| Abstract: |
| In this talk, we study aggregation properties of the swarm sphere model equipped with an adaptive coupling law on a sphere. The temporal evolution of coupling strength is determined by the balance between a feedback law due to relative positions and damping. For the analytical treatment, we employ two types of adaptive feedback laws, namely anti- Hebbian and Hebbian laws. For the anti-Hebbian law, we provide a sufficient condition leading to the complete aggregation in which all particles aggregate together and behave like one big cluster asymptotically. Our conditions are given in terms of initial positions and coupling strengths. For the Hebbian law, we provide subsets of basin of attractions to the complete synchronization and bi-polar synchronization in which the particles aggregate toward the north pole and south pole separately. We also present a uniform lp-stability of the swarm sphere model with adaptive couplings with respect to initial data when the complete aggregation occurs exponentially fast. |
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