Lab 12 Expectations


Submit Plots: 1a, 1b, 2a, 2b, 3b, 3c, 4a, 5b, 5d

1.            a. Use pplane to plot several orbits for (*). What kind of solution does the 
               origin seem to be?
               b. Plot these solutions on the square |x,y|<0.1
               c. Let A be the 2x2 matrix for (*). Find the eigenvalues and eigenvectors for
               A and use these to prove what you found in a. is correct.
 
2.            a. Repeat parts a. and b. from part 1 for this system.
 
3.            a. Determine the approximating linear system for each of the systems listed. 
               Find the eigenvalues for these systems and use them to prove what kind of 
               point the origin is.
               b. Plot some orbits for the linear system to support your claim in part a.
               c. Plot some orbits for the corresponding non-linear system to show that 
               these predictions still hold.
 
4.            a. Use pplane to plot several orbits for the system (B). Find the 
               approximating linear system for (B*) about (0,0). Plot several 
               orbits for this approximate system. Show that when a small enough scale is 
               used, the uv plot around the origin looks like the xy plot around (.5, .5).
               b. Use Matlab to find the eigenvalues and eigenvectors for the linear 
               approximation to (B*) about (0,0). Then, using the general solution to the 
               system, prove your origin is a sink.
 
5.            a. Plot a few orbits of the corresponding linear system. They should appear 
               to be circular about the origin. Prove this is mathematically true.
               b. Use pplane to plot the system (C) in the interval |x,y|<2.
               c. Choose the smaller scale so that you can really see what’s happening 
               around the origin. Use (C) to prove (***). How does it follow that the orbits 
               move constantly toward the origin? How does the sign of the derivative function 
               relate to the behavior of the function? How can you explain the fact that the 
               computer shows you that the orbits are loops when in fact they are spirals?
               d. Use pplane to plot the specific orbit which starts at x = 0, y = -0.2, use 
               the intervals |x,y|<3 and the zoom-in feature several times near the origin to 
               approximate this radius.