arXiv subjects

Georgi S. Medvedev

Publications and source records attributed to Georgi S. Medvedev.

At least 19 recordsLinked to original sources

arXiv2026

The Kuramoto model on the Sierpinski Gasket II: Twisted states

We study the Kuramoto model (KM) of coupled phase oscillators on graphs approximating the Sierpinski gasket (SG). As the size of the graph tends to infinity, the limit points of the sequence of stable equilibria in the KM correspond to the minima of the Dirichlet energy, i.e., to harmonic maps from the SG to the circle. We provide a complete description of the stable equilibria of the continuum limit of the KM on graphs approximating the SG, under both Dirichlet and free boundary conditions. We show that there is a unique stable equilibrium in each homotopy class of continuous functions from the SG to the circle. These equilibria serve as generalizations of the classical twisted states on ring networks. Furthermore, we extend the analysis to the KM on post-critically finite fractals. The results of this work reveal the link between self-similar organization and network dynamics.

math-ph↗

arXiv2026

Graphon Spin Systems as Exactly Solvable Models

Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erdős-Rényi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.

cond-mat.dis-nn↗

arXiv2026

Kuramoto model on Sierpinski Gasket I: Harmonic maps

Motivated by the study of attractors in the Kuramoto model (KM) on graphs approximating the Sierpinski gasket (SG), we revisit the problem of harmonic maps (HMs) from SG to the circle, first considered by Strichartz. We provide a geometric proof of Strichartz's theorem, which states that for a prescribed degree and suitable boundary conditions, there exists a unique HM from SG to the circle. We extend this result to HMs on post-critically finite (p.c.f.) fractals. For continuous functions on SG, we define a degree given by vector of integers of arbitrary finite length. We show that the degree determines a homotopy class on SG with values in the circle. This provides an analog of the Hopf degree theorem on SG. We move on to analyze HMs. At the heart of our method lies an original construction of covering spaces. After lifting continuous functions on SG with values in the unit circle to continuous real-valued functions on the covering space, we use the harmonic extension algorithm to obtain a harmonic function on the covering space. The desired HM is obtained by restricting the domain of the harmonic function to the fundamental domain and projecting the range to the circle. Each covering space is constructed separately for HMs of a given homotopy class, capturing its intrinsic topology. We show that with suitable modifications the method applies to p.c.f. fractals, a large class of self-similar domains. We illustrate our method using numerical examples of HMs from SG to the circle and discuss the construction of covering spaces for several representative p.c.f. fractals, including the 3-level SG, hexagasket, and pentagasket. The results of this paper provide the foundation for a follow-up work where we give a complete description of attractors in the KM on graphs approximating p.c.f. fractals. Specifically, we show that all HMs identified in this paper are stable steady states of the KM.

math-ph↗

arXiv2026

Interacting dynamical systems on networks and fractals: discrete and continuous models, mean-field limit, and convergence rates

We develop a continuum limit and mean-field theory for interacting particle systems (IPS) on self-similar networks, a new class of discrete models whose large-scale behavior gives rise to nonlocal evolution equations on fractal domains. This work extends the graphon-based framework for IPS, used to derive continuum and mean-field limits in the non-exchangeable setting, to situations where the spatial domain is fractal rather than Euclidean. The motivation arises from both physical models naturally formulated on fractals and real-world networks exhibiting hierarchical or quasi-self-similar structure. Our analysis relies on tools from fractal geometry, including Iterated Function Systems and self-similar measures. A central result is an explicit isomorphism between self-similar IPS and graphon IPS, which allows us to justify the continuum and mean-field limits in the self-similar setting. This connection reveals that macroscopic dynamics on fractal domains emerge naturally as limits of dynamics on appropriate discretizations of fractal sets. Another contribution of the paper is the derivation of optimal convergence rates for the discrete self-similar models. We introduce a scale of generalized Lipschitz spaces on fractals, extending the Nikolskii-Besov spaces used in the Euclidean setting, and obtain convergence estimates for discontinuous Galerkin approximations of nonlocal equations posed on fractal domains. These results apply to kernels with minimal regularity addressing models relevant in applications.

