Chapter 7.2, Problem 26P

In this problem, we show how small changes in the coefficients of a system of linear equations can affect the nature of a critical point when the eigenvalues are equal. Consider the system

$\text{x'}=\left(\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right)\text{x}$

Show that the eigenvalues are ${\lambda }_1=-1$, ${\lambda }_2=-1$ so that the critical point $\left(0,0\right)$ is an asymptotically stable node. Now consider the system

$\text{x'}=\left(\begin{array}{cc}-1& 1\\ -\epsilon & -1\end{array}\right)\text{x}$,

where $\left|\epsilon \right|$ is arbitrarily small. Show that if $\epsilon >0$, then the eigenvalues are $-1\pm i\sqrt{\epsilon }$, so that the asymptotically stable node becomes an asymptotically stable spiral point. If $\epsilon <0$, then the eigenvalues are $-1\pm i\sqrt{\left|\epsilon \right|}$, and the critical point remains an asymptotically stable node.