Cucker-Smale Model Research Articles

Stochastic flocking dynamics of the Cucker–Smale model driven by Lévy noise

Lévy noise is a broader type of white noise, which describes discontinuous noise and is more closely to simulate the realistic environment. In this paper, we introduce Lévy noise into the Cucker–Smale model to investigate its effect on the flocking dynamics of the system. The well-posedness of the system is guaranteed by defining an appropriate stopping time and constructing a Lyapunov function. Through the strong law of large numbers for martingales, a sufficient framework of the strong stochastic flocking is obtained without the strict assumption that the communication rate function has a positive lower bound due to the introduction of Lévy noise. We find that Lévy noise can accelerate flocking and become good noise in some cases. Finally, several numerical simulations are presented to validate our results and further observe effects of Lévy noise.

From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019. In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by [Formula: see text], is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker([Formula: see text]) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator [Formula: see text] is not symmetric, i.e. [Formula: see text], where [Formula: see text] is the dual of [Formula: see text]. Moreover, the collision invariants lies in Ker([Formula: see text]), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear. In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

On the comparison between phenomenological and kinetic theories of gas mixtures with applications to flocking

We study the comparison between the phenomenological and kinetic models for a mixture of gases from the viewpoint of collective dynamics. In the case in which constituents are Eulerian gases, balance equations for mass, momentum, and energy are the same in the main differential part, but production terms due to the interchanges between constituents are different. They coincide only when the thermal and mechanical diffusion are sufficiently small. In this paper, we first verify that both models satisfy the universal requirements of conservation laws of total mass, momentum, and energy, Galilean invariance and entropy principle. Following the work of Ha and Ruggeri (ARMA 2017), we consider spatially homogeneous models which correspond to the generalizations of the Cucker Smale model with thermal effect. In these circumstances, we provide analytical results for the comparison between two resulting models and also present several numerical simulations to complement analytical results.

A strong form of propagation of chaos for Cucker–Smale model

A strong form of propagation of chaos for Cucker–Smale model

A fuzzy q-closest alignment model

The paper examines the issue of well-posedness of the Cucker-Smale model with communication restricted to the q-closest neighbors, known also as the Cucker-Dong model. With agents oscillating on the boundary of different clusters, the system becomes difficult to precisely define, which leads to further problems with kinetic limits as the number of agents tends to infinity. We introduce the fuzzy q-closest system, which circumvents the issues with well-posedness. For such a system we prove a stability estimate for measure-valued solutions and perform the kinetic mean-field limit.

On weak solutions to the kinetic Cucker–Smale model with singular communication weights

We establish the local-in-time existence of weak solutions to the kinetic Cucker–Smale model with singular communication weights ϕ ( x ) = | x | − α \phi (x) = |x|^{-\alpha } with α ∈ ( 0 , d ) \alpha \in (0,d) . In the case α ∈ ( 0 , d − 1 ] \alpha \in (0, d-1] , we also provide the uniqueness of weak solutions extending the work of Carrillo et al [MMCS, ESAIM Proc. Surveys, vol. 47, EDP Sci., Les Ulis, 2014, pp. 17–35] where the existence and uniqueness of weak solutions are studied for α ∈ ( 0 , d − 1 ) \alpha \in (0,d-1) .

Flocking dynamics of agents moving with a constant speed and a randomly switching topology

Abstract In this paper, we present a sufficient framework to exhibit the sample path-wise asymptotic flocking dynamics of the Cucker–Smale model with unit-speed constraint and the randomly switching network topology. We employ a matrix formulation of the given equation, which allows us to evaluate the diameter of velocities with respect to the adjacency matrix of the network. Unlike the previous result on the randomly switching Cucker–Smale model, the unit-speed constraint disallows the system to be considered as a nonautonomous linear ordinary differential equation on velocity vector, which forces us to get a weaker form of the flocking estimate than the result for the original Cucker–Smale model.

