2021/05/24 by Carl Kolon, Kolon, Carl, Kostya Medynets +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #34D06 #34D35 #37B25 #37C75 #Diffusion and Search Dynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Micro and Nano Robotics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2105.11419
openalex publication_date 2021/05/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper, we study the dynamics of a system of n coupled, self-propelled particles: rk = (α-β| rk|2) rk - \fracγn∑m=1n(rk-rm), rk∈ \mathbb R2. 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 (1)/(√ 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.