arXiv subjects

Seung-Yeal Ha

Publications and source records attributed to Seung-Yeal Ha.

At least 37 records · Page 2Linked to original sources

arXiv2025

Inertia perturbation theory for the inertial Kuramoto model

In this work, we study the inertial Kuramoto model, which is a second-order extension of the classical first-order Kuramoto model, as an inertial perturbation of the first-order Kuramoto model. We develop a quantitative Tikhonov theorem, from which we derive a new synchronization statement in the small inertia regime, with strong bounds on the limiting order parameter. We also explore the determinability of phase velocities from phase positions, which shows that the perturbation viewpoint must be limited to the small inertia regime. This paper complements our recent work (2025), where we established asymptotic phase-locking of inertial Kuramoto oscillators under generic initial conditions in the low inertia-high coupling regime.

math.DS↗

arXiv2025

Emergent dynamics of spatially extended relativistic kinetic Cucker-Smale model

We study the emergent dynamics of the relativistic kinetic Cucker-Smale (RKCS) model without assuming compactness in spatial and velocity support. In this setting, the lower bound of the kernel function in the nonlocal velocity alignment force can be zero so that the previous approach based on the energy method does not provide a quantitative flocking estimate. To overcome this difficulty, we introduce a suitable decay ansatz for the one-particle distribution function and an {\it effective domain} by identifying a time-varying region in which the total mass outside of it decays to zero asymptotically. Using these two ingredients, we show that weak flocking dynamics emerges asymptotically in the sense that the second moment for the velocity fluctuation around the velocity average tends to zero asymptotically, whereas the second moment for spatial fluctuations around the center of mass remains bounded uniformly in time. Our results demonstrate the robustness of the emergent dynamics in the RKCS model across various non-compact physically important distributions, including Gaussian, sub-Gaussian, and $D$-th moment integrable distributions.

math.AP↗

arXiv2025

Phase transition of the kinetic Justh-Krishnaprasad type model for nematic alignment

We present a stochastic Justh-Krishnaprasad flocking model and study the phase transition of the Vlasov-McKean-Fokker-Planck (VMFP) equation, which can be obtained in the mean-field limit. To describe the alignment, we use order parameters in terms of the distribution function of the kinetic model. For the constant noise case, we study the well-posedness of the VMFP equation on the torus. Based on regularity, we show that the phenomenon of phase transition is only related to the ratio between the strengths of noise and coupling. In particular, for the low-noise case, we derive an exponential convergence to the von-Mises type equilibrium, which shows a strong evidence for the nematic alignment. The multiplicative noise is also studied to obtain a non-symmetric equilibrium with two different peaks on the torus.

math.AP↗

arXiv2025

Uniform-in-time asymptotic limits of generalized Kuramoto models

We study two uniform-in-time asymptotic limits for generalized Kuramoto (GK) models. For these GK type models, we first derive the uniform stability estimates with respect to initial data, natural frequency and communication network under a suitable framework, and then as direct applications of this uniform stability estimate, we establish two asymptotic limits which are valid in the whole time interval, namely uniform-in-time continuum and mean-filed limit to the continuum and kinetic GK models, respectively. In the mean-field limit setting (the number of particles tends to infinity), we show global-in-time existence of measure-valued solutions to the corresponding kinetic equation. On the other hand, in a continuum limit setting (the lattice size tends to zero), we show that the lattice GKM solutions converge to a classical solution to the continuum GK model in supremum norm. Two asymptotic limits improve earlier results for the generalized GK type models.

math.AP↗

arXiv2025

Measure-valued death state and local sensitivity analysis for Winfree models with uncertain high-order couplings

We study the measure-valued death state and local sensitivity analysis of the Winfree model and its mean-field counterpart with uncertain high-order couplings. The Winfree model is the first mathematical model for synchronization, and it can cast as the effective approximation of the pulse-coupled model for synchronization, and it exhibits diverse asymptotic patterns depending on system parameters and initial data. For the proposed models, we present several frameworks leading to oscillator death in terms of system parameters and initial data, and the propagation of regularity in random space. We also present several numerical tests and compare them with analytical results.

math.AP↗

arXiv2024

Optimal convergence speed in the classical limits of relativistic Cucker-Smale models

We study quantitative estimates for the flocking and uniform-time classical limit to the relativistic Cucker-Smale (in short RCS) model introduced in \cite{Ha-Kim-Ruggeri-ARMA-2020}. Different from previous works, we do not neglect the relativistic effect on the presence of the pressure in momentum equation. For the RCS model, we provide a quantitative estimate on the uniform-time classical limit with an optimal convergence rate which is the same as in finite-time classical limit under a relaxed initial condition. We also allow corresponding initial data for the RCS and Cucker-Smale (CS) model to be different in the classical limit. This removes earlier constraints employed in the previous classical limit. As a direct application of this optimal convergence rate in the classical limit of the RCS model, we derive an optimal convergence rate for the corresponding uniform-time classical limit for the kinetic RCS model.

