Consider the nonlinear system: \begin{align} \dot x &= -x - \frac{y}{\log \sqrt{x^2 + y^2}} \\ \dot y &= -y + \frac{x}{\log \sqrt{x^2 + y^2}} \end{align} a) Show that the origin is a stable node for linearized system:
Attempt at the solution:
The Jacobian is $$ J(x, y) = \begin{bmatrix} \frac{x y}{(x^2 + y^2) \log^2{\sqrt{x^2 + y^2}}} - 1 & \frac{y^2}{(x^2 + y^2) \log^2{\sqrt{x^2 + y^2}}}- \frac{1}{\log\sqrt{x^2 + y^2}} \\ \frac{1}{\log\sqrt{x^2 + y^2}} - \frac{x^2}{(x^2 + y^2) \log^2\sqrt{x^2 + y^2}} & -\frac{x y}{(x^2 + y^2) \log^2\sqrt{x^2 + y^2}} - 1 \\ \end{bmatrix}. $$ As we approach $(0, 0)$ it appears that $J$ approaches $$ \begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix}, $$ if this is true then it shows $(0, 0)$ is a stable node. However, I'm unsure how to show this rigorously.
b) Show that the origin of the nonlinear system is not a stable node but rather a stable spiral.
Attempt at the solution: I am unsure how to proceed for this part.
Thanks for your help.