Seung-Yeal Ha

Cucker-Smale Flocking With Inter-Particle Bonding Forces

The Cucker-Smale (CS) flocking model is an interacting particle system, in which each particle updates its velocity by adding to it a weighted average of the differences of its velocity with those of other particles. It has been shown that by using the C-S model, the velocities of particles converge to a common value despite the absence of a central command. In this note, we make an extension of the C-S model by introducing additional interaction terms between agents which we refer to as the inter-particle bonding force, in order to incorporate collision avoidance between agents, and at the same time achieve tighter spatial configurations. The proposed inter-particle bonding force makes use of position and velocity information of other agents in order to achieve such separation and cohesion. With the inter-particle bonding forces and the velocity-alignment term of the original C-S model, we show the emergent behavior of asymptotic flocking to spatial equilibrium configurations.

Emergent Behavior of a Cucker-Smale Type Particle Model With Nonlinear Velocity Couplings

In this note, we present a Cucker-Smale type flocking model with nonlinear velocity couplings, and derive sufficient conditions for the formation of flocking in terms of communication weight and initial spatial, velocity standard deviations.

Stochastic flocking dynamics of the Cucker–Smale model with multiplicative white noises

We present a strong asymptotic stochastic flocking estimate for the stochastically perturbed Cucker–Smale model. We characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states. When the strength of multiplicative noises is sufficiently large, we show that the strong stochastic flocking occurs even for negative communication weights.

Flocking and synchronization of particle models

In this note, we present a multi-dimensional flocking model rigorously derived from a vector oscillatory chain model and study the connection between the Cucker-Smale flocking model and the Kuramoto synchronization model appearing in the statistical mechanics of nonlinear oscillators. We provide an alternative direct approach for frequency synchronization to the Kuramoto model as an application of the flocking estimate for the Cucker-Smale model.

Time-asymptotic interaction of flocking particles and an incompressible viscous fluid

We present a new coupled kinetic-fluid model for the interactions between Cucker–Smale (C–S) flocking particles and incompressible fluid on the periodic spatial domain . Our coupled system consists of the kinetic C–S equation and the incompressible Navier–Stokes equations, and these two systems are coupled through the drag force. For the proposed model, we provide a global existence of weak solutions and a priori time-asymptotic exponential flocking estimates for any smooth flow, when the kinematic viscosity of the fluid is sufficiently large. The velocity of individual C–S particles and fluid velocity tend to the averaged time-dependent particle velocities exponentially fast.

A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid

We present a new hydrodynamic model for the interactions between collision-free Cucker–Smale flocking particles and a viscous incompressible fluid. Our proposed model consists of two hydrodynamic models. For the Cucker–Smale flocking particles, we employ the pressureless Euler system with a non-local flocking dissipation, whereas for the fluid, we use the incompressible Navier–Stokes equations. These two hydrodynamic models are coupled through a drag force, which is the main flocking mechanism between the particles and the fluid. The flocking mechanism between particles is regulated by the Cucker–Smale model, which accelerates global flocking between the particles and the fluid. We show that this model admits the global-in-time classical solutions, and exhibits time-asymptotic flocking, provided that the initial data is appropriately small. In the course of our analysis for the proposed system, we first consider the hydrodynamic Cucker–Smale equations (the pressureless Euler system with a non-local flocking dissipation), for which the global existence and the time-asymptotic behavior of the classical solutions are also investigated.

Emergent Dynamics of the Cucker–Smale Flocking Model and Its Variants

In this chapter, we present the Cucker–Smale-type flocking models and discuss their mathematical structures and flocking theorems in terms of coupling strength, interaction topologies, and initial data. In 2007, two mathematicians Felipe Cucker and Steve Smale introduced a second-order particle model which resembles Newton’s equations in N-body system and present how their simple model can exhibit emergent flocking behavior under sufficient conditions expressed only in terms of parameters and initial data. After Cucker–Smale’s seminal works in [31, 32], their model has received lots of attention from applied math and control engineering communities. We discuss the state of the art for the flocking theorems to Cucker–Smale-type flocking models.

A quest toward a mathematical theory of the dynamics of swarms

This paper addresses some preliminary steps toward the modeling and qualitative analysis of swarms viewed as living complex systems. The approach is based on the methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlocal, nonlinearly additive and modeled by theoretical tools of stochastic game theory. Collective learning theory can play an important role in the modeling approach. We present a kinetic equation incorporating the Cucker–Smale flocking force and stochastic game theoretic interactions in collision operators. We also present a sufficient framework leading to the asymptotic velocity alignment and global existence of smooth solutions for the proposed kinetic model with a special kernel. Analytic results on the global existence and flocking dynamics are presented, while the last part of the paper looks ahead to research perspectives.

Emergent behaviour of a generalized Viscek-type flocking model

We present a planar agent-based flocking model with a distance-dependent communication weight. We derive a sufficient condition for the asymptotic flocking in terms of the initial spatial and heading-angle diameters and a communication weight. For this, we employ differential inequalities for the spatial and phase diameters together with the Lyapunov functional approach. When the diameter of the agents initial heading-angles is sufficiently small, we show that the diameter of the heading-angles converges to the average value of the initial heading-angles exponentially fast. As an application of flocking estimates, we also show that the Kuramoto model with a connected communication topology on the regular lattice for identical oscillators exhibits a complete-phase-frequency synchronization, when coupled oscillators are initially distributed on the half circle.

APPLICATION OF FLOCKING MECHANISM TO THE MODELING OF STOCHASTIC VOLATILITY

In this study, we present a new stochastic volatility model incorporating a flocking mechanism between individual volatilities of assets. Collective phenomena of asset pricing and volatilities in financial markets are often observed; these phenomena are more apparent when the market is in critical situations (market crashes). In the classical Heston model, the constant theoretical mean of the square of the volatility was employed, which can be assumed a priori. Our proposed model does not assume this mean value a priori, we instead use the flocking effect to continuously update the theoretical mean value using the local weighted average of individual volatility values. To perform this function, we use the Cucker–Smale flocking mechanism to calculate the local mean. For some classes of interaction weights such as all-to-all and symmetric coupling with a positive lower bound, we show that the fluctuations of the square process of volatility are uniformly bounded, such that the overall dynamics are mainly dictated by the averaged process. We also provide several numerical examples showing the dynamics of volatility.