Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit

Ha, Seung‐Yeal; Kim, Jeong-ho; Zhang, Xiongtao

doi:10.3934/krm.2018045

Cited by 64 publications

(57 citation statements)

References 32 publications

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“…We now apply the exponential stability estimate obtained in Proposition 6.1 to prove Theorem 3.6 the uniform-in-time mean-field limit. As we mentioned, we use the strategy proposed in [26]. We also refer to [6] for the stochastic case.…”

Section: 3mentioning

confidence: 99%

Emergent dynamics of the Kuramoto ensemble under the effect of inertia

Choi¹,

Morales

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.Contents 1 (Seung-Yeal Ha)

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Section: 3mentioning

confidence: 99%

Emergent dynamics of the Kuramoto ensemble under the effect of inertia

Choi¹,

Morales

2018

Discrete &Amp; Continuous Dynamical Systems - A

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“…Lemma 3.4. [15] Suppose that two nonnegative Lipschitz functions X and V satisfy the coupled differential inequalities:…”

Section: )mentioning

confidence: 99%

Local sensitivity analysis for the Cucker-Smale model with random inputs

Ha¹,

Jin²

2018

Kinetic &Amp; Related Models

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We present pathwise flocking dynamics and local sensitivity analysis for the Cucker-Smale(C-S) model with random communications and initial data. For the deterministic communications, it is well known that the C-S model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents are not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the C-S model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random C-S model with uncertain communication.Date: December 13, 2017. 1991 Mathematics Subject Classification. 15B48, 92D25.

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“…In 2007, Cucker and Smale [14] introduced a simple second-order Netwon-like model in the spirit of Vicsek's model [44] as an analytically manageable model, and they derived sufficient frameworks leading to the global and local asymptotic flocking on the complete graph. After their work, the Cucker-Smale(CS) model has been extensively studied in applied math community from several aspects, e.g., stochastic noises [1,4,13,27], discrete approximation [17,23], random failures [19,20,30,41], time-delayed communications [8,9,21,34], collision avoidance [10], network topologies [11,12,18,32,33,39,42], mean-field limit [26,28], kinetic and hydrodynamic descriptions [5,6,22,29,40], nonsymmetric communication networks [36], bonding force [38], extra internal variables [24,25], etc (see recent survey articles [2,7]).…”

Section: Introductionmentioning

confidence: 99%

Emergence of mono-cluster flocking in the thermomechanical Cucker–Smale model under switching topologies

Dong

Kim

2020

Anal. Appl.

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We study emergent dynamics of the discrete Cucker-Smale (in short, DCS) model with randomly switching network topologies. For this, we provide a sufficient framework leading to the stochastic flocking with probability one. Our sufficient framework is formulated in terms of an admissible set of network topologies realized by digraphs and probability density function for random switching times. As examples for the law of switching times, we use the Poisson process and the geometric process and show that these two processes satisfy the required conditions in a given framework so that we have a stochastic flocking with probability one. As a corollary of our flocking analysis, we improve the earlier result [15] on the continuous C-S model.

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Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit

Cited by 64 publications

References 32 publications

Emergent dynamics of the Kuramoto ensemble under the effect of inertia

Emergent dynamics of the Kuramoto ensemble under the effect of inertia

Local sensitivity analysis for the Cucker-Smale model with random inputs

Emergence of mono-cluster flocking in the thermomechanical Cucker–Smale model under switching topologies

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