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 .

中文翻译：

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梯度流在不同势能单位球面上的渐近行为

摘要 我们考虑一个多智能体系统，其动力学由单位球体上的梯度流控制，该梯度流与通过加权距离测量的所有智能体位置之间的相互作用势相关联。xi − xj | p + 2 对于任何 p ≠ 0 。在本文中，我们同时使用吸引和排斥耦合来研究伴随 p > 0（正范围）和 p 0（负范围）的系统的渐近行为，这能够产生更丰富的动力学现象。首先，在一个有吸引力的制度中，我们专注于完全聚合的出现；然而，正范围内朝向聚合状态的松弛动力学与负范围内的松弛动力学不同。更准确地说，当 p > 0 时，完全聚合以代数速率 O (t − 1 / p) 发生。另一方面，对于 p 0 ，全局存在的问题是由于奇异相互作用而产生的，并且与聚合估计密切相关。为此，我们展示了完整的聚合在有限时间内出现，因此直到这样的时间才存在解决方案。最后，在排斥机制中，我们主要考虑正负范围的展开状态，并提供了几个案例研究以获得定性洞察力。最后，我们将我们的结果与 p = 0 的情况进行比较。并提出了几个案例研究以获得定性的见解。最后，我们将我们的结果与 p = 0 的情况进行比较。并提供了几个案例研究以获得定性的见解。最后，我们将我们的结果与 p = 0 的情况进行比较。