Messing around with this it becomes clear that
i) if a=0, then (0,0) is a center
ii) if a>0, the trajectory is an asymptotically unstable spiral about (0,0)
iii) if a<0, the trajectory is an asymptotically stable spiral about (0,0)
I'm not seeing from the linearization why this is so yet. My first cut had $a$ dropping out of the system matrix at (0,0).
I'll continue to look at this.