For the system $$x'=y(1+r^2),~~y'=-x(1+r^2),$$ where $r=x^2+y^2,$ determine whether this is asymptotically stable or not.
My approach: Writing in terms of Polar coordinates, $x=r \cos \theta,~y=r \sin \theta$ and simplifying, I got $$r'=0,~~\theta'=-1-r^2.$$ By integrating, $$r=r_0,~~\theta=\theta_0 -(r_0^2 + 1)t.$$ From this it's clear these form a family of periodic orbits with each being a circle in the phase plane. So they are orbitally stable. But what can I say about the asymptotic stability ?
The reason why there are no other stable orbits is that the angular speed $\theta'$ varies with $r$ and so from orbit to orbit. Moreover, for asymptotic stability all orbits in a sufficiently small neighborhood of the origin should converge to it.
Note that it doesn't make sense to ask whether an equation is asymptotically stable or not, only an orbit.