A New Model for Self-organized Dynamics and Its Flocking Behavior

By Sébastien Motsch, Eitan Tadmor. 2011

445 citations recorded by the source corpus.

We introduce a model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The proposed model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, our model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. The use of relative distances destroys the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems, which covers systems with non-symmetric influence matrices. In particular, we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with “leaders”.The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our proposed model, and prove its unconditional flocking for slowly decaying influence functions.

Subfields: Computer Networks and Communications, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/f2cdf1dee1a68f19e70b80f09c2ba91be20e242b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

In Modeling and Simulation

Citations in the corpus

Flocking and asymptotic velocity of the Cucker–Smale model with processing delay (2014, methodology): “1 in [11] for the case with delay, we relied heavily on the use of a Lyapunov functional.”
Emergent dynamics of the Cucker-Smale flocking model and its variants (2016, methodology): “As discussed in previous section, after Cucker-Smale’s seminal works in [31, 32], several variants of the C-S model have been introduced for better modelings including local and nonsymmetric interactions, collision avoidance and formation control, etc in [52, 56].”
Ring patterns and their bifurcations in a nonlocal model of biological swarms (2015, background): “Introduction Mathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10, 6, 2, 26, 25, 14, 7, 13, 27, 19, 33, 32, 23, 11, 17, 37, 38, 34, 3”
Flocking estimates for the Cucker–Smale model with time lag and hierarchical leadership (2017, background): “The extension of the flocking result to the case of non symmetric communication rates has been proposed by Motsch and Tadmor [39].”
A review of the mean field limits for Vlasov equations (2014, unknown): “This is a time continuous variant, derived in [128], of the so-called Krause model, [111], [89] (see also the review [158]).”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “1) retains simple additive interactions, contrary to the more complex structure found for example in [40, 41]; but it still includes a large range of first order models, such as swarming, opinion dynamics or aggregation equations, see for instance [2, 5, 10, 13] or [32].”
Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight (2013, background): “for alignment models similar to C-S (both in continuous and discrete cases) were extensively studied, both from physical and biological point of view [13] -[16], [30], [33] - [35] and from more theoretical point of view [1, 2], [4] - [11], [17] - [25], [29, 31].”
Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0 (2017, methodology): “The model is argued in [12] as more realistic in both—close to and away from equilibrium regimes, but its lack of symmetry is less amenable to the spectral analysis available in the symmetric Cucker–Smale model (1.”
Cucker-Smale model with normalized communication weights and time delay (2016, methodology): “Motsch and Tadmor [23] scaled the communication rates in terms of the relative distance between the agents, so that their model does not involve any explicit dependence on the number of agents.”