Synchronization for the Rough Kuramoto Model

Abstract: We study the local synchronization of phases and frequencies for the Kuramoto model driven by rough noise. In particular, we prove exponential convergence towards synchronization and we give the explicit rate of convergence and quantify the size of the random basin of attraction. Furthermore, we show that the long time behavior of the system is determined by the evolution of phases' mean. Our result relies on the use of a Lyapunov function, capable of overriding the particular structure of the noise, taking in account only its intensity. Finally, we illustrate our analytical results and possible extensions with the help of numerical simulations.

• The paper establishes local exponential phase synchronization in Kuramoto oscillator networks using rough path methods under non-semimartingale noise.
• Methodologies include a Doss–Sussmann transformation and Lyapunov function analysis that links convergence rates to graph Laplacian properties and noise roughness.
• Implications extend to various network types—connected, disconnected, and signed graphs—with applications in physics, neuroscience, and engineering.

Synchronization Phenomena in the Kuramoto Model with Rough Noise

Introduction and Context

The paper addresses synchronization for the Kuramoto model under rough (non-semimartingale) stochastic perturbations, representing a significant extension from prior works concerned primarily with classical Brownian or white noise perturbations. The authors rigorously analyze the local phase and frequency synchronization for oscillator networks, deriving explicit exponential convergence rates and quantifying stochastic basins of attraction. The framework leverages rough path theory, addressing the well-posedness and stability of rough differential equation (RDE) driven Kuramoto systems, and systematically connects graph-theoretic properties (in particular, connectivity and balanced signed graphs) to synchronization robustness.

Model Formulation

The core model studied is

$$d\theta_i(t)=\left[\varpi_i+\frac{K}{N}\sum_{j=1}^N a_{ij}\sin{(\theta_j(t)-\theta_i(t))}\right]dt + (G(\theta(t))dW_t)_i, \quad i=1,\ldots,N,$$

where

• $\theta_i\in\mathbb{S}^1$ are the oscillator phases,
• $K$ is the coupling strength,
• $\varpi_i$ their intrinsic frequencies,
• $a_{ij}$ the adjacency matrix codifying the network topology,
• $G$ a noise matrix function,
• $W$ a multi-dimensional stationary increment stochastic process with H\"older regularity $\gamma>1/3$,
• and $N$ the network size.

The generality of the noise includes, but is not limited to, fractional Brownian motion with $H>\tfrac{1}{3}$, substantially broadening the applicability beyond Itô/Stratonovich settings.

Main Results: Synchronization Under Rough Noise

Lyapunov Analysis and Exponential Convergence

A central achievement is a rigorous proof of local exponential synchronization in phase for connected graphs with identical frequencies. The methodology centers on the Doss–Sussmann transformation—crucial to reducing RDEs to ODEs with random coefficients for Lyapunov analysis. The Lyapunov function $\theta_i\in\mathbb{S}^1$0 (in the orthogonal complement of the mean phase direction) is shown to satisfy a drift inequality controlled by the positive spectrum of the graph Laplacian and the roughness and intensity of the noise. The explicit convergence rate $\theta_i\in\mathbb{S}^1$1 derived satisfies

$\theta_i\in\mathbb{S}^1$2

where

• $\theta_i\in\mathbb{S}^1$3 is the algebraic connectivity (smallest nonzero eigenvalue of the Laplacian),
• $\theta_i\in\mathbb{S}^1$4 quantifies contraction around the equilibrium domain,
• $\theta_i\in\mathbb{S}^1$5 bounds $\theta_i\in\mathbb{S}^1$6 and its derivatives,
• $\theta_i\in\mathbb{S}^1$7 counts roughness-induced "greedy" intervals.

The result holds robustly for noise intensities below a threshold $\theta_i\in\mathbb{S}^1$8 that depends explicitly on system parameters and roughness of $\theta_i\in\mathbb{S}^1$9, and the explicit size of the attraction basin is established.

