Computer Assignment 3    Math 375  Uhlenbeck/Buck

You are encouraged to hand in joint work (up to three people).

1-2.  Graph a couple of solutions to the equations and describe the
    equilibrium point at zero as a sink, spiral sink, source, spiral
    source, saddle or center.  It is suggested that you use the
    plots of direction fields to do this problem.
                                         x(t)
          dX/dt =  A (X)   where X(t) = (    )
                                         y(t)

   a.           -1  1
            A =(      )
                -1  -1


   b.             -1 1
            A  = (    )
                   1 1

   c.             0 4
            A  =(    )
                 -1 0

   d.           5  1
            A =(    ) 
                1  5

3.  Use one of the programs under systems to find the eigenvalues of
   the four matrices in problems 1-2. You may also do the computation
   by hand if you like. Check your answers in 1-2.

4.  Experiment numerically  with the 3X3 system belonging to the matrix

          1 1  0
        ( 1 2  1)
          1 -1 4

   in the section of programs.  Try a couple of initial values. Are
   the solutions you plot stable or unstable? The best program for doing this
   is under "higher order equations converted to systems and solved".
   Use the last program under systems  to find the eigenvalues of this
   matrix.  Classify zero as a critical point. Do your answers agree?

5. Make a direction field plot of the non-linear equation

      dx/dt = .2* x*(1- .25*x  +  .1*y) 
      dy/dt = .3*y*(1-.25*y + .1*x).

Find the critical points and classify them (either by computational
methods or by looking at the plots).