Problem: Consider the system $\displaystyle \dot{x} = -y - x^3$, $\displaystyle \dot{y} = x$. Show that the origin is a spiral, although the linearization predicts a center.

I tried converting to polar coordinates by doing $\displaystyle x = r\cos{\theta}$, $\displaystyle y = r\sin{\theta}$. But the equations I got were:

$\displaystyle \dot{r} = -r^3\cos^4{\theta}$ and $\displaystyle \dot{\theta} = 1 + r^2\cos^3{\theta}\sin{\theta}$, which aren't independent equations, so it didn't really help in analyzing the system.

The two identities used were: $\displaystyle x\dot{x} + y\dot{y} = r\dot{r}$ and $\displaystyle \dot{\theta} = \frac{x\dot{y} - y\dot{x}}{r^2}$

Any hints or suggestions would be greatly appreciated!