Kuramoto Oscillators Research Articles

Noise-to-State Stability of Random Coupled Kuramoto Oscillators via Feedback Control

This paper is intended to study noise-to-state stability in probability (NSSP) for random coupled Kuramoto oscillators with input control (RCKOIC). A feedback control is designed, which makes us give the existence and uniqueness of a solution for RCKOIC. Based on Kirchhoff’s matrix tree theorem in graph theory, an original and appropriate Lyapunov function for RCKOIC is established. With the help of the Lyapunov method and by resorting to some analysis skills, NSSP for RCKOIC with an arbitrarily coupled topological structure and second-order moment process stochastic disturbance is acquired. Finally, the effectiveness of the obtained results is verified by a numerical test and its simulation process.

Chimera states in pulse-coupled oscillator systems

Coupled oscillator systems can lead to states in which synchrony and chaos coexist. These states are called “chimera states.” The mechanism that explains the occurrence of chimera states is not well understood, especially in pulse-coupled oscillators. We study a variation of a pulse-coupled oscillator model that has been shown to produce chimera states, demonstrate that it reproduces several of the expected chimera properties, like the formation of multiple heads and the ability to control the natural drift that Kuramoto's chimera states experience in a ring, and explain how chimera states emerge. Our contribution is defining the model, analyzing the mechanism leading to chimera states, and comparing it with examples from the field of Kuramoto oscillators. Published by the American Physical Society 2024

Synchronized States of Power Grids and Oscillator Networks by Convex Optimization

Synchronization is essential for the operation of ac power systems: all generators in the power grid must rotate with fixed relative phases to enable a steady flow of electric power. Understanding the conditions for and the limitations of synchronization is of utmost practical importance. In this article, we propose a novel approach to computing and analyzing the stable stationary states of a power grid or a network of Kuramoto oscillators in terms of a convex optimization problem. This approach allows us to systematically compute stable states where the phase difference across an edge does not exceed π/2. Furthermore, the optimization formulation allows us to rigorously establish certain properties of synchronized states and to bound the error in the widely used linear power flow approximation. Published by the American Physical Society 2024

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins.

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: They stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by markedly reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives.

Chimera-inspired dynamics: When higher-order interactions are expressed differently.

The exploration of chimera-inspired dynamics in nonlocally coupled networks of Kuramoto oscillators with higher-order interactions is still in its nascent stages. Concurrently, the investigation of collective phenomena in higher-order interaction networks is gaining attraction. Here, we observe that hypergraph networks tend to synchronize through lower-order interactions, whereas simplicial complex networks exhibit a preference for higher-order interactions. This observation suggests that higher-order representations manifest substantial differences in chimera-inspired synchronization regions. Moreover, we introduce an explicit expression for identifying the chimera state. With a comprehensive basin stability analysis and the interplay of pairwise and higher-order interaction strengths, the emergence of the chimera state is inherent in high-order interaction networks. Our findings contribute to the understanding of chimera-inspired dynamics in higher-order interaction networks.

Neuromodulatory effects on synchrony and network reorganization in networks of coupled Kuramoto oscillators.

Neuromodulatory processes in the brain can critically change signal processing on a cellular level, leading to dramatic changes in network level reorganization. Here, we use coupled nonidentical Kuramoto oscillators to investigate how changes in the shape of phase response curves from Type 1 to Type 2, mediated by varying ACh levels, coupled with activity-dependent plasticity may alter network reorganization. We first show that, when plasticity is absent, the Type 1 networks with symmetric adjacency matrix, as expected, exhibit asynchronous dynamics with oscillators of the highest natural frequency robustly evolving faster in terms of their phase dynamics. However, interestingly, Type 1 networks with anasymmetric connectivity matrix can produce stable synchrony (so-called splay states) with complex phase relationships. At the same time, Type 2 networks synchronize independent ofthe symmetry of their connectivity matrix, with oscillators locked so that those with higher natural frequency have a constant phase lead as compared to those with lower natural frequency. This relationship establishes a robust mapping between the frequency and oscillators' phases in the network, leading to structureandfrequency mapping when plasticity is present. Finally, we show that biologically realistic, phase-locking dependent, connection plasticity naturally produces splay states in Type 1 networks that do not display the structure-frequency reorganization observed in synchronized Type II networks. These results indicate thatthe formation of splay states in the brain could be a common phenomenon.

