Problem 1: Edelstein-Keshet p.364, #7.

Problem 2: Ut oh.  Another handwritten problem.  Consider the following reaction:
 

,

.

a.   Write down the ODE that corresponds to this chemical reaction.  Rescale X and Y and time to write the equations as

b.  Find the steady state(s) of the nondimensional equation given in a.

c.  Comment on the stability of the steady state with respect to the parameters.  In particular, find a relationship between the parameters which will imply stability.  Further, derive a relationship between a and b which will imply that the steady state is both unstable and also a spiral.  (E.g. Something like, "If b-a < (b+a)³ then .... ")

d.  Hopefully b=0.5, a=0.1 satisfies your conditions for an unstable spiral.  For these parameters, use pplane5 to plot the direction field and null clines of the system (not individual trajectories).  Can you find a region around the critical point so that all the arrows in the direction field along the boundary are pointing into the region?  Print out the direction field and draw (by hand) such a region.

e.  What can you conclude (maybe using the Poincare-Bendixson theorem)?  Use pplane5 to verify your conclusions.

Problem 3: Edelstein-Keshet p.364, #8.