Stability of Translating States for Self-propelled Swarms with Quadratic Potential

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.