math.DS↗

arXiv2025

Phase transitions in the Ising model on random graphs

We study phase transitions in the Ising model on random graphs using graph limits. We show that the critical temperatures are determined by the eigenvalues of the kernel operator associated with the graph limit. Bifurcation diagrams for Erdos-Renyi, small-world, and power-law graphs illustrate the theory. In the small-world case, we identify metastable behavior in both ferromagnetic and antiferromagnetic regimes.

math-ph↗

arXiv2025

Graphon Signal Processing for Spiking and Biological Neural Networks

Graph Signal Processing (GSP) extends classical signal processing to signals defined on graphs, enabling filtering, spectral analysis, and sampling of data generated by networks of various kinds. Graphon Signal Processing (GnSP) develops this framework further by employing the theory of graphons. Graphons are measurable functions on the unit square that represent graphs and limits of convergent graph sequences. The use of graphons provides stability of GSP methods to stochastic variability in network data and improves computational efficiency for very large networks. We use GnSP to address the stimulus identification problem (SIP) in computational and biological neural networks. The SIP is an inverse problem that aims to infer the unknown stimulus s from the observed network output f. We first validate the approach in spiking neural network simulations and then analyze calcium imaging recordings. Graphon-based spectral projections yield trial-invariant, lowdimensional embeddings that improve stimulus classification over Principal Component Analysis and discrete GSP baselines. The embeddings remain stable under variations in network stochasticity, providing robustness to different network sizes and noise levels. To the best of our knowledge, this is the first application of GnSP to biological neural networks, opening new avenues for graphon-based analysis in neuroscience.

eess.SP↗

arXiv2025

Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators

The Kuramoto model (KM) of $n$ coupled phase-oscillators is analyzed in this work. The KM on a Cayley graph possesses a family of steady state solutions called twisted states. Topologically distinct twisted states are distinguished by the winding number $q\in\mathbb{Z}$. These states are known to be stable for small enough $q$. In the presence of small noise, the KM exhibits metastable transitions between $q$-twisted states: A typical trajectory remains in the basin of attraction of a given $q$-twisted state for an exponentially long time, but eventually transitions to the vicinity of another such state. In the course of this transition, it passes in close proximity of a saddle of Morse index $1$, called a relevant saddle. In this work, we provide an exhaustive analysis of metastable transitions in the stochastic KM with nearest-neighbor coupling. We start by analyzing the equilibria and their stability. First, we identify all equilibria in this model. Using the discrete Fourier transform and eigenvalue estimates for rank-1 perturbations of symmetric matrices, we classify the equilibria by their Morse indices. In particular, we identify all stable equilibria and all relevant saddles involved in the metastable transitions. Further, we use Freidlin-Wentzell theory and the potential-theoretic approach to metastability to establish the metastable hierarchy and sharp estimates of Eyring-Kramers type for the transition times. The former determines the precise order, in which the metastable transitions occur, while the latter characterizes the times between successive transitions. The theoretical estimates are complemented by numerical simulations and a careful numerical verification of the transition times. Finally, we discuss the implications of this work for the KM with other coupling types including nonlocal coupling and the continuum limit as $n$ tends to infinity.

math.PR↗

arXiv2024

The Large Deviation Principle for $W$-random spectral measures

The $W$-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for $W$-random graphs from [9], we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by the large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the underlying graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of the integral operators corresponding to certain random graph sequences. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces of the integral operators are affected by large deviations. Potential implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing are indicated.

math.PR↗

arXiv2023

Turing bifurcation in the Swift-Hohenberg equation on deterministic and random graphs

