Chapter 7.2, Problem 25P

In this problem, we show how small changes in the coefficients of a system of linear equations can affect a critical point that is a centre. Consider the system

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

Show that the eigenvalues are $\pm i$ so that $\left(0,0\right)$ is a centre. Now consider the system

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

where $\left|\epsilon \right|$ is arbitrarily small. Show that the eigenvalues are $\epsilon \pm i$. Thus no matter how small $\left|\epsilon \right|\ne 0$ is, the centre becomes a spiral point. If $\epsilon <0$, the spiral point is asymptotically stable; if $\epsilon >0$, the spiral point is unstable.