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How to determine the direction a stable focus (source) or unstable focus (sink) is rotating, given the eigenvalues $\lambda=\alpha\pm\beta i$ ?

I know that if $\alpha > 0$ then it is source and if $\alpha < 0$, then it is a sink. This poses no problem.

However, for instance, in the book Dynamical Systems with Applications using Maple (S. Lynch), p.38 is stated that $\dot \theta = -\beta$ and if $\dot \theta > 0$, then the spiral is rotating counter-clockwise and if $\dot \theta < 0$, then the spiral is rotating clockwise. The problem is the $\pm$ sign - there are two $\beta$s - one positive and one negative. How to determine the direction of the spiral motion?

peter.babic
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  • "How to determine the direction a stable focus (source) or unstable focus (sink) is rotating, given the eigenvalues $\lambda=\alpha\pm\beta i$ ?" One cannot, compare the two systems $x'=-x\pm y$, $y'=\mp x-y$. – Did Jun 19 '15 at 09:13
  • Hello @Did, I have never encountered the relation you are describing, unfortunately. Anyway, the exam is over. But thank you for a reply. – peter.babic Jun 21 '15 at 07:33
  • I described no relation, whatever this could mean, rather I provided a pair of telling examples. Did you check them? – Did Jun 21 '15 at 08:39

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