# Stability of dynamical system described in polar coordinates

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?

• If the system of polar coordinates is centered at the fixed point, stability should probably be equivalent to $A_{11}\leqslant0$ and $A_{12}=0$. – Did Jan 31 '15 at 7:42
• @Did, Could you show (or prove) this in an answer? – wdg Feb 1 '15 at 12:03
• 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. – Did Feb 1 '15 at 12:07