Chapter 7.3, Problem 8P

Consider the system (2) in the text, and assume that

${\sigma }_1{\sigma }_2-{\alpha }_1{\alpha }_2=0.$

a) Find all the critical points of the system. Observe that the result depends on whether ${\sigma }_1{\epsilon }_2-{\alpha }_2{\epsilon }_1$ is zero.

b) If ${\sigma }_1{\epsilon }_2-{\alpha }_2{\epsilon }_1>0,$ classify each critical points and determine whether it is asymptotically stable, stable, unstable. Note that Problem 5 is of this type. Then do the same if ${\sigma }_1{\epsilon }_2-{\alpha }_2{\epsilon }_1<0$

c) Analyse the qualitative behaviour of the trajectories when ${\sigma }_1{\epsilon }_2-{\alpha }_2{\epsilon }_1=0$