

Irina Popovici (United States Naval Academy)
Title:
Stability of Flocking States for Selfpropelled Swarms with Quadratic Potential
Abstract:
The main result of this paper is proving the stability of translating states (flocking states) for the system of ncoupled selfpropelled agents governed by r_{k}''=(1r_{k}'^{2}) r_{k}'(r_{k}R), where R is the position of the center of mass, and the position vectors r_{k} are in the plane. A translating state is a solution where all agents move with identical velocity, at unit speed. Numerical explorations have shown that for a large set of initial conditions, after some drift, the particles' velocities synchronize directions and magnitudes, with the distance between agents going to zero. We prove that every solution that starts near a translating state asymptotically approaches some translating state nearby, an asymptotic behavior exclusive to swarms in the plane. We quantify the rate of convergence as being as slow as 1/
√ t
, and give rates of convergence for the directional drift and the mean field speed. The slow rate of convergence matches the rate of milling states with even number of agents. We overcome the technical difficulties associated with having nonisolated equilibrium states and a 2n+1 dimensional central manifold by employing statedepended projections that decouple the slowest three variables from the other 2n2 of the central manifold. We show that in average the magnitudes of the oscillations in the direction normal to the motion decrease.


