Asymptotic collective behaviors of a delayed multi‐particle system with multiple processes

Abstract

It is a significant issue to explore the collective behaviors of the delayed complex systems. This paper studies asymptotic collective behaviors of a delayed multi‐particle system with multiple processes while particles interact directionally and locally. Up to now, the studies on dynamic behaviors of multi‐particle systems do not involve limited processing capacity, that is, the information queuing phenomenon is ignored. To this aim, we introduce multiple processes to formulate the evolution of a delayed multi‐particle system. To optimize the delay parameters, we choose the greedy principle to set the processing order. The main idea is to decompose the velocity into each process and use functional differential equation theory and matrix theory to constrain the velocity variation rate while the processing order is fixed. As the new results, in the non‐critical and general situation, when the time delay is less than or equal to the threshold value and the eigenvalue 1 of the average matrix is semi‐simple, we achieve the criterion of flocking and multi‐cluster with exponential convergence rate involving multiple processes. Furthermore, we get the lower bound estimate of the processing capacity.