Emergence of bi-cluster flocking for the Cucker–Smale model
Emergence of bi-cluster flocking for the Cucker–Smale model
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DOI：
10.1142/s0218202516500287
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发表时间：
2016-04
影响因子：
3.5
通讯作者：
Jung-Kyun Cho;Seung‐Yeal Ha;F. Huang;Chunyin Jin;Dongnam Ko
中科院分区：
文献类型：
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作者：
Jung-Kyun Cho;Seung‐Yeal Ha;F. Huang;Chunyin Jin;Dongnam Ko
我们利用一些准备充分的初始数据，对短程 Cucker-Smale 模型的双簇聚集现象进行了数学分析。为此，我们推导了测量局部空间和速度波动以及局部速度平均值差异的泛函的微分不等式系统，然后估计空间波动的上限和局部速度平均值差异的下限。我们明确提出了一类可接受的初始配置，导致双簇聚集现象的渐进出现。与收敛速率始终呈指数关系的全局集群（速度上的单集群集群配置）不同，双集群集群配置的渐近收敛受到通信权重的远场衰减率的影响，因此它可以是代数的。
We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.