Synchronization Structure and Long-Time Behavior

Notably, the authors prove that for symmetric $K$0 and connected graphs, the mean phase asymptotically governs the dynamics. When the perturbation $K$1 satisfies a zero-mean (rotational invariance) condition across oscillators, the synchronized phase cluster is contained in a deterministic hyperplane. In contrast, more general $K$2 leads to a stochastic random attractor governed by the evolution of the mean of the noise functional.

Generalization to Disconnected and Signed Graphs

A substantial technical contribution is extending local exponential synchronization to each connected component in disconnected graphs. For balanced signed graphs, the analysis reveals splitting into exactly two synchronized clusters (phases separated by $K$3). The authors use a nontrivial change of variables, leveraging the balance structure to reduce the dynamics to tractable, positively coupled form.

Regarding the frequency synchronization with non-identical $K$4, the paper proves—under mild conditions and provided the phases remain within a contraction domain—that the (possibly distributional) frequencies synchronize in the sense of convergence of the time-averaged increments, even though the phases do not synchronize.

Technical Aspects

Rough Path Framework

The use of rough path theory is essential to handle non-semimartingale drivers such as fBm with $K$5 and more general noise classes, ensuring global existence and uniqueness for the RDE system. This includes a careful treatment of controlled rough paths, Gubinelli derivatives, and the Greedy partitioning method for rough norm control.

Graph-Theoretic Bounds

The explicit dependencies of synchronization rates and attraction basin on topological invariants (algebraic connectivity and Cheeger constant) are leveraged, and the authors discuss the computational implications and bounds for commonly used network motifs.

Explicit Quantitative Bounds and Ergodicity

The basin of attraction is quantitatively expressed in terms of the initial condition spread, rough path moments, and contraction rates. Assumptions of ergodicity of the noise process underpin almost-sure statements for typical realizations, with pathwise estimates provided where ergodicity is dropped.

Numerical Validation

The numerical section demonstrates sharpness and limitations of the theoretical conditions, e.g., that synchronization fails for initial splits exceeding phase difference $K$6, but can be more robust in practice—indicating that current conservative bounds for basin size may be improved with refined Lyapunov constructions. Importantly, the authors present evidence that synchronization survives violations of rotational invariance assumptions, instead resulting in convergence to a random attractor rather than a deterministic value.

Simulations further confirm the splitting effect for balanced signed networks and the criticality of noise intensity (controlled by $K$7) for the persistence or destruction of synchronization. The authors also analyze the behavior of instantaneous frequencies, clarifying the distinction between classical and rough settings.

Implications and Future Directions

The results establish the robust persistence of synchronized collective behavior in coupled oscillator networks even under highly irregular, non-Markovian perturbations, provided network connectivity and noise intensity-magnitude are adequate. This has immediate implications for applications in physics, neuroscience, and engineering where environmental stochasticity or endogenous fluctuations are non-smooth and not well captured by semimartingale models.

From a theoretical perspective, the analysis strengthens connections between synchronization, stability theory for non-Markovian RDEs, and spectral graph theory. The framework is positioned for further extensions, including:

• development of less conservative Lyapunov functions accounting for the algebraic structure of the noise,
• extension to global (rather than local) synchronization regimes for broad initial data,
• rigorous treatment of random attractors in the absence of rotational invariance,
• investigation of multifrequency clusters, frustration, and multi-cluster attractors in general signed or directed networks,
• specialized probabilistic analysis in the classical Brownian case leveraging Markov and martingale tools.

Conclusion

The paper provides a rigorous, quantitative, and general analytical framework for the synchronization of rough-noise-driven Kuramoto oscillator networks. By making explicit the precise impact of both network structure and roughness properties of the external perturbations, the authors lay a foundation for further exploration of robustness and bifurcation phenomena in network synchronization beyond the classical stochastic and deterministic paradigms.