Smale Model Research Articles

Global existence of strong solutions to the kinetic Cucker–Smale model coupled with the Stokes equations

In this paper, we investigate existence of global-in-time strong solutions to the kinetic Cucker–Smale model coupled with the Stokes equations in the whole space. By introducing a weighted Sobolev space and using space–time estimates for the linear non-stationary Stokes equations, we present a complete analysis on the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on initial data.

Phase Transition and Asymptotic Behavior of Flocking Cucker--Smale Model

In this paper, we study a continuously flocking Cucker--Smale model with noise, which has isotropic and polarized stationary solutions depending on the intensity of the noise. The first result esta...

A Simple Proof of Asymptotic Consensus in the Hegselmann--Krause and Cucker--Smale Models with Normalization and Delay

A Simple Proof of Asymptotic Consensus in the Hegselmann--Krause and Cucker--Smale Models with Normalization and Delay

The delayed Cucker-Smale model with short range communication weights

<p style='text-indent:20px;'>Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cucker<inline-formula><tex-math id="M2">\begin{document}$ - $\end{document}</tex-math></inline-formula>Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.</p>

Emergence of stochastic flocking for the discrete Cucker–Smale model with randomly switching topologies

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 [J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim: On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies. arXiv:1911.07390.] on the continuous C-S model.

Leader–Follower Flocking for Discrete-Time Cucker–Smale Models With Lossy Links and General Weight Functions

In this article, we investigate leader–follower flocking for the Cucker–Smale model with lossy links and general weight functions. Here, the loss phenomenon of the control packages from the controller to the actuator is characterized by a Bernoulli stochastic variable and the edge weights of the interaction network are determined by a general update rule based on the distance between agents. A method based on the products of substochastic matrices is developed to analyze the flocking behavior with lossy links. By means of this method, a sufficient condition which depends on the agents’ initial states, the topology structure, the successful information transmission rate and the weight function is established. Compared with the previous Cucker–Smale flocking results, which are only applicable to positive and decreasing weight functions with specific forms, our result is more general and can be applied to arbitrary positive and decreasing weight functions with nonzero lower bounds. Finally, our result is illustrated through a simulation example.

Convergence of some Mean Field Games systems to aggregation and flocking models

For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modelling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the evolution of the mass density. The first model is a 2nd order MFG system with vanishing viscosity, and the limit is an Aggregation Equation. The result has an interpretation for models of collective animal behaviour and of crowd dynamics. The second class of problems are 1st order MFGs of acceleration and the limit is the kinetic equation associated to the Cucker–Smale model. The first problem is analysed by PDE methods, whereas the second is studied by variational methods in the space of probability measures on trajectories.

New conditions to avoid collisions in the discrete Cucker–Smale model with singular interactions

In this paper, we consider the discrete Cucker–Smale model with singular interactions. Firstly, we use the inductive method and discrete energy method to prove there is no collision between any two agents in finite time. Then we show that collision-avoiding flocking would occur with appropriate initial conditions. At last, we give an example to show the collision might occur for some different positions initially.

Asymptotic analysis of a stochastic Cucker–Smale model with multiplicative excitation

In this paper, the asymptotic behavior of a stochastic Cucker–Smale model with multiplicative excitation is investigated. Firstly, it is proven that stochastic flocking occurs for the communication weight originally proposed by Cucker and Smale if the noise strength is sufficiently small. Then, a non-flocking result under the effect of large noise is provided. In addition, the kinematics of the macroscopic variables is addressed which can be regarded as an effective supplement of consensus dynamics.

Density-induced consensus protocol

The paper introduces a model of collective behavior where agents receive information only from sufficiently dense crowds in their immediate vicinity. The system is an asymmetric, density-induced version of the Cucker–Smale model with short-range interactions. We prove the basic mathematical properties of the system and concentrate on the presentation of interesting behaviors of the solutions. The results are illustrated by numerical simulations.

On the critical exponent of the one-dimensional Cucker–Smale model on a general graph

We study a critical exponent of the flocking behavior to the one-dimensional 1D Cucker–Smale (C–S) model with a regular inverse power law communication on a general network with a spanning tree. For this, we propose a new nonlinear functional which can control the velocity diameter and decays exponentially fast as time goes on. As an application of the time-evolution of the nonlinear functional, we show that the C–S model on a line exhibits a unique critical exponent for unconditional flocking on a general network so that this improves an earlier result [S.-Y. Ha and J.-G. Liu, A simple proof of Cucker–Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297–325.] on the all-to-all network. Our result also resolves the critical exponent conjecture posed in Cucker–Dong’s work [On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci. 19 (2009) 1391–1404] for 1D setting. Emergent behavior of the C–S model is independent of the special structure of the underlying network, as long as it contains a spanning tree.

Flocking Behavior of Cucker–Smale Model with Processing Delay

The dynamics of a delay multiparticle swarm, which contains symmetric and asymmetric pairwise influence functions, are analyzed. Two different sufficient conditions to achieve conditional flocking are obtained. One does not have a clear relationship with this delay, and the other proposes a range of processing delays that affect the emergence of a flock. It is also pointed out that if the interparticle communication function has tail dissipation, unconditional flocking can be guaranteed. Compared with the previous results, the range of the communication rate β that allows a flock to emerge has been expanded from 1/4 to 1/2.

