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From particle to kinetic and hydrodynamic descriptions of flocking
 Kinetic and Related Methods
"... Abstract. We discuss the CuckerSmale’s (CS) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasovtype kinetic model for the CS particle model and prove it exhibits timeasymptotic floc ..."
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Cited by 65 (5 self)
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Abstract. We discuss the CuckerSmale’s (CS) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasovtype kinetic model for the CS particle model and prove it exhibits timeasymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the CS Vlasovtype kinetic model and prove flocking behavior without closure of higher moments. 1. Introduction. Collective
Asymptotic Flocking Dynamics for the kinetic CuckerSmale model
, 2009
"... Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting ..."
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Cited by 61 (14 self)
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Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmanntype equation. The largetime behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.
A simple proof of the CuckerSmale flocking dynamics and meanfield limit
 Comm. Math. Sci
"... Abstract. We present a simple proof on the formation of flocking to the CuckerSmale system based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to flocking state will occur. For large par ..."
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Cited by 40 (3 self)
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Abstract. We present a simple proof on the formation of flocking to the CuckerSmale system based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to flocking state will occur. For large particle systems, we give a rigorous justification for the meanfield limit from the many particle CuckerSmale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity. Key words. Flocking, swarming, emergence, selfdriven particles system, autonomous agents, Vlasov equation, Lyapunov functional, measure valued solution, KantorovichRubinstein distance. Subject classifications. Primary 92C17; secondary 82C22, 82C40.
A new model for selforganized dynamics and its flocking behavior
, 2011
"... We introduce a model for selforganized dynamics which, we argue, addresses several drawbacks of the celebrated CuckerSmale (CS) 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 dis ..."
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Cited by 27 (4 self)
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We introduce a model for selforganized dynamics which, we argue, addresses several drawbacks of the celebrated CuckerSmale (CS) 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 CS model, which was the key for the various recent studies of CS 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 nonsymmetric 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.
Particle, Kinetic, and Hydrodynamic Models of Swarming
"... We review the stateoftheart in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individualbased models based on “particle”like assumptions, we connect to hydrodynamic/macroscopic descriptions ..."
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Cited by 26 (9 self)
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We review the stateoftheart in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individualbased models based on “particle”like assumptions, we connect to hydrodynamic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the role of the kinetic viewpoint in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system.
Heterophilious dynamics enhances consensus
, 2013
"... We review a general class of models for selforganized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents”, with the tendency to adjust to their ‘environmental averages’. This, in turn, leads to the formation of clusters, e.g., ..."
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Cited by 15 (2 self)
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We review a general class of models for selforganized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents”, with the tendency to adjust to their ‘environmental averages’. This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, etc. A natural question which arises in this context is to understand when and how clusters emerge through the selfalignment of agents, and what type of “rules of engagement ” influence the formation of such clusters. Of particular interest to us are cases in which the selforganized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking or concentration of other positions intrinsic to the dynamics. Many standard models for selforganized dynamics in social, biological and physical science assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, “Similarity breeds connection,” reflects our intuition that increasing the intensity of alignment as the difference of positions decreases, is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily — the tendency to bond more with those who are different rather than with those who are similar, plays a decisive rôle in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.
CuckerSmale Flocking Under Hierarchical Leadership and Random Interactions
"... Consider a flock of birds that fly interacting between them. The interactions are modelled through a hierarchical system in which each bird, at each time step, adjusts its own velocity according to his past velocity and a weighted mean of the relative velocities of its superiors in the hierarchy. We ..."
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Cited by 5 (1 self)
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Consider a flock of birds that fly interacting between them. The interactions are modelled through a hierarchical system in which each bird, at each time step, adjusts its own velocity according to his past velocity and a weighted mean of the relative velocities of its superiors in the hierarchy. We consider the additional fact, that each of the birds can fail to see any of its superiors with certain probability, that can depend on the distances between them. For this model with random interactions we prove that the flocking phenomena, obtained for similar deterministic models, holds true.