Swarming is a near-universal phenomenon in nature. Many mathematical models of swarms exist, both to model natural processes and to control robotic agents. We study a swarm of agents with spring-like at-traction and nonlinear self-propulsion. Swarms of this type have been studied numerically, but to our knowledge, no proofs of stability yet exist. We are motivated by a desire to understand the system from a mathematical point of view. Previous numerical experiments have shown that the system either converges to a rotating circular limit cycle with a fixed center of mass, or the agents clump together and move along a straight line. We show that this is not always the case, and the behavior is sometimes more nuanced. Our specific goal is to investigate stability of the system's circular rotating state. The system is translation-invariant, and when the center of mass comes to a halt, the agents decouple from each other. We apply methods from the stability theory of dynamical systems, including Lienard's Theorem, Lasalle's Invariance Principle, and Lyapunov's direct and indirect methods, to globally characterize the behavior of these decoupled systems, and to locally characterize the desired behavior of the entire swarm. We confirm our theoretical findings with numerical experiments.
Trident Scholar Report No. 471
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