Finite-size effect in Kuramoto oscillators with higher-order interactions.

Finite-size systems of a Kuramoto model display intricate dynamics, especially in the presence of multi-stability where both coherent and incoherent states coexist. We investigate such a scenario in globally coupled populations of Kuramoto phase oscillators with higher-order interactions and observe that fluctuations inherent to finite-size systems drive the transition to the synchronized state before the critical point in the thermodynamic limit. Using numerical methods, we plot the first exit-time distribution of the magnitude of a complex order parameter and obtain numerical transition probabilities across various system sizes. Furthermore, we extend this study to a two-population oscillator system, and, using the velocity field of the associated order parameters, show the emergence of a new fixed point corresponding to a partially synchronized state arising due to the finite-size effect, which is absent in the thermodynamics limit.

Third order interactions shift the critical coupling in multidimensional Kuramoto models

The study of higher order interactions in the dynamics of Kuramoto oscillators has been a topic of intense research. Previous works have demonstrated that such interactions can give rise to interesting new phenomena such as multi-stability and synchronization even if the interaction between oscillators is repulsive. Here we consider higher order interactions in the multidimensional Kuramoto model where pairs (1-simplex), triplets (2-simplex) and quadruples (3-simplex) of oscillators interact simultaneously with different coupling strengths, k1, k2 and k3, respectively. For the types of asymmetric interactions considered, we show that three body terms shift the critical coupling for synchronization towards higher values, except in 2 dimensions where a cancellation occurs. However, after the transition, three and four body interactions combine to facilitate synchronization. We also show that, for fixed values of k2 and k3, and fully connected networks, the behavior of the order parameter r(k1) is described by a universal curve given by its value at k2=k3=0, shifted along the k1 axis. Similar to the 2-dimensional case with asymmetric interactions, bi-stability and hysteresis develop for large enough higher order interactions. Multi-stability, typical of symmetric higher order interactions, is not found. We show simulations in three and four dimensions to illustrate the dynamics.

Synchronization of high-dimensional Kuramoto-oscillator networks with variable-gain impulsive coupling on the unit sphere

Kuramoto models (KMs) in scalar or high-dimensional form can describe the synchronization phenomenon for large populations of coupled oscillators in networks of dynamical systems such as power grids, satellite mobile sensing networks, etc. However, these models are developed based on continuous-time coupling among oscillators, which is not applicable to networks where the coupling between oscillators occurs only at impulsive instants. Herein, we propose for the first time a generalized high-dimensional Kuramoto oscillator network (HDKON) with variable-gain impulsive coupling on the unit sphere. The proposed HDKON can be reduced to a scalar form comprising a sinusoidal function, thereby generalizing the scalar KM in both temporal and spatial domains. Furthermore, we provide some variation coefficients of the synchronization errors for the oscillator pairs at impulsive instants, and derive a sufficient condition for the exponential synchronization of the HDKON with identical natural frequency. Moreover, we consider an HDKON with a central oscillator and demonstrate that peripheral oscillators almost globally exponentially synchronize to the central oscillator under a sufficient condition. Finally, numerical simulations are performed to verify the main theoretical results.

