Emergence of Multi-Cluster Configurations From Attractive and Repulsive Interactions

Abstract

We discuss a first-order Cucker-Smale-type consensus model with attractive and repulsive interactions and present upper and lower bound estimates on the number of asymptotic point-clusters depending on the relative ranges of interactions and coupling strength. When the number of agents approaches infinity, we introduce a scalar conservation law with a non-local flux for a macroscopic description. We show that the corresponding conservation law admits a classical solution for sufficiently smooth initial data, which illustrates the shock avoidance effect due to the non-locality of the interactions. We also study the dynamics of special Dirac-Comb-type solutions consisting of two and three point-clusters.There have been many previous literature on the opinion formation. 2,7,14,15,21,22,26,32,36,40,42,44 However most of these models deal with symmetric interactions or only attractive interactions. For the symmetric and all-to-all interactions c(j, i) = c(i, j), the first moment of ζ i is conserved along the dynamics of (1.1), hence the standard global analysis tools, e.g. energy or Lyapunov estimates in Refs. 18, 19 and 29, can be employed to establish the asymptotic emergence of synchronized opinion groups, whereas for the non-symmetric and repulsive interactions, the first moment of ζ i is not conserved, hence the aforementioned standard tools are not applicable. We instead estimate the difference L i := ζ i+1 − ζ i directly using the system (1.1) and study the asymptotic behavior of L i .The novelty of this paper is threefold. First, we consider non-symmetric communication weights which simultaneously include the positive coupling (assimilation) and negative coupling (repulsion or distinction). Most available literature [14][15][16]18,19,27,[29][30][31]37 on the consensus problem deals with symmetric coupling such as attractive mean-field coupling, finite-range interaction coupling, and the nearest neighbor coupling such as toda lattice, etc. However when the communication weights are not symmetric, the first moment of information under consideration is no longer conserved, which greatly complicates the rigorous analysis. As far as we know, there is no rigorous mathematical results for non-symmetric coupling. We address this non-symmetric difficulty via detailed estimate of the evolution of immediate distance between neighbors.Secondly, as was mentioned at the opening of this paper, global diversity due to local synchronization recently attracted attention in various disciplines, and we aim to provide a succinct quantification of its mechanisms. We first formalize the non-symmetric communication weight (1.3) using the range and relative strengths between attractive and repulsive interactions and then provide quantitative estimates on the number of asymptotic synchronized clusters (see Definition 4.1). From a mathematical viewpoint, there has been no systematic approach to analyze flocking models with attractive and repulsive interactions. Most existing analytical results 18,19,29,31 deal with mean-field coupling. ...

Sharp conditions to avoid collisions in singular Cucker–Smale interactions

“…[8,22]), pattern formation (see e.g. [21,33]), models with additional deterministic (see e.g. [9,20]) or stochastic (see e.g.…”

The Cucker–Smale Equation: Singular Communication Weight, Measure-Valued Solutions and Weak-Atomic Uniqueness

“…[21]) and pattern formation (see e.g. [20,34]) or analysis of the models with additional forces that simulate various natural factors (see e.g. [10,17] -deterministic forces or [13] -stochastic forces).…”

Regular solutions to the fractional Euler alignment system in the Besov spaces framework

“…Proposition 4.4. For all n = 1, 2, ..., given u n−1 and σ n−1 in C(R + ; W ∞ d ), there exists a unique solution u n ∈ C 1 (R + ; W ∞ d ) to (16) such that for all T ≥ 0,…”

“…Step 4: Higher-order estimates. In order to obtain higher-order estimates we simply differentiate equations (16) and (17) with respect to the space variable. Denoting by ∂ k the derivative with respect to the variable x k , we get for k = 1, · · · , d,…”

“…The CS model is a simple example of interacting particles' models that has been extensively studied in various, mostly qualitative, directions such as collision avoidance [1,6,4] and asymptotic and pattern formation [5,16]. These two directions lead to further branching of research into the CS model with singular communication weight [31,32], or various additional forces that ensure specific asymptotic pattern formation [15,17] or under a leadership of selected individuals [7,35].…”