.
**Important:** I am not talking about the trace-determinant plane here;
the horizontal axis in your picture
should be the **a**-axis, and the vertical axis
should be the **b**-axis.

You should first find all of the eigenvalues for the corresponding
matrix (which, of course, depend on both **a** and **b**).
Then you should determine all of the
curves for which the system undergoes
a bifurcation, i.e., the parameters in the **a,b**-plane for which
there are repeated eigenvalues, zero eigenvalues, or pure imaginary
eigenvalues. Next, draw each of these curves in the **a,b**-plane.
Finally, in each region between these curves, you should
indicate the type
of phase plane that occurs, e.g., a saddle, spiral sink, source, etc.
Also indicate where there are repeated zero eigenvalues in your picture.
You need not display the direction that solutions travel around the origin
when the eigenvalues are complex, and you do not need to compute/display
eigenvectors. You will lose one point for each erroneous computation,
incorrectly drawn
curve, and incorrectly identified region in the parameter plane.

**
System 1:
**

**
dx/dt = ay
dy/dt = bx
**

**
**

System 2:

dx/dt = ax - by

dy/dt = (b/4)x

System 3:

dx/dt = 2bx + (a + 1)y

dy/dt = (a - 1)x

System 4:

dx/dt = 2bx - y

dy/dt = (sin^{2}(a)) x

**Remark:** You
do not need to use technology for this lab, though
you are free to do so if you wish. The Linear Phase Portraits applet
will be useful here. This can help you to check
whether your parameter plane pictures are correct.
Enter various **a** and **b** values into this applet and
view the corresponding phase plane.