Research on Group Behavior Modeling and Individual Interaction Modes with Informed Leaders

This study investigates coordinated behaviors and the underlying collective intelligence in biological groups, particularly those led by informed leaders. By establishing new convergence condition based on experiments involving real biological groups, this research introduces the concept of a volitional term and heterogeneous networks, constructing a coupled-force Cucker–Smale model with informed leaders. Incorporating informed leaders into the leader-follower group model enables a more accurate representation of biological group behaviors. The paper then extracts the Flock Leadership Hierarchy Network (FLH), a model reflecting real biological interactions. Employing time slicing and rolling time windows, the study methodically analyzes group behavior stages, using volatility and convergence time as metrics to examine the relationship between group consistency and interactions. Comparative experiments show the FLH network’s superior performance. The Kolmogorov-Smirnov test demonstrates that the FLH network conforms to a power-law distribution, a prevalent law in nature. This result further illuminates the crucial role that power-law distribution plays in the evolutionary processes of biological communities. This study offers new perspectives on the evolution of biological groups, contributing to our understanding of the behaviors of both natural and artificial systems, such as animal migration and autonomous drone operations.

Flocking Behavior of the Cucker–Smale Model on Infinite Graphs with a Central Vertex Group

Flocking Behavior of the Cucker–Smale Model on Infinite Graphs with a Central Vertex Group

Leader-Following Rendezvous Control for Generalized Cucker-Smale Model on Riemannian Manifolds

Leader-Following Rendezvous Control for Generalized Cucker-Smale Model on Riemannian Manifolds

Flocking of a thermodynamic Cucker-Smale model with local velocity interactions

Flocking of a thermodynamic Cucker-Smale model with local velocity interactions

Broadcasting-based Cucker–Smale flocking control for multi-agent systems

This paper investigates the flocking control issue of multi-agent systems, where each agent utilizes the broadcasting communication manner and takes the Cucker–Smale style dynamic model. In the setting of broadcasting scenario, each agent broadcasts the state information only at certain moments determined by its own clock, and recalculates the control input only when its clock ticks or it receives information from the neighbors. In the Cucker–Smale model, a nonlinear weight function about the communication distance is set to portray the mutual influence degree between agents. With the help of the properties of sub-stochastic matrix, a comprehensive analysis of the asymptotic convergence of the model is performed and the algebraic conditions for achieving flocking control are established. In addition, the obtained theoretical results are further extended to the bipartite flocking control issue for multi-agent systems with both cooperative and competitive interactions. At last, numerical simulations are provided to verify the theoretical results.

Multi-cluster flocking of the thermodynamic Cucker-Smale model with a unit-speed constraint under a singular kernel

<abstract><p>This paper presents several sufficient frameworks for multi-cluster flocking of the thermodynamic Cucker-Smale model with a unit-speed constraint (in short, TCSUS) under a singular kernel. By providing precise estimates and deriving the dissipative structure of TCSUS, it was proved that under specific well-prepared conditions for particle positions and fully separated initial velocities, multi-cluster flocking occurs in the TCSUS system under a strong singular kernel. Furthermore, the velocities and temperatures converge to the average final data for each cluster group.</p></abstract>

Inevitable monokineticity of strongly singular alignment

We prove that certain types of measure-valued mappings are monokinetic i.e. the distribution of velocity is concentrated in a Dirac mass. These include weak measure-valued solutions to the strongly singular Cucker–Smale model with singularity of order greater or equal to the dimension of the ambient space. Consequently, we are able to answer a couple of open questions related to the singular Cucker–Smale model. First, we prove that weak measure-valued solutions to the strongly singular Cucker–Smale kinetic equation are monokinetic, under very mild assumptions that they are uniformly compactly supported and weakly continuous in time. This can be interpreted as a rigorous derivation of the macroscopic fractional Euler-alignment system from the kinetic Cucker–Smale equation without the need to perform any hydrodynamical limit. This suggests the superior suitability of the macroscopic framework to describe large-crowd limits of strongly singular Cucker–Smale dynamics. Second, we perform a direct micro- to macroscopic mean-field limit from the Cucker–Smale particle system to the fractional Euler-alignment model. This leads to the final result—the existence of weak solutions to the fractional Euler-alignment system with almost arbitrary initial data in 1D, including the possibility of a vacuum. Existence can be extended to 2D under the a priori assumption that the density of the mean-field limit has no atoms.

