Consider the linear system $x^{\text{'}}=Ax,$ where $A$ is a $22$ real constant matrix. Let $p=$trA, $q=\frac{d}{e}tA$ and $=p^2-4q.$

$($a$)$ Show that the critical point $(0,0)$ is

a node if $q>0$ and $0$;

a saddle point if $q<mn>0</mn>$;

a spiral point if $p0$ and $<mn>0</mn>$;

a center if $p=0$ and $q>0.$

$($b$)$ Show that the critical point $(0,0)$ is

asymptotically stable if $q>0$ and $p<mn>0</mn>$;

stable if $q>0$ and $p=0$;

$1$

unstable if $q<mn>0</mn>$ or $p>0.$

See Fig. $9\mathrm{.}1\mathrm{.}9$ in the book for the diagram illustrating these results.