Asymptotic behavior of gradient flows on the unit sphere with various potentials

Abstract We consider a multi-agent system whose dynamics is governed by a gradient flow on the unit sphere associated with the interaction potential between positions of all agents measured by a weighted distance | x i − x j | p + 2 for any p ≠ 0 . In this paper, we employ both attractive and repulsive couplings to study the asymptotic behavior of the system accompanied by both p > 0 (positive range) and p 0 (negative range), and this enables to yield richer dynamical phenomena. Firstly in an attractive regime, we focus on the emergence of the complete aggregation; however, the relaxation dynamics towards the aggregated state for the positive range differs from the one for the negative range. More precisely for p > 0 , the complete aggregation occurs with an algebraic rate O ( t − 1 / p ) . On the other hand for p 0 , the issue of global existence arises due to the singular interaction and is crucially related to the aggregation estimate. To this end, we show that the complete aggregation emerges in finite time and thus a solution exists until such a time. Lastly in a repulsive regime, we mainly consider the splay state for both positive and negative ranges, and several case studies are presented to obtain the qualitative insight. Finally, we compare our results with the case of p = 0 .

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