How to understand the dynamical collective performances is of particular significance in both theories and applications. In this paper, we are interested in investigating the combined influences of local interaction and processing delay on the asymptotic behavior in a particle model with local communication weights. With new observations, we show that the desired particle system undergoes both periodic flocking and periodic clustering behaviors when the processing delay crosses a threshold value and the eigenvalue 1 of average matrix is semi‐simple. In this case, the connectedness of the particle system may be absent. Also, the number of clusters is discussed by using subspace analysis. In results, some criteria on flocking and clustering emergence with exponential convergent rate are established by the standard functional differential equations analysis when the processing delay is small. When the processing delay reaches the threshold value, the system undergoes periodic flocking and periodic clustering emergence. It also shows that the processing time lags qualitatively change the emergent performances in a nonlinear way. Finally, we conclude this study with several numerical simulations that intuitively illustrate the validity of the theoretical results and address some discussions for both variable communication weight and distributed processing delay.

中文翻译：

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涉及处理延迟的多粒子模型中的集体周期性运动

如何理解动态集体绩效在理论和应用中都具有特别重要的意义。在本文中，我们有兴趣研究局部相互作用和处理延迟对具有局部通信权重的粒子模型中渐近行为的综合影响。通过新的观察，我们表明，当处理延迟超过阈值并且平均矩阵的特征值1为半简单时，所需的粒子系统会同时经历周期性的植绒和周期性的聚类行为。在这种情况下，可能没有粒子系统的连接性。此外，通过使用子空间分析来讨论聚类的数量。结果，当处理时延较小时，通过标准泛函微分方程分析，建立了具有指数收敛速率的植绒和聚类出现的一些准则。当处理延迟达到阈值时，系统会经历周期性的植绒和周期性的聚类出现。它还表明，处理时间滞后以非线性方式定性地改变了紧急情况的性能。最后，我们用几个数值模拟结束了本研究，这些数值模拟直观地说明了理论结果的有效性，并讨论了可变通信权重和分布式处理延迟的一些讨论。它还表明，处理时间滞后以非线性方式定性地改变了紧急情况的性能。最后，我们用几个数值模拟结束了本研究，这些数值模拟直观地说明了理论结果的有效性，并讨论了可变通信权重和分布式处理延迟的一些讨论。它还表明，处理时间滞后以非线性方式定性地改变了紧急情况的性能。最后，我们用几个数值模拟结束了本研究，这些数值模拟直观地说明了理论结果的有效性，并讨论了可变通信权重和分布式处理延迟的一些讨论。