arXiv subjects

Seung-Yeon Ryoo

Publications and source records attributed to Seung-Yeon Ryoo.

9 recordsLinked to original sources

arXiv2026

Relaxation dynamics of the Inertial Winfree model

We prove two synchronization theorems for the second-order (inertial) Winfree model of coupled oscillators. The first result is a pathwise oscillator-death theorem with explicit smallness thresholds on the natural frequencies, initial velocities, and inertia, scaling as $R_0^{3/2}$ in the initial order parameter $R_0$. The second result is a qualitative zero-inertia synchronization statement: under generic initial data, if the intrinsic and initial velocity spreads are small compared to $κ$ and the inertia $m$ is small, then the limiting order parameter can be made arbitrarily close to 2. The proof of the first result is organized around three mechanisms, namely inertial gradient flow and the Łojasiewicz theorem, an initial layer argument, and an order-parameter bootstrapping argument. The proof of the second result involves approximation to the first-order case via a quantitative Tikhonov theorem.

math.AP↗

arXiv2026

Emergent behaviors of Winfree oscillators on special orthogonal group

We propose a generalized matrix-valued synchronization model which can be regarded as matrix generalization of the classical Winfree model to the special orthogonal group, and we provide several sufficient frameworks leading to the emergent behaviors of the Winfree matrix model. For $SO(2)$ case, the proposed model reduces to the classical Winfree model. For the general (non-identical) case, we prove the existence of a positively invariant trapping region, establish a leader--follower mechanism in which sufficiently strong coupling draws all oscillators into a neighborhood of the identity whenever at least one oscillator is initially nearby, and show $\ell^1$-exponential stability of solutions, from which we deduce existence, uniqueness, and exponential convergence to an equilibrium. In the identical-oscillator regime, we show that complete state synchronization and oscillator death both occur exponentially fast with an explicit decay rate, and we classify all equilibrium configurations as solutions to a fixed-point equation for the mean influence.

math.DS↗

arXiv2026

On oscillator death in the Winfree model

We show that for the standard sinusoidal Winfree model, a coupling strength exceeding twice the maximal magnitude of the intrinsic frequencies guarantees the convergence of the system for Lebesgue almost every initial data. This is proven by first showing, via an order parameter bootstrapping argument, that the pathwise critical coupling strength is upper bounded by a function of the order parameter, and then showing by a volumetric argument that for Lebesgue almost every data the order parameter cannot stay below and be bounded away from 1 for all time; this is a Winfree model counterpart of the analysis of Ha and the author (2020) performed for the Kuramoto model. Using concentration of measure and the aforementioned volumetric argument, we show that, except possibly on a set of very small measure, oscillator death is observed in finite time; this rigorously demonstrates the existence of the oscillator death regime numerically observed by Ariaratnam and Strogatz (2001). These results are robust under many other choices of interaction functions often considered for the Winfree model. We demonstrate that the asymptotic dynamics described in this paper are sharp by analyzing the equilibria of the Winfree model, and we bound the total number of equilibria using a polynomial description.

math.CA↗

arXiv2025

Asymptotics of Riemannian Lie groups with nilpotency step 2

We derive sharp estimates comparing asymptotic Riemannian or sub-Riemannian metrics in 2-step nilpotent Lie groups. For each metric, we construct a Carnot metric whose square remains at bounded distance from the square of the original metric. In particular, we deduce the analogue of a conjectire by Burago-Margulis: every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone. As a consequence, we obtain a refined estimate of the error term in the asymptotic expansion of the volume of the (sub-)Riemannian metric balls. To achive this, we develop a novel technique to efficiently perturb rectifiable curves modifying their endpoints in a prescribed vertical direction.

math.DG↗

arXiv2025

Quantitative relaxation dynamics from generic initial configurations in the inertial Kuramoto model

We study the relaxation dynamics of the inertial Kuramoto model toward a phase-locked state from a generic initial phase configuration. For this, we propose a sufficient framework in terms of initial data and system parameters for asymptotic phase-locking. It can be roughly stated as set of conditions such as a positive initial order parameter, a coupling strength sufficiently larger than initial frequency diameter and intrinsic frequency diameter, but less than the inverse of inertia. Under the proposed framework, generic initial configuration undergoes three dynamic stages (initial layer, condensation and relaxation stages) before it reaches a phase-locked state asymptotically. The first stage is the initial layer stage in analogy with fluid mechanics, during which the effect of the initial natural frequency distribution is dominant, compared to that of the sinusoidal coupling between oscillators. The second stage is the condensation stage, during which the order parameter increases, and at the end of which a majority cluster is contained in a sufficiently small arc. Finally, the third stage is the persistence and relaxation stage, during which the majority cluster remains stable (persistence) and the total configuration relaxes toward a phase-locked state asymptotically (relaxation). The intricate proof involves with several key tools such as the quasi-monotonicity of the order parameter (for the condensation stage), a nonlinear Grönwall inequality on the diameter of the majority cluster (for the persistence stage), and a variant of the classical Łojasiewicz gradient theorem (for the relaxation stage).

math.DS↗

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

Quantitative nonembeddability of groups of polynomial growth into uniformly convex spaces

Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of radius $r\ge 2$ in the aforementioned groups must incur bilipschitz distortion at least a constant multiple of $(\log r)^{1/q}$ into a $q(\ge 2)$-uniformly convex Banach space. This bound is sharp for the $L^p$ ($1<p<\infty$) spaces. We prove this by establishing ``vertical versus horizontal inequalities'' for functions from the aforementioned groups into uniformly convex spaces, using the vector-valued Littlewood--Paley--Stein theory approach of Lafforgue and Naor (2012). These inequalities are quantitative nonembeddability statements that any Lipschitz mapping from the aforementioned groups into a uniformly convex space quantitatively collapses along certain central subgroups. In the special case of mappings of Carnot groups into the $L^p$ ($1<p<\infty$) spaces, we prove that the quantitative collapse occurs on the commutator subgroup; this is in line with the qualitative Pansu--Semmes nonembeddability argument given by Cheeger and Kleiner (2006) and Lee and Naor (2006). We prove this by establishing a version of the classical Dorronsoro theorem on Carnot groups. Previously, in the setting of Heisenberg groups, Fässler and Orponen (2019) established a one-sided Dorronsoro theorem with a restriction $0<α<2$ on the range of exponents $α$ of the Laplacian; this restriction does not appear in the commutative setting and is caused by their use of horizontal polynomials as approximants. We identify the correct class of approximant polynomials and prove the two-sided Dorronsoro theorem with the full range $0<α<\infty$ of exponents in the general setting of Carnot groups, thus strengthening and extending the work of Fässler and Orponen.

math.MG↗

arXiv2025

Asymptotic formation and orbital stability of phase-locked states in Kuramoto--Lohe type synchronization models on Lie groups

Some mathematical models of synchronization, such as the Kuramoto model (1975) and its generalizations pioneered by Lohe (2009), are formulated as ordinary differential equations describing populations of particles on Lie groups with locally attractive interactions. We suggest a model of synchronization on Lie groups and present a framework to understand the formation of phase-locked states and their orbital stability. This is a sequel to a previous joint work with Ha and Ko (2017).

math.DS↗

arXiv2023

Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any $0<\varepsilon<1/2$ the metric space $(G,d_G^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^{D_G}$ with distortion $O_G(\varepsilon^{-1/2})$. This is done by building on the approach of T. Tao (2021), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash--Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lovász local lemma and the concentration of measure phenomenon on the sphere in place of Tao's use of a quantitative homotopy argument.

math.MG↗