math.AP↗

arXiv2024

Mono-cluster flocking and uniform-in-time stability of the discrete Motsch-Tadmor model

The Motsch-Tadmor (MT) model is a variant of the Cucker-Smale model with a normalized communication weight function. The normalization poses technical challenges in analyzing the collective behavior due to the absence of conservation of momentum. We study three quantitative estimates for the discrete-time MT model considering the first-order Euler discretization. First, we provide a sufficient framework leading to the asymptotic mono-cluster flocking. The proposed framework is given in terms of coupling strength, communication weight function, and initial data. Second, we show that the continuous transition from the discrete MT model to the continuous MT model can be made uniformly in time using the finite-time convergence result and asymptotic flocking estimate. Third, we present uniform-in-time stability estimates for the discrete MT model. We also provide several numerical examples and compare them with analytical results.

math.NA↗

arXiv2024

Discrete Consensus-Based Optimization

We propose Discrete Consensus-Based Optimization (DCBO), a fully discrete version of the Consensus-Based Optimization (CBO) framework. DCBO is a multi-agent method for the global optimization of possibly non-convex and non-differentiable functions. It aligns with the CBO paradigm, which promotes a consensus among agents towards a global optimum through simple stochastic dynamics amenable to rigorous mathematical analysis. Despite the promises, there has been a gap between the analysis of CBO and the actual behavior of the agents from its time-discrete implementation, as the former has focused on the system of continuous stochastic differential equations defining the model or its mean-field approximation. In particular, direct analysis of CBO-type algorithms with heterogeneous stochasticity is very challenging. DCBO distinguishes itself from these approaches in the sense that it has no continuous counterpart, thanks to the replacement of the "softmin" operator with the "hardmin" one, which is inherently discrete. Yet, it maintains the operational principles of CBO and allows for rich mathematical analysis. We present conditions, independent of the number of agents, for achieving a consensus or convergence and study the circumstances under which global optimization occurs. We test DCBO on a large number of benchmark functions to show its merits. We also demonstrate that DCBO is applicable to a diverse range of real-world problems, including neural network training, compressed sensing, and portfolio optimization, with competitive performance.

math.OC↗

arXiv2024

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

We study the compression 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 the 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.

math.DS↗

arXiv2023

On the emergent dynamics of the infinite set of Kuramoto oscillators

We propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row(or column)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc.

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

Emergent dynamics of the generalized Winfree model]{Emerging asymptotic patterns in a Winfree ensemble with higher-order couplings

The Winfree model is a phase-coupled synchronization model which simplifies pulse-coupled models such as the Peskin model on pacemaker cells. It is well-known that the Winfree ensemble with the first-order coupling exhibits discrete asymptotic patterns such as incoherence, locking and death depending on the coupling strength and variance of natural frequencies. In this paper, we further study higher-order couplings which makes the dynamics more close to the behaviors of the Peskin model. For this, we propose several sufficient frameworks for asymptotic patterns compared to the first-order coupling model. Our proposed conditions on the coupling strength, natural frequencies and initial data are independent of the number of oscillators so that they can be applied to the corresponding mean-field model. We also provide several numerical simulations and compare them with analytical results.

math.DS↗

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

An algebraic approach for the weak coupling of multiple Lohe tensor models

We present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely "{\it fusion operation}" as a binary operation between the characteristic symbols. It turns out that the fusion operation satisfies the associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multiple LT models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the LT models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide a sufficient framework leading to emergent dynamics to the proposed weakly coupled model.

math.DS↗

arXiv2021

Emergent behaviors of discrete Lohe aggregation flows

The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.

math-ph↗

arXiv2021

Relaxation dynamics of SIR-flocks with random epidemic states

We study the collective dynamics of a multi-particle system with three epidemic states as an internal state. For the collective modeling of active particle system, we adopt modeling spirits from the swarmalator model and the SIR epidemic model for the temporal evolution of particles' position and internal states. Under suitable assumptions on system parameters and non-collision property of initial spatial configuration, we show that the proposed model does not admit finite-time collisions so that the standard Cauchy-Lipschitz theory can be applied for the global well-posedness. For the relaxation dynamics, we provide several sufficient frameworks leading to the relaxation dynamics of the proposed model. The proposed sufficient frameworks are formulated in terms of system parameters and initial configuration. Under such sufficient frameworks, we show that the state configuration relaxes to the fixed constant configuration via the exponentially perturbed gradient system and explicit dynamics of the SIR model. We present explicit lower and upper bounds for the minimal and maximal relative distances.

math.DS↗

arXiv2021

Emergence of a periodically rotating one-point cluster in a thermodynamic Cucker-Smale ensemble

We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating one-point cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.

math.AP↗