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We study the emergent behaviors of the weak solutions to the kinetic Cucker-Smale (in short, KCS) model in a non-compact spatial-velocity support setting. Unlike the compact support situation, non-compact support of a weak solution can…

Analysis of PDEs · Mathematics 2026-04-21 Seung-Yeal Ha , Xinyu Wang

The strong well-posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki (VFPDK) equation with correlated noise is established. This equation can be interpreted as the fluctuating mean-field limit of second-order Newtonian particle systems,…

Probability · Mathematics 2026-05-05 Zimo Hao , Zhengyan Wu , Johannes Zimmer

We investigate a general class of models for swarming/self-collective behaviour in domains with boundaries. The model is expressed as a stochastic system of interacting particles subject to both reflecting boundary condition and common…

Probability · Mathematics 2025-10-21 Razvan C. Fetecau , Hui Huang , Jinniao Qiu

In a system of noisy self-propelled particles with interactions that favor directional alignment, collective motion will appear if the density of particles increases beyond a certain threshold. In this paper, we argue that such a threshold…

Soft Condensed Matter · Physics 2010-03-29 Chiu Fan Lee

Stochastic uncertainties in complex dynamical systems lead to variability of system states, which can in turn degrade the closed-loop performance. This paper presents a stochastic model predictive control approach for a class of nonlinear…

Optimization and Control · Mathematics 2016-11-18 Edward A. Buehler , Joel A. Paulson , Ali Akhavan , Ali Mesbah

This paper focuses on the long-term behavior of solutions to nonlinear stochastic Fokker-Planck equations driven by common noise, where the drift term has a linear dependence on the measure. These equations, which describe the evolution of…

Analysis of PDEs · Mathematics 2025-03-07 Raphael Maillet

We present the relativistic analogue of the Cucker-Smale model with a bonding force on Riemannian manifold, and study its emergent dynamics. The Cucker-Smale model serves a prototype example of mechanical flocking models, and it has been…

Dynamical Systems · Mathematics 2023-01-19 Hyunjin Ahn , Junhyeok Byeon , Seung-Yeal Ha , Jaeyoung Yoon

Consider a system of autonomous interacting agents moving in space, adjusting each own velocity as a weighted mean of the relative velocities of the other agents. In order to test the robustness of the model, we assume that each pair of…

Probability · Mathematics 2014-05-05 Eduardo Canale , Federico Dalmao , Ernesto Mordecki , Max Souza

The Cucker-Smale flocking model belongs to a wide class of kinetic models that describe a collective motion of interacting particles that exhibit some specific tendency e.g. to aggregate, flock or disperse. The paper examines the kinetic…

Analysis of PDEs · Mathematics 2017-07-19 Piotr B. Mucha , Jan Peszek

We present some recent development in modeling concurrent emergence of collective behaviors, namely, the emergence of clustering, flocking and synchronization at the same time. We derive two new models, namely Swarmalator-Vicsek and…

Dynamical Systems · Mathematics 2023-02-15 Trenton Gerew , Ming Zhong

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…

Dynamical Systems · Mathematics 2019-12-30 Jiu-Gang Dong , Seung-Yeal Ha , Jinwook Jung , Doheon Kim

We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis.…

Chaotic Dynamics · Physics 2017-09-13 Igor Franovic , Oleg V. Maslennikov , Iva Bacic , Vladimir I. Nekorkin

In this paper, we investigate a Cucker-Smale flocking model with varying time delay. We establish exponential asymptotic flocking without requiring smallness assumptions on the time delay size and the monotonicity of the influence function.

Optimization and Control · Mathematics 2024-07-08 Elisa Continelli

The usual Langevin approach to describe systems driven by noise fails to describe the long time behavior of systems with multiple attractors. The solution of the associated linear Fokker-Planck equation is always unique, even though it…

Statistical Mechanics · Physics 2017-06-20 M. Morillo , J. M. Casado

We introduce a stochastic agent-based model for the flocking dynamics of self-propelled particles that exhibit velocity-alignment interactions with neighbours within their field of view. The stochasticity in the dynamics of the model arises…

Statistical Mechanics · Physics 2019-07-24 Trilochan Bagarti , Shakti N. Menon

We analyze Cucker-Smale flocking particles with delayed coupling, where different constant delays are considered between particles. By constructing a system of dissipative differential inequalities together with a continuity argument, we…

Dynamical Systems · Mathematics 2017-12-05 Young-Pil Choi , Zhuchun Li

We present a quantitative continuum theory of ``flocking'': the collective coherent motion of large numbers of self-propelled organisms. Our model predicts the existence of an ``ordered phase'' of flocks, in which all members of the flock…

Statistical Mechanics · Physics 2009-10-31 John Toner , Yuhai Tu

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…

Probability · Mathematics 2009-12-24 Federico Dalmao , Ernesto Mordecki

A stochastic treatment yielding to the derivation of a general Fokker-Planck equation is presented to model the slow convergence towards equilibrium of mean-field systems due to finite-N effects. The thermalization process involves notably…

Mathematical Physics · Physics 2011-09-28 W. Ettoumi , M. -C. Firpo

Despite the huge number of neurons composing a brain network, ongoing activity of local cell assemblies composing cortical columns is intrinsically stochastic. Fluctuations in their instantaneous rate of spike firing $\nu(t)$ scale with the…

Neurons and Cognition · Quantitative Biology 2024-04-15 Gianni V. Vinci , Roberto Benzi , Maurizio Mattia