Abstract
We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.
Highlights
We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees
We studied the emergent dynamics of the continuous and discrete CuckerSmale models on digraph connection topology under the effect of time-varying time delays
For a mono-cluster flocking behavior of the ensemble, we assume that our digraph contains a spanning tree, i.e., there is a root whose information flows into any particle along a finite length path
Summary
The terminology “flocking” represents a collective phenomenon in which the ensemble of flocking particles organize into an ordered motion using only limited information and simple rules. Our main interest in this paper lies on the Newton type particle system proposed by Cucker and Smale [16] in a decade ago In this model, the interaction weight between particles is assumed to be an algebraically decaying function depending on the Euclidean distance between particles. Β ≥ 0, system (1.1) was first proposed in [16, 17] and the threshold phenomena between global and local flocking have been observed Since these flocking estimates have been generalized for a general non-increasing communication weight in [26] which corresponds to the zerotime delayed case (1.1) with τij = 0, χij = 1.
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