Abstract
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 behaviour in a particle model
with local communication weights. As 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 the subspace analysis. In results,
some criterion of 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.