Chapter 3.5, Problem 16P

Continuing Problem $15$, Show that the critical point $\left(0,0\right)$ is a

a) Asymptotically stable if $q>0$ and $p<0$;

b) Stable if $q>0$ and $p=0$;

c) Unstable if $q<0$ and $p>0$.

The result of Problem $15$ and $16$ are summarized visually in Figure $\mathrm{3.5.7}$.

Figure $\mathrm{3.5.7}$ Stability diagram.

15. Consider the linear system $\frac{dx}{dt}=a_{11}x+a_{12}y,\frac{dy}{dt}=a_{21}x+a_{22}y$, where $a_{11},a_{12},a_{21}$ and $a_{22}$ are real constants. Let $p=a_{11}+a_{22},q=a_{11}{a}_{22}-a_{12}{a}_{21}$ and $\Delta =p^2-4q$. Observe that $p$ and $q$ are the trace and determinant, respectively, of the coefficient matrix of the given system. Show that the critical point $\left(0,0\right)$ is a

a) Node if $q>0$ and $\Delta \ge 0$;

b) Saddle point if $q<0$;

c) Spiral point if $p\ne 0$ and $\Delta <0$;

d) Center if $p=0$ and $q>0$.

Hint: These conclusion can be obtained by studying the eigenvalues ${\lambda }_1$ and ${\lambda }_2$. It may also be helpful to establish, and then to use, the relation ${\lambda }_1{\lambda }_2=q$ and ${\lambda }_1+{\lambda }_2=p$.