On the stability of Rotating States in Second-Order Self-Propelled Multi-Particle Systems

In this paper, we study the dynamics of a system of $n$ coupled, self-propelled particles: $\ddot r_k = (\alpha-\beta |\dot r_k|^2)\dot r_k - \frac{\gamma}{n}\sum_{m=1}^n(r_k-r_m)$, $r_k\in \mathbb R^2.$ Numerical experiments indicate that, for a large set of initial conditions, after an initial drift, the center of mass converges to a stationary point, with each particle eventually rotating around it with constant angular velocity. The distribution of particles on the circle need not be uniform. These limit configurations, where all particles rotate in the same direction, are termed {\it rotating states} . We prove that rotating states are stable and that every solution that starts sufficiently close, asymptotically approaches a rotating state, exponentially fast if $n$ is odd, or at a rate that may be exponential or $\frac{1}{\sqrt t}$ if $n$ is even. The proof uses a new approximation technique for the flow on the center manifold in the presence of non-isolated fixed points.