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 = (sin2(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.