  The study of emergent behavior of swarms is of great interest for applied
sciences. In this paper, we study the stability of the system of $n$-coupled
self-propelled particle systems $\ddot r_k = (1-|\dot r_k|^2)\dot r_k -
\frac{1}{n}\sum_{m=1}^n(r_k-r_m)$, $r_k\in \mathbb R^2$, with all-to-all
spring-like coupling. Previous numerical experiments have shown that for a
large set of initial conditions, after an initial drift, the center of mass of
the system converges to a stationary point and each particle eventually rotates
with unit angular velocity around the stationary center of mass. The
distribution of the particles on the circle need not be uniform. Such limit
states are dubbed {\it rotating states}. We prove that a rotating state with
particles spinning in the same direction is stable and we show that every
solution that starts near a rotating state asymptotically approaches a rotating
state. The proof uses new slow manifold ideas to improve the approximations of
the flow on the center manifold in the presence of non-isolated fixed points.
