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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.
On Krause’s MultiAgent Consensus Model With StateDependent Connectivity
"... Abstract—We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by a real number, and updates its opinion by averaging all agent opinions that differ from its own by less than one. We give a new proof of convergence into clusters of agents, with all agents i ..."
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Cited by 57 (8 self)
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Abstract—We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by a real number, and updates its opinion by averaging all agent opinions that differ from its own by less than one. We give a new proof of convergence into clusters of agents, with all agents in the same cluster holding the same opinion. We then introduce a particular notion of equilibrium stability and provide lower bounds on the intercluster distances at a stable equilibrium. To better understand the behavior of the system when the number of agents is large, we also introduce and study a variant involving a continuum of agents, obtaining partial convergence results and lower bounds on intercluster distances, under some mild assumptions. Index Terms—Consensus, decentralized control, multiagent system, opinion dynamics.
DOUBLE MILLING IN SELFPROPELLED SWARMS FROM KINETIC THEORY
"... (Communicated by Tong Yang) Abstract. We present a kinetic theory for swarming systems of interacting, selfpropelled discrete particles. Starting from the Liouville equation for the manybody problem we derive a kinetic equation for the single particle probability distribution function and the rela ..."
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Cited by 47 (13 self)
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(Communicated by Tong Yang) Abstract. We present a kinetic theory for swarming systems of interacting, selfpropelled discrete particles. Starting from the Liouville equation for the manybody problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other nontrivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.
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.
Nonlocal interactions by repulsiveattractive potentials: radial ins/stability
, 2011
"... Abstract. We investigate nonlocal interaction equations with repulsiveattractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some condit ..."
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Cited by 32 (10 self)
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Abstract. We investigate nonlocal interaction equations with repulsiveattractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a nonradially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsiveattractive power law potential as well as numerical results. 1.
Continuoustime averagepreserving opinion dynamics with opiniondependent communications
, 2010
"... We study a simple continuoustime multiagent system related to Krause’s model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence t ..."
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Cited by 28 (7 self)
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We study a simple continuoustime multiagent system related to Krause’s model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multiagent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a nontrivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents.
CuckerSmale flocking under hierarchical leadership
 SIAM J. Appl. Math
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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.