arXiv subjects

Valeriia Zhidkova

Publications and source records attributed to Valeriia Zhidkova.

3 recordsLinked to original sources

arXiv2026

Emergent behaviors of the singular continuum Kuramoto model and its graph limit

We study the emergent dynamics of the singular continuum Kuramoto model (in short, SCKM) and its graph limit. The SCKM takes the form of an integro-differential equation exhibiting two types of nonlocal singularities: a nonlocal singular interaction weight and a nonlocal singular alignment force. The natural frequency function determines the emergent dynamics of the SCKM, and we emphasize that singularity plays a crucial role in the occurrence of sticking phenomena. For the identical natural frequency function, we derive the complete phase synchronization in finite time under a suitable set of conditions for system parameters and initial data. In contrast, for a nonidentical natural frequency function, we show the emergence of practical synchronization, meaning that the phase diameter is proportional to the inverse of coupling strength asymptotically. Furthermore, we rigorously establish a graph limit from the singular Kuramoto model with a finite system size to the SCKM. We also provide several numerical simulations to illustrate our theoretical results.

math.DS↗

arXiv2026

Relaxation dynamics of the continuum Kuramoto model with non-integrable kernels

We study the asymptotic behavior of the continuum Kuramoto model with a fractional Laplacian-type kernel. For this, we construct global weak solutions via a two-parameter regularization procedure using a kernel truncation with fractional dissipation. Using a priori uniform estimates derived in fractional Sobolev spaces, we employ compactness arguments to construct global weak solutions to the singular continuum Kuramoto model. Furthermore, we also establish an exponential relaxation toward the initial phase average in $L^2$-norm under suitable assumptions on initial data and system parameters. These findings provide a rigorous characterization of the existence of solutions and the emergent dynamics of Kuramoto ensembles under physically important strongly singular interactions, including power-law singular kernels and Coulomb-type kernels.

math.AP↗

arXiv2025

Optimal Control for Kuramoto Model: from Many-Particle Liouville Equation to Diffusive Mean-Field Problem

In this paper, we investigate the mean-field optimal control problem of a swarm of Kuramoto oscillators. Using the notion of wrapped distribution, we explain the connection between the stochastic particle system and the mean-field PDE on the periodic domain. In the limit of an infinite number of oscillators the collective dynamics of the agents' density is described by a diffusive mean-field model in the form of a non-local PDE, where the non-locality arises from the synchronization mechanism. We prove the existence of the optimal control of the mean-field model by using $Γ$-convergence strategy of the cost functional corresponding to the Liouville equation on the particle level. In the discussion of propagation of chaos for fixed control functions we complete the relative entropy estimate by using large deviation estimate given by \cite{MR3858403}.

math.AP↗