This lab is due Thursday, November 14 at 5 PM. Turn it in to your TF or to the TF in the lab.
In this lab you will discuss the phase portraits and bifurcations for the linear system of differential equations
dy/dt = x - 0.5 y
where A is a parameter and -0.4 < A < 0.4. If you click on the following icon, you will see a QuickTime animation of the phase planes for this system as A changes.
Click here to download the animation.
Incidentally, contrary to what is printed in the animation, you cannot click on the buttons to control the movie. Depending upon which Quick Time viewer you have, you can stop and start the motion by using the control panel at the bottom of the screen, or by double clicking in the window.
In this animation, we have selected eight different initial points (marked by asterisks) and plotted the corresponding solution curves through these points. Note that the behavior of these solutions changes as A varies. Your job is to explain the bifurcations that this system undergoes as the parameter changes.
Answer each of the following questions about this system.
1. List below all values of A for which there are bifurcations for this system. Sketch the phase portraits just before, at, and just after each bifurcation. Use one color pen to indicate solutions that tend toward an equilibrium point, another for those that tend away from an equilibrium point, and a third for solutions that do neither. Also, indicate what happens to the eigenvalues of the system as the parameter passes through the bifurcation.
2. Sketch the path in the trace-determinant plane that this family of matrices is following as A increases from -0.4 to 0.4. Indicate on this path where the two bifucations have occurred.