Interplay of unit-speed constraint and singular communication in the thermodynamic Cucker-Smale model.

We study collision avoidance resulting from unit-speed constraint and singular communication weight in a thermo-mechanical Cucker-Smale (TCS) flocking ensemble moving in a Euclidean space. The TCS model governs the flocking dynamics of the CS particles in the presence of a self-consistent temperature field. The issue of collision avoidance has been addressed in the realistic modeling of collective particle models, and it has been investigated in the CS and TCS models without unit-speed constraint. In this paper, we take one further step by examining how unit-speed constraint and singular communication weights can result in collision avoidance between particles. For this, we introduce a sufficient framework to ensure collision avoidance in terms of system parameters and initial data.

Asymptotic Flocking for the Cucker-Smale Model with Time Variable Time Delays

Asymptotic Flocking for the Cucker-Smale Model with Time Variable Time Delays

Distributed swarm system with hybrid-flocking control for small fixed-wing UAVs: Algorithms and flight experiments

This paper presents a distributed swarm system for small fixed-wing unmanned aerial vehicles (UAVs). In particular, to perform various missions with multiple UAVs that are densely gathered and collision free, a hybrid-flocking control algorithm is synthesized by using three types of control protocols: vector field guidance (for path following/loitering), augmented Cucker-Smale (ACS) model (for collective flocking behavior), and potential field (for collision avoidance). In particular, to address the issue of conflicts between different control protocols, the adaptive ACS model is proposed and the optimization problem is formulated to determine the suitable mixing weights of control protocols. We also design the transition of multiple operation modes and communication architecture for the swarm system. The system is evaluated using the proposed hybrid-flocking control algorithm by proof-of-concept real flight experiments using 18 small fixed-wing UAVs as well as extensive numerical simulations. Flight experiments are successfully performed for multiple consecutive tasks including the individual task, circular path loitering and elliptical path loitering while avoiding collisions among UAVs.

Asymptotic analysis of the linear formation model with an undirected connected topology

This paper introduces linear formation into an undirected connected Cucker–Smale model. Firstly, the projection system corresponding to the original system is established, and the mutual control relationship between the displacement difference between the original system and the projection system is given. Secondly, the boundedness of the projection system’s displacement is derived by using graph theory and matrix theory, and the convergence of the system speed is given by using Lyapunov stability theory. The research results indicate that under certain conditions, for any given direction, the multi-agent system can asymptotically converge to a flock and form a straight line in the presented direction. Moreover, the velocity of the agent converges to the average of the initial velocity. At last, the validity of the results is verified by numerical simulations.

A collisionless singular discrete Cucker-Smale model with deterministic perturbations

In this paper, we investigate two non-linearly perturbed extensions of the discrete Cucker-Smale model with singular coupling weights. The first perturbation is that all agents have non-identical free-will accelerations, and the second is that all agents have identical intrinsic dynamics with the Lipschitz property. For the first model, we apply the induction method and discrete energy method to show that agents avoid collisions for any time and flocking occurs under some initial conditions, if the diameter of agents’ free-will accelerations is summable. For the second model, we obtain collision-avoiding flocking occurrence under suitable initial data and the Lipschitz constant of the function for the intrinsic dynamics. We also provide several numerical examples to illustrate our main results.

Pattern formation in the Cucker-Smale model

In this paper, we study the pattern formation phenomenon in the Cucker-Smale model with short-range communication weight under the interaction with introduced driving force. We show that the multi-cluster flocking and multi-cluster alignment emerge, and each flock achieves a line-shaped motion pattern for a given class of admissible initial configurations if the coupling strength and driving force strengths satisfy certain conditions. We also show the emergence of the global flocking, the global alignment, and the line-shaped motion pattern in the Cucker-Smale model with short-range communication weight if the coupling strength has a lower bound.