For the system

$\displaystyle \dot{x} = -y + ax(x^{2}+y^{2}) $

$\displaystyle \dot{y} = x + ay(x^{2}+y^{2}) $

linearise the system around the fixed point and comment on the stability properties for $\displaystyle a < 0, a = 0, a > 0. $

The system can be written in polar coordinates as

$\displaystyle \dot{r} = ar^{3} $

$\displaystyle \dot{\theta} = 1 $

By solving these equations, comment on the stability properties of the fixed point for $\displaystyle a < 0, a = 0, a > 0. $

I've found the fixed point to be (0,0). I found that the fixed point is a centre, and so stable but not asymptotically stable for all a - but this seems to simple when the question specifically mentions $\displaystyle a < 0, a = 0, a > 0. $.

I've converted the equations to polar coords and solved to get $\displaystyle r = \sqrt{\frac{r_{0}}{1-2ar_{0}t}} $ but don't know where to go from here to comment on stability.

Any help would be really appreciated