关键词: semi-simple eigenvalue, flocking emergence, clustering emergence, distributed delay, exponential convergence

Abstract: For a collective system, the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature, such as flocking or multi-clustering. In this paper, we study a nonsymmetric multi-particle system with a constant and local cut-off weight. A distributed communication delay is also introduced into both the velocity adjoint term and the cut-off weight. As a new observation, we show that the desired multi-particle system undergoes both flocking and clustering behaviors when the eigenvalue 1 of the adjacency matrix is semi-simple. In this case, the adjacency matrix may lose the connectedness. In particular, the number of clusters is discussed by using subspace analysis. In terms of results, for both the non-critical and general neighbourhood situations, some criteria of flocking and clustering emergence with an exponential convergent rate are established by the standard matrix analysis for when the delay is free. As a distributed delay is involved, the corresponding criteria are also found, and these small time lags do not change the emergent properties qualitatively, but alter the final value in a nonlinear way. Consequently, some previous works [14] are extended.

Key words: semi-simple eigenvalue, flocking emergence, clustering emergence, distributed delay, exponential convergence