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David Poyato

Publications and source records attributed to David Poyato.

15 recordsLinked to original sources
arXiv2026

Graph-Induced Rotational Twisted States in Systems of Identical Oscillators

In this article, we study a new class of collective motion for identical coupled Stuart-Landau oscillators on graphs. This model was previously known to converge to the synchronized state for a certain class of initial data. Here, we show that when the interaction matrix is circulant, there exists another class of attractors, in which the particles are uniformly distributed on a circle, reminiscent of the \textit{twisted states} known for the Kuramoto model, but which rotate around the origin. However, contrary to the classical Kuramoto case, here the rotation is caused by the asymmetry of the graph structure, and not by the natural frequencies. We identify conditions for the existence and local stability of the \textit{rotational twisted states}, both in the Stuart-Landau model and in the Kuramoto model. We also provide sufficient conditions for the existence and stability of the synchronized state, which can co-exist with the rotational twisted state in a certain parameter region. We provide a generalization of the class of interaction matrices able to generate rotational twisted states via leader-follower interaction matrices. We show that heterogeneous rotational twisted states can also exist for a system of heterogeneous oscillators, attracted to different individual target amplitudes. This study is accompanied by numerical simulations that illustrate the possible behaviors of the system, which also include metastable dynamics and a chimera state.

math.DS↗
arXiv2026

Mean field limit of non-exchangeable interacting diffusions on co-evolutionary networks

Systems in which the network structure and particle states co-evolve in mutual influence are increasingly recognized as essential for modeling complex adaptive systems. However, traditional models of non-exchangeable interacting particle systems frequently assume a fixed network topology, a simplification that fails to capture the dynamical nature of many real-world phenomena. In this paper, we rigorously establish the mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The primary analytical challenge arises from the coupling between the network dynamics and the agents' states, which induces non-Markovian dynamics where the system's evolution depends on its entire history. Consequently, the macroscopic limit is not governed by a classical partial differential equation, but rather by a coupled system of path-dependent McKean-Vlasov SDEs for the particles' states together with a transport equation for the distribution of the weights. A further difficulty stems from the non-linear weight dynamics, which requires an adequate choice for the limiting network structure. To overcome the structural limitations of classical graphon theory, we employ the framework of \(\mathcal{K}\)-graphons (Lov\'asz and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025). To the best of our knowledge, this is the first example in the literature in which the natural topology of probability-graphons is used in the context of mean-field limits, providing a natural and rigorous framework that is fully compatible with non-linear network adaptivity.

math.PR↗
arXiv2025

Synchronization of coupled Stuart-Landau oscillators: How heterogeneity can facilitate synchronization

We study the collective dynamics of coupled Stuart--Landau oscillators, which model limit-cycle behavior near a Hopf bifurcation and serve as the amplitude-phase analogue of the Kuramoto model. Unlike the well-studied phase-reduced systems, the full Stuart--Landau model retains amplitude dynamics, enabling the emergence of rich phenomena such as amplitude death, quenching, and multistable synchronization. We provide a complete analytical classification of asymptotic behaviors for identical natural frequencies, but heterogeneous inherent amplitudes in the finite-$N$ setting. In the two-oscillator case, we classify the asymptotic behavior in all possible regimes including heterogeneous natural frequencies and inherent amplitudes, and in particular we identify and characterize a novel regime of \emph{leader-driven synchronization}, wherein one active oscillator can entrain another regardless of frequency mismatch. For general $N$, we prove exponential phase synchronization under sectorial initial data and establish sharp conditions for global amplitude death. Finally, we analyze a real-valued reduction of the model, connecting the dynamics to nonlinear opinion formation and consensus processes. Our results highlight the fundamental differences between amplitude-phase and phase-only Kuramoto models, and provide a new framework for understanding synchronization in heterogeneous oscillator networks.

math.DS↗
arXiv2024

Mean-field limit of non-exchangeable multi-agent systems over hypergraphs with unbounded rank

