Emergence of aggregation in the swarm sphere model with adaptive coupling laws

Seung-Yeal Ha; Dohyun Kim; Jaeseung Lee; Se Eun Noh

doi:10.3934/krm.2019018

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2019, Volume 12, Issue 2: 411-444. Doi: 10.3934/krm.2019018

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Emergence of aggregation in the swarm sphere model with adaptive coupling laws

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2.
Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea
3.
Seoul National University, Dept. of Math. Sciences, Seoul 08826, Republic of Korea
4.
Myongji University, Dept. of Mathematics, Yong-In 17058, Republic of Korea

* Corresponding author: Se Eun Noh
* Corresponding author: Se Eun Noh

Received: May 31, 2018

Published: March 2019

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Abstract

We present aggregation estimates for the swarm sphere model equipped with the adaptive coupling laws on a sphere. The temporal evolution of coupling strength is determined by a feedback rule incorporating the balance between relative spatial variations and linear damping. For the analytical treatment, we employ two adaptive feedback laws, namely anti-Hebbian and Hebbian laws. For the anti-Hebbian law, we provide a sufficient framework leading to the complete aggregation in which all particles aggregate to the same position and behave like one big point cluster asymptotically. Our frameworks are given in terms of the initial positions and the coupling strengths. For the Hebbian law, we provide proper subsets of the basin of attractions for the complete aggregation and bi-polar aggregation where particles aggregate to the north pole and south pole simultaneously. We also present a uniform $\ell_p$-stability of the swarm sphere model with an adaptive coupling with respect to the initial data when the complete aggregation occurs exponentially fast.

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Mathematics Subject Classification: Primary: 82C10, 82C22, 35B37.

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Figure 1. Vector field of (60) for $\mu = \gamma = 1$

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