  The main result of this paper is proving the stability of translating states
(flocking states) for the system of $n$-coupled self-propelled agents governed
by $\ddot r_k = (1-|\dot r_k|^2)\dot r_k - \frac{1}{n}\sum_{j=1}^n(r_k-r_j)$,
$r_k\in \mathbb R^2$. A flocking state is a solution where all agents move with
identical velocity, of magnitude one. Numerical explorations have shown that
for a large set of initial conditions, after some drift, the particles'
velocities align, and the distance between agents tends to zero. We prove that
every solution starting near a translating state asymptotically approaches a
translating state nearby, an asymptotic behavior exclusive to swarms in the
plane. We quantify the rate of convergence for the directional drift, the mean
field speed, and the oscillations in the direction normal to the motion. The
latter decay at a rate of $1/ \sqrt t$, mimicking the oscillations of some
systems with almost periodic coefficients and cubic nonlinearities. We give
sufficient conditions for that class of systems to have an asymptotically
stable origin.