Interacting particle systems are known for their ability to generate large-scale self-organized structures from simple local interaction rules between each agent and its neighbors. In addition to studying their emergent behavior, a main focus of the mathematical community has been concentrated on deriving their large-population limit. In particular, the mean-field limit consists of describing the limit system by its population density in the product space of positions and labels. The strategy to derive such limits is often based on a careful combination of methods ranging from analysis of PDEs and stochastic analysis, to kinetic equations and graph theory. In this article, we focus on a generalization of multi-agent systems that includes higher-order interactions, which has largely captured the attention of the applied community in the last years. In such models, interactions between individuals are no longer assumed to be binary (i.e. between a pair of particles). Instead, individuals are allowed to interact by groups so that a full group jointly generates a non-linear force on any given individual. The underlying graph of connections is then replaced by a hypergraph, which we assume to be dense, but possibly non uniform and of unbounded rank. For the first time in the literature, we show that when the interaction kernels are regular enough, then the mean-field limit is determined by a limiting Vlasov-type equation, where the hypergraph limit is encoded by a so-called UR-hypergraphon (unbounded-rank hypergraphon), and where the resulting mean-field force admits infinitely-many orders of interactions.

math.AP↗
arXiv2023

Uniform contractivity of the Fisher infinitesimal model with strongly convex selection

The Fisher infinitesimal model is a classical model of phenotypic trait inheritance in quantitative genetics. Here, we prove that it encompasses a remarkable convexity structure which is compatible with a selection function having a convex shape. It yields uniform contractivity along the flow, as measured by a $L^\infty$ version of the Fisher information. It induces in turn asynchronous exponential growth of solutions, associated with a well-defined, log-concave, equilibrium distribution. Although the equation is non-linear and non-conservative, our result shares some similarities with the Bakry-Emery approach to the exponential convergence of solutions to the Fokker-Planck equation with a convex potential. Indeed, the contraction takes place at the level of the Fisher information. Moreover, the key lemma for proving contraction involves the Wasserstein distance $W_\infty$ between two probability distributions of a (dual) backward-in-time process, and it is inspired by a maximum principle by Caffarelli for the Monge-Amp\`ere equation.

math.PR↗
arXiv2022

Measure solutions to a kinetic Cucker-Smale model with singular and matrix-valued communication

We introduce a multi-dimensional variant of the kinetic Cucker-Smale model with singular and matrix-valued communication weight, which reduces to the singular kinetic Cucker-Smale equation in the one-dimensional case. We propose an appropriate notion of weak measure-valued solution to this second-order system and a suitable first-order reduction, which persist beyond the blow-up time in classical norms. The core of the paper is to show that both formulations are equivalent in this singular regime, thus extending the previous results from regular to weakly singular communication weights. As a consequence, we obtain: (i) global-in-time well-posedness of weak measure-valued solutions, (ii) quantitative convergence rates to equilibrium, and (iii) uniform-in-time mean-field limits thanks to the above-mentioned equivalence and our recent work on fibered gradient flows.

math.AP↗
arXiv2022

Heterogeneous gradient flows in the topology of fibered optimal transport

We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case $\mathbb{R}^d\times \mathbb{R}^d$, where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures $(\mathcal{P}_{2,\nu}(\mathbb{R}^{2d}),W_{2,\nu})$ with fixed marginal $\nu\in \mathcal{P}(\mathbb{R}^d)$ in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.

math.AP↗
arXiv2021

Mean-field limit of non-exchangeable systems

This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.

math.PR↗
arXiv2021

Ergodicity of the Fisher infinitesimal model with quadratic selection

We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher's infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogues, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.

math.PR↗
arXiv2021

Hydrodynamic limit of a coupled Cucker-Smale system with strong and weak internal variable relaxation

In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker--Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes' law. Whilst the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions

math.AP↗
arXiv2019

On the trend to global equilibrium for Kuramoto Oscillators

In this paper, we study the convergence to the stable equilibrium for Kuramoto oscillators. Specifically, we derive estimates on the rate of convergence to the global equilibrium for solutions of the Kuramoto-Sakaguchi equation in a large coupling strength regime from generic initial data. As a by-product, using the stability of the equation in the Wasserstein distance, we quantify the rate at which discrete Kuramoto oscillators concentrate around the global equilibrium. In doing this, we achieve a quantitative estimate in which the probability that the oscillators will concentrate at the given rate tends to one as the number of oscillators increases. Among the essential steps in our proof are: 1) An entropy production estimate inspired by the formal Riemannian structure of the space of probability measures, first introduced by F. Otto in [35]; 2) A new quantitative estimate on the instability of equilibria with antipodal oscillators based on the dynamics of norms of the solution in sets evolving by the continuity equation; 3) The use of generalized local logarithmic Sobolev and Talagrand type inequalities, similar to the ones derived by F. Otto and C. Villani in [36]; 4) The study of a system of coupled differential inequalities, by a treatment inspired by the work of L. Desvillettes and C. Villani [13]. Since the Kuramoto-Sakaguchi equation is not a gradient flow with respect to the Wasserstein distance, we derive such inequalities under a suitable fibered transportation distance.

