arXiv subjects

Hyunjin Ahn

Publications and source records attributed to Hyunjin Ahn.

7 recordsLinked to original sources

arXiv2026

Logarithmic Velocity Alignment on Riemannian Manifolds

We introduce a logarithmic velocity alignment model for collective motion on Riemannian manifolds. The interaction law is defined by the covariant time derivative of logarithmic displacement vectors and therefore provides an intrinsic geometric analogue of pairwise velocity-difference coupling. Under two-sided nonpositive sectional-curvature bounds and a compatibility condition between the interaction kernel and the curvature-induced growth of logarithmic terms, we derive a kinetic-energy dissipation estimate and prove interaction-weighted asymptotic velocity alignment, assuming that the logarithmic interactions remain globally well-defined and that transported velocity discrepancies have uniformly controlled time variation. In hyperbolic space, both assumptions are verified directly from the geometry and the energy estimates, yielding an unconditional interaction-weighted alignment result within the stated class of global solutions.

math.DS↗

arXiv2026

Geometric Control of Moving Parallel Transport in Riemannian Cucker--Smale Dynamics with Bonding Forces

We study a Cucker--Smale type system with bonding forces on complete Riemannian manifolds with uniformly bounded curvature. On general manifolds, the time variation of parallel transport between moving agents produces curvature-dependent terms, so the standard energy-dissipation argument does not directly yield asymptotic velocity alignment. The bonding energy confines all pairwise distances below the injectivity radius, providing global well-posedness and time integrability of the transported velocity discrepancies. To overcome the remaining geometric obstruction, we combine the variation formula for parallel transport with a uniform endpoint estimate for Jacobi fields along moving minimizing geodesics. This yields the uniform regularity needed to convert energy dissipation into asymptotic alignment. Under an energy-dependent injectivity condition and a positive communication bound on the dynamically relevant distance range, we establish asymptotic flocking. Numerical simulations illustrate the resulting dynamics in a nonconstant-sectional-curvature setting.

math.DS↗

arXiv2023

Thermodynamic Cucker-Smale ensemble with unit speed and its sufficient framework for collision avoidance

We investigate a Cucker-Smale-type flocking model for multi-agent systems that move with constant speed. The model incorporates both kinematic observables and internal energy (temperatures) in the agents' interactions. Traditionally, collision avoidance in the absence of speed limitation is achieved by introducing singularities into the communication rule. However, when a unit speed constraint is applied, the mechanism of collision avoidance can differ, and the singularity may not necessarily prevent collisions. In this paper, we propose a framework that generates collision avoidance, asymptotic flocking, thermal equilibrium, and strict spacing between agents, subject to sufficient conditions expressed by the initial condition, system parameters, and degree of singularity.

math.CA↗

arXiv2023

Interplay of geometric constraint and bonding force in the emergent behaviors of relativistic Cucker-Smale flocks

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 extensively studied from various points of view. Recently, the authors studied collision avoidance and asymptotic flocking of the Cucker-Smale model with a bonding force on the Euclidean space. In this paper, we provide an analytical framework for collision avoidance and asymptotic flocking of the proposed model on Riemannian manifolds. Our analytical framework is explicitly formulated in terms of system parameters, initial data and the injectivity radius of the ambient manifold, and we study how the geometric information of an ambient manifold can affect the flocking dynamics.

math.DS↗

arXiv2022

Collective behaviors of second-order nonlinear consensus models with a bonding force

We study the collective behaviors of two second-order nonlinear consensus models with a bonding force, namely the Kuramoto model and the Cucker-Smale model with inter-particle bonding force. The proposed models contain feedback control terms which induce collision avoidance and emergent consensus dynamics in a suitable framework. Through the cooperative interplays between feedback controls, initial state configuration tends to an ordered configuration asymptotically under suitable frameworks which are formulated in terms of system parameters and initial configurations. For a two-particle system on the real line, we show that the relative state tends to the preassigned value asymptotically, and we also provide several numerical examples to analyze the possible nonlinear dynamics of the proposed models, and compare them with analytical results.

math.DS↗

arXiv2021

On a generalized Kuramoto model with relativistic effects and emergent dynamics

We propose a generalized Kuramoto model with relativistic effects and investigate emergent asymptotic behaviors. The proposed generalized Kuramoto model incorporates relativistic Kuramoto(RK) type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere under suitable approximations. We present several sufficient frameworks leading to complete synchronization in terms of initial data and system parameters. For the relativistic Kuramoto model, we show that it can be reduced to the Kuramoto model in any finite time interval in a non-relativistic limit. We also provide several numerical examples for two approximations of the relativistic Kuramoto model, and compare them with analytical results.

math.DS↗

arXiv2020

Emergent behaviors of Cucker-Smale flocks on the hyperboloid

We study emergent behaviors of Cucker-Smale(CS) flocks on the hyperboloid $\mathbb{H}^d$ in any dimensions. In a recent work \cite{H-H-K-K-M}, a first-order aggregation model on the hyperboloid was proposed and its emergent dynamics was analyzed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of Cucker-Smale flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds by explicitly calculating the geodesic and parallel transport. Velocity alignment has been shown by combining general {velocity alignment estimates} for the abstract CS model on manifolds and verifications of a priori estimate of second derivative of energy functional. For the two-dimensional case $\mathbb{H}^2$, similar to the recent result in \cite{A-H-S}, asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on $\mathbb{H}^2$.

math-ph↗