Near a fixed point, a dynamical system $\dot{\bf{x}}=\bf{F}(\bf{x})$ can be approximated by $\dot{\bf{x}}=A\bf{x}$, where $A$ is the Jacobian matrix. From the trace and determinant of the Jacobian $A$, we can determine the property of the fixed point, i.e., stable/unstable node/spiral, etc. However, usually this is shown in Cartesian coordinates. If I write the dynamical system in

$$ \begin{pmatrix} \dot{r}\\ \dot{\theta}\\ \end{pmatrix} =A\begin{pmatrix} r\\ \theta\\ \end{pmatrix} $$ near a fixed point. My question is, what does $A$ tell us about the property of the fixed point? Do the things that hold for Cartesian coordinates also hold for polar coordinate (e.g. trace and determinant of the Jacobian $A$ determines the property of the fixed point)? Why?

  • $\begingroup$ If the system of polar coordinates is centered at the fixed point, stability should probably be equivalent to $A_{11}\leqslant0$ and $A_{12}=0$. $\endgroup$ – Did Jan 31 '15 at 7:42
  • $\begingroup$ @Did, Could you show (or prove) this in an answer? $\endgroup$ – wdg Feb 1 '15 at 12:03
  • $\begingroup$ That $A_{11}\lt0$ and $A_{12}=0$ imply stability should be within your reach. The stronger statement in my first comment is probably slightly too strong to be true. $\endgroup$ – Did Feb 1 '15 at 12:07

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