math.AP↗
arXiv2019

Filippov flows and mean-field limits in the kinetic singular Kuramoto model

The agent-based singular Kuramoto model was proposed in [60] as a singular version of the Kuramoto model of coupled oscillators that is consistent with Hebb's rule of neuroscience. In such paper, the authors studied its well-posedness via the concept of Filippov solutions. Interestingly, they found some new emergent phenomena in the paradigm of Kuramoto model: clustering into subgroups and emergence of global phase synchronization taking place at finite time. This paper aims at introducing the associated kinetic singular Kuramoto model along $\mathbb{T}\times\mathbb{R}$, that is reminiscent of the classical Kuramoto-Sakaguchi equation. Our main goal is to propose a well-posedness theory of measure-valued solutions that remains valid after eventual phase collisions. The results will depend upon the specific regime of singularity: subcritical, critical and supercritical. The cornerstone is the existence of Filippov characteristic flows for interaction kernels with jump discontinuities. Our second goal is to study stability with respect to initial data, that in particular will provide quantitative estimates for the mean-field limit of the agent-based model towards the kinetic equation in quadratic Wasserstein-type distances. Finally, we will recover global phase synchronization at the macroscopic scale under appropriate generic assumptions on the initial data. The most singular regime will be tackled separately with an alternative method, namely, a singular hyperbolic limit on a regularized second order model with inertia. Note that we will work within the manifold $\mathbb{T}\times\mathbb{R}$ and will avoid resorting on Euclidean approximations. Also, no gradient-type structure will be needed, as opposed to some preceding literature.

math.AP↗
arXiv2018

Filippov trajectories and clustering in the Kuramoto model with singular couplings

We study the synchronization of a generalized Kuramoto system in which the coupling weights are determined by the phase differences between oscillators. We employ the fast-learning regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by the approach of phases. First, we study the well-posedness problem for the singular weighted Kuramoto systems in which the Lipschitz continuity is deprived. We present the dynamics of the system equipped with singular weights in all the subcritical, critical and supercritical regimes of the singularity. A key fact is that solutions in the most singular cases must be considered in Filippov's sense. We characterize sticking of phases in the subcritical and critical case and we exhibit a continuation criterion of classical solutions after any collision state in the supercritical regime. Second, we prove that strong solutions to these systems of differential inclusions can be recovered as singular limits of regular weights. We also provide the emergence of synchronous dynamics for the singular and regular weighted Kuramoto models.

math.AP↗
arXiv2016

Stability results, almost global generalized Beltrami fields and applications to vortex structures in the Euler equations

Strong Beltrami fields have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field itself times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) have been recently shown to be rare, in the sense that for "most" proportionality factors there are no nontrivial Beltrami fields of high enough regularity (e.g., of class $C^{6,α}$), not even locally. We show that, nevertheless, there are "many" Beltrami fields with non-constant factor, even realizing arbitrarily complicated vortex structures. The core results are an "almost global" stability theorem for strong Beltrami fields, which ensures that a global strong Beltrami field with suitable decay at infinity can be perturbed to get "many" Beltrami fields with non-constant factor of arbitrarily high regularity and defined in the exterior of an arbitrarily small ball, and a "local" stability theorem for generalized Beltrami fields, which is an analogous perturbative result that is valid for any kind of Beltrami field (not just with a constant factor) but only applies to small enough domains. The proof relies on an iterative scheme of Grad-Rubin type. For this purpose, we study the Neumann problem for the inhomogeneous Beltrami equation in exterior domains via a boundary integral equation method and we obtain Hölder estimates, a sharp decay at infinity and some compactness properties for these sequences of approximate solutions.

math.AP↗