We're given the following dynamical system $$\dot{x} = -x - \frac{y}{\log(\sqrt{x^2 + y^2})} $$ $$\dot{y} = -y + \frac{x}{\log(\sqrt{x^2 + y^2})} $$ And then asked to show that the origin is a stable node for the linearized system, and a stable spiral for the nonlinear system.
First, I converted the system to polar, giving me the following $$\dot{r} = -r $$ $$\dot{\theta} = \frac{1}{\log(r)} $$ When I plotted the phase plane in wolfram, I do in fact see that the origin is a stable node/spiral, despite the dynamical system not being defined there. What I'm not sure how to proceed is dealing with the linearization. We have that $\frac{\partial \dot{\theta}}{\partial r} = \frac{-1}{r(\log(r))^2} $, which is neither defined at the origin nor is its limit. How do I consider the linearization about the origin then?