On mathematical analysis of synchronization of bidirectionally coupled Kuramoto oscillators under inertia effect

In this article, we analyze the phase synchronization and frequency synchronization for the bidirectionally coupled Kuramoto model under the effect of inertia. Unlike the classical Kuramoto model equipped with all-to-all coupled interaction, in the setting of this model, each oscillator θi only interacts directly with θi+1 and θi−1. The bidirectional interaction is a typical setting of the concatenation in power systems. Additionally, it is necessary to impose the effect of inertia in the Kuramoto model in the applications such as power systems and Josephson junction array. In this article, we first present a theory of the global convergence for frequency synchronization for the identical case. For the non-identical case, we prove that the second-order bidirectionally coupled Kuramoto model exhibits a frequency synchronization if the coupling strength is large, inertia is small, and all oscillators are initially confined to a sector. We emphasize that the arc length of this sector possesses a positive lower bound which is independent of the number of oscillators. If, in addition, all natural frequencies are identical, we further show that the phase synchronization emerges. Moreover, we demonstrate the numerical simulations to support the main results. On the other hand, we observe that the model equipped with large inertia can exhibit the synchronization. Exploring the synchronization theory for large inertia case is left as the future work.

Synchronization transitions in adaptive Kuramoto-Sakaguchi oscillators with higher-order interactions.

Coupled oscillators models help us in understanding the origin of synchronization phenomenon prevalent in both natural and artificial systems. Here, we study the coupled Kuramoto oscillator model having phase lag and adaptation in higher-order interactions. We find that the type of transition to synchronization changes from the first-order to second-order through tiered synchronization depending on the adaptation parameters. Phase lag enables this transition at a lower exponent of the adaptation parameters. Moreover, an interplay between the adaptation and phase lag parameters eliminates tiered synchronization, facilitating a direct transition from the first to second-order. In the thermodynamic limit, the Ott-Antonsen approach accurately describes all stationary and (un)stable states, with analytical results matching those obtained from numerical simulations for finite system sizes.

Critical points, stability, and basins of attraction of three Kuramoto oscillators with isosceles triangle network

We investigate the Kuramoto model with three oscillators interconnected by an isosceles triangle network. The characteristic of this model is that the coupling connections between the oscillators can be either attractive or repulsive. We list all critical points and investigate their stability. We furthermore present a framework studying convergence towards stable critical points under special coupled strengths. The main tool is the linearization and the monotonicity arguments of oscillator diameter.

Early Predictor for the Onset of Critical Transitions in Networked Dynamical Systems

Numerous natural and human-made systems exhibit critical transitions whereby slow changes in environmental conditions spark abrupt shifts to a qualitatively distinct state. These shifts very often entail severe consequences; therefore, it is imperative to devise robust and informative approaches for anticipating the onset of critical transitions. Real-world complex systems can comprise hundreds or thousands of interacting entities, and implementing prevention or management strategies for critical transitions requires knowledge of the exact condition in which they will manifest. However, most research so far has focused on low-dimensional systems and small networks containing fewer than ten nodes or has not provided an estimate of the location where the transition will occur. We address these weaknesses by developing a deep-learning framework which can predict the specific location where critical transitions happen in networked systems with size up to hundreds of nodes. These predictions do not rely on the network topology, the edge weights, or the knowledge of system dynamics. We validate the effectiveness of our machine-learning-based framework by considering a diverse selection of systems representing both smooth (second-order) and explosive (first-order) transitions: the synchronization transition in coupled Kuramoto oscillators; the sharp decline in the resource biomass present in an ecosystem; and the abrupt collapse of a Wilson-Cowan neuronal system. We show that our method provides accurate predictions for the onset of critical transitions well in advance of their occurrences, is robust to noise and transient data, and relies only on observations of a small fraction of nodes. Finally, we demonstrate the applicability of our approach to real-world systems by considering empirical vegetated ecosystems in Africa. Published by the American Physical Society 2024

Complete and Partial Synchronization of Two-Group and Three-Group Kuramoto Oscillators

Complete and Partial Synchronization of Two-Group and Three-Group Kuramoto Oscillators

Phase holonomy underlies puzzling temporal patterns in Kuramoto models with two sub-populations.