Fixed-Time Flocking and Collision Avoidance Problem of a Cucker–Smale Model

Multiagent systems are used in artificial intelligence, control theory, and social sciences. In this article, we studied a Cucker–Smale model with a continuous non-Lipschitz protocol. The methodology presented in the current paper is based on the explicit construction of a Lyapunov functional. By using the fixed-time control technology, we show that the flocking can occur in fixed-time and collision avoiding when a singular communication function with a weighted sum of sign functions of the relative velocities among agents, and we can obtain the estimation of the converging time which is independent of the initial states of agents. Theoretical results are supported by numerical simulations.

Emergent Behaviors of Thermodynamic Kuramoto Ensemble on a Regular Ring Lattice

The temporal evolution of Kuramoto oscillators influenced by the temperature field often appears in biological oscillator ensembles. In this paper, we propose a generalized Kuramoto type lattice model on a regular ring lattice with the equal spacing assuming that each oscillator has an internal energy (temperature). Our lattice model is derived from the thermodynamical Cucker–Smale model for flocking on the 2D free space under the assumption that the ratio between velocity field and temperature field at each lattice point has a uniform magnitude over lattice points. The proposed model satisfies an entropy principle and exhibits emergent dynamics under some sufficient frameworks formulated in terms of initial data and system parameters. Moreover, the phase-field tends to the Kuramoto phase-field asymptotically.

Fixed-Time Flocking Problem of a Cucker–Smale Model

In this paper, we studied a Cucker–Smale model with continuous non-Lipschitz protocol. The methodology presented in the current work is based on the explicit construction of a Lyapunov functional. By using the fixed-time control technology, we show that the flocking can occur in fixed time if the communication rate function is locally Lipschitz continuous and has a lower bound, and we can obtain the estimation of the converging time which is independent of the initial states of agents. Theoretical results are supported by numerical simulations.

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

We study the emergent dynamics of the thermomechanical Cucker–Smale (TCS) model with switching network topologies. The TCS model is a generalized CS model with extra internal dynamical variable called “temperature” in which isothermal case exactly coincides with the CS model for flocking. In previous studies, emergent dynamics of the TCS model has been mostly restricted to some static network topologies such as complete graph, connected graph with positive in and out degrees at each node, and digraphs with spanning trees. In this paper, we consider switching network topologies with a spanning tree in a sequence of time-blocks, and present two sufficient frameworks leading to the asymptotic mono-cluster flocking in terms of initial data and system parameters. In the first framework in which the sizes of time-blocks are uniformly bounded by some positive constant, we show that temperature and velocity diameters tend to zero exponentially fast, and spatial diameter is uniformly bounded. In the second framework, we admit a situation in which the sizes of time-blocks may grow mildly by a logarithmic function. In latter framework, our temperature and velocity diameters tend to zero at least algebraically slow.

Complete characterization of flocking versus nonflocking of Cucker–Smale model with nonlinear velocity couplings

We consider the Cucker–Smale model with a regular communication rate and nonlinear velocity couplings, which can be understood as the parabolic equations for the discrete p-Laplacian (p ≥ 1) with nonlinear weights involving a parameter β( > 0). For this model, we study the initial data and the ranges of p and β to characterize when flocking and nonflocking occur. Specifically, we analyze the nonflocking case, subdividing it into semi-nonflocking (only velocity alignment holds) and full nonflocking (group formation and velocity alignment do not hold). More precisely, we show that if β ∈ (0, 1], p ∈ [1, 3), then flocking occurs for any initial data. If β ∈ (0, 1], p ∈ [3, ∞), then semi-nonflocking occurs for any initial data. If β ∈ (1, ∞), p ∈ [1, 3), then flocking occurs for some initial data. In the case β ∈ (1, ∞) and p ∈ [3, ∞), we observe alternative states. Finally, we have numerically verified the conclusions obtained by analytical calculations.

Asymptotic Flocking Behavior of the General Finite-Dimensional Cucker–Smale Model with Distributed Time Delays

In this paper, we study a Cucker–Smale-type flocking model with distributed time delays, in which individuals interact with each other through general communication weights, and delays are information processing and reactions of individuals. By constructing two dissipative differential inequalities on position and velocity together with a continuity argument, and using the Lyapunov functional approach, we present some sufficient conditions for the existence of asymptotic flocking solutions to two classes of different communication weights. All results are novel and can be illustrated by numerical simulations using some concrete influence functions.

On the Euler-alignment system with weakly singular communication weights

We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of the Cucker–Smale model on animal flocks. For the Euler-alignment system with bounded interactions, a critical threshold phenomenon is proved in Tadmor and Tan (2014 Phil. Trans. R. Soc. A 372 20130401), where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in Do et al (2018 Arch. Ration. Mech. Anal. 228 1–37), for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain

We study local existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker–Smale (in short TCS) model coupled with incompressible Navier–Stokes (NS) equations in the whole space. The coupled system consists of the kinetic TCS equation for particle ensemble and the incompressible NS equations for a fluid via a drag force. For the strong solution, we investigate the blow-up mechanism for the coupled system, and we also study the global existence of a weak solution in the whole space.