The Swift-Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger and Holzer [2], we consider discrete SHE on deterministic and random graphs. The two families of the discrete models share the same continuum limit in the form of a nonlocal SHE on a circle. The analysis of the continuous system, parallel to the analysis of the classical SHE, shows bifurcations of spatially periodic solutions at critical values of the control parameters. However, the proximity of the discrete models to the continuum limit does not guarantee that the same bifurcations take place in the discrete setting in general, because some of the symmetries of the continuous model do not survive discretization. We use the center manifold reduction and normal forms to obtain precise information about the number and stability of solutions bifurcating from the homogeneous state in the discrete models on deterministic and sparse random graphs. Moreover, we present detailed numerical results for the discrete SHE on the nearest-neighbor and small-world graphs.

nlin.PS↗

arXiv2023

Galerkin method for nonlocal diffusion equations on self-similar domains

Integro-differential equations, analyzed in this work, comprise an important class of models of continuum media with nonlocal interactions. Examples include peridynamics, population and opinion dynamics, the spread of disease models, and nonlocal diffusion, to name a few. They also arise naturally as a continuum limit of interacting dynamical systems on networks. Many real-world networks, including neuronal, epidemiological, and information networks, exhibit self-similarity, which translates into self-similarity of the spatial domain of the continuum limit. For a class of evolution equations with nonlocal interactions on self-similar domains, we construct a discontinuous Galerkin method and develop a framework for studying its convergence. Specifically, for the model at hand, we identify a natural scale of function spaces, which respects self-similarity of the spatial domain, and estimate the rate of convergence under minimal assumptions on the regularity of the interaction kernel. The analytical results are illustrated by numerical experiments on a model problem.

math.NA↗

arXiv2023

Bifurcations and patterns in the Kuramoto model with inertia

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov-Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity.

nlin.PS↗

arXiv2022

A global bifurcation organizing rhythmic activity in a coupled network

We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before the bifurcation from noncontractibile ones after the bifurcation. Both families are stable for the model at hand.

nlin.AO↗

arXiv2021

Stability and bifurcation of mixing in the Kuramoto model with inertia

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erdős--Rényi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.

math.DS↗

arXiv2021

Chimeras unfolded

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to coupled lasers, to power grids. In this paper, we describe a codimension-2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras. We use a combination of linear stability analysis and Penrose diagrams to identify and analyze a variety of spatiotemporal patterns including stationary and traveling coherent clusters and twisted states, as well as their combinations with regions of incoherent behavior called chimera states. The linear stability analysis is used to estimate of the velocity distribution within these structures. Penrose diagrams, on the other hand, predict accurately the basins of their existence. Furthermore, we show that network topology can endow chimera states with nontrivial spatial organization. In particular, we present twisted chimera states, whose coherent regions are organized as stationary or traveling twisted states. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model with uni-, bi-, and tri-modal frequency distributions and all-to-all and nonlocal nearest-neighbor connectivity.

nlin.CD↗

arXiv2020

Instability of mixing in the Kuramoto model: From bifurcations to patterns

We study patterns observed right after the loss of stability of mixing in the Kuramoto model of coupled phase oscillators with random intrinsic frequencies on large graphs, which can also be random. We show that the emergent patterns are formed via two independent mechanisms determined by the shape of the frequency distribution and the limiting structure of the underlying graph sequence. Specifically, we identify two nested eigenvalue problems whose eigenvectors (unstable modes) determine the structure of the nascent patterns. The analysis is illustrated with the results of the numerical experiments with the Kuramoto model with unimodal and bimodal frequency distributions on certain graphs.

nlin.CD↗

arXiv2020

Stability of clusters in the second-order Kuramoto model on random graphs

The Kuramoto model of coupled phase oscillators with inertia on Erdos-Renyi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources.

nlin.AO↗

arXiv2019

Sparse Monte Carlo method for nonlocal diffusion problems

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media flow, peridynamics, models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. An important feature of our method is sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows to use fewer discretization points without compromising the accuracy. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates.

math.NA↗

arXiv2018

The continuum limit of the Kuramoto model on sparse random graphs

In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erdős-R{\' e}nyi, small-world, and power law graphs.

math.DS↗