Stability of higher-order fixed points for systems of ordinary differential equations In the book by Strogatz, Nonlinear Dynamics and Chaos (1994), the author discusses examples of higher-order fixed points for systems of ordinary differential equations in polar coordinates:


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*$\dot{r}=ar^3, \dot{\theta}=1$, $a\ne0$

*$\dot{r}=-r, \dot{\theta}=1/\ln(r)$
In the above cases, the linearized systems show a non-isolated fixed point at the origin. However, the nonlinear systems are spirals at the origin. Is there an analytical (non-graphical) method to deduce this result ?

You can solve these examples explicitly. In the first system, the equations for $\dot{r}$ and $\dot{\theta}$ are decoupled, so you can solve the equations individually. In the second equation, you can solve $\dot{r} = -r$ first, and then plug the result into the second equation to get an explicit function for $\dot{\theta}$, which you can then integrate in $t$ to obtain $\theta(t)$. Once you have the explicit solutions you should be able to deduce the spiral behavior.