We present a geometric investigation of curious dynamical behaviors previously reported in Kuramoto models with two sub-populations. Our study demonstrates that chimeras and traveling waves in such models are associated with the birth of geometric phase. Although manifestations of geometric phase are frequent in various fields of physics, this is the first time (to our best knowledge) that such a phenomenon is exposed in ensembles of Kuramoto oscillators or, more broadly, in complex systems.

Local exponential synchronization rate of commutative Kuramoto oscillators on spheres

For proportional Kuramoto oscillators on spheres, the local exponential synchronization rate for complete phase synchronization was obtained in existing literature. But for the general nonidentical Kuramoto oscillators on spheres, the problem of local exponential synchronization rate has not been solved. In this paper, for commutative Kuramoto oscillators on spheres, the matrix differential equations involving the synchronization errors are constructed, and the local exponential synchronization rate is obtained by calculating the spectrum of the linearized error system limited to an invariant subspace.

Stability of twisted states in power-law-coupled Kuramoto oscillators on a circle with and without time delay.

Other than the fully synchronized state, a twisted state can also be an equilibrium solution in the Kuramoto model and its variations. In the present work, we explore the stability of the twisted state in Kuramoto oscillators put on a rim of a planar circle in the two-dimensional space in the presence of power-law decaying interaction strength (∼r^{-α} with the distance r) and time delays due to a finite speed of information transfer. For example, our model can phenomenologically mimic a large sports stadium where many people try to sing or clap their hands in unison; the sound intensity decays with the distance and there can exist a time delay proportional to the distance due to the finiteness of sound speed. We first consider the case without the time delay effect and numerically find that stable twisted states emerge when the exponent α exceeds a critical value of α_{c}≈2. In other words, for α<α_{c}, the fully synchronized state, not the twisted state, is the only stable fixed point of the dynamics. In our analytic approach, we also derive an equation for α_{c} and discuss its solutions. In the presence of time delay, we find that it is possible that the synchronized state becomes unstable while twisted states are stable.

On synchronization of third-order power systems governed by second-order networked Kuramoto oscillators incorporating first-order magnitude dynamics

This study explores the sufficient conditions for achieving synchronization in the third-order one-axis generator models in modern power systems, characterized by the second-order networked Kuramoto oscillators integrating the first-order amplitude dynamics. First, we provide detailed descriptions of dynamic models for this particular class of third-order power systems, which are governed by second-order networked Kuramoto oscillators incorporating first-order voltage magnitude dynamics. Then, we show that if the phase difference of oscillators is less than π2 and the initial amplitudes are positive, then the amplitudes are bounded and possess a positive lower bound. Furthermore, under specific conditions on the coupling strength, inertia, and oscillator parameters, we prove that the third-order model exhibits a frequency synchronization and the derivatives of amplitudes converge to zero. Numerical simulations are performed to support our main results.

Binary pattern retrieval with Kuramoto-type oscillators via a least orthogonal lift of three patterns

Abstract Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.

Neuronal activity induces symmetry breaking in neurodegenerative disease spreading

Dynamical systems on networks typically involve several dynamical processes evolving at different timescales. For instance, in Alzheimer’s disease, the spread of toxic protein throughout the brain not only disrupts neuronal activity but is also influenced by neuronal activity itself, establishing a feedback loop between the fast neuronal activity and the slow protein spreading. Motivated by the case of Alzheimer’s disease, we study the multiple-timescale dynamics of a heterodimer spreading process on an adaptive network of Kuramoto oscillators. Using a minimal two-node model, we establish that heterogeneous oscillatory activity facilitates toxic outbreaks and induces symmetry breaking in the spreading patterns. We then extend the model formulation to larger networks and perform numerical simulations of the slow-fast dynamics on common network motifs and on the brain connectome. The simulations corroborate the findings from the minimal model, underscoring the significance of multiple-timescale dynamics in the modeling of neurodegenerative diseases.