Synchronization of Kuramoto model in a high-dimensional linear space

In this Letter, Kuramoto model in a high-dimensional linear space is investigated. Some results on the equilibria and synchronization of the classical Kuramoto model are generalized to the high-dimensional Kuramoto model. It is proved that, if the interconnection graph is connected and all the initial states lie in a half part of the state space, the synchronization can be achieved. Finally, numerical simulations are given to validate the obtained theoretical results.     

Complete synchronization of Kuramoto oscillators with finite inertia

We present an approach based on Gronwall’s inequalities for the asymptotic complete phase-frequency synchronization of Kuramoto oscillators with finite inertia. For given finite inertia and coupling strength, we present admissible classes of initial configurations and natural frequency distributions, which lead to the complete phase-frequency synchronization asymptotically. For this, we explicitly identify invariant regions for the Kuramoto flow, and derive second-order Gronwall’s inequalities for the evolution of phase and frequency diameters. Our detailed time-decay estimates for phase and frequency diameters are independent of the number of oscillators. We also compare our analytical results with numerical simulations.     

Local effects in synchronization on an extended network model

We investigate collective behaviors of coupled phase oscillators on an extended network model which can develop two fundamentally different topologies, scale-free or exponential. Each component of the network is assumed as an oscillator and that each interacts with the others following the Kuramoto model. The order parameters that measure synchronization of phases and frequencies are computed by means of dynamic simulations. It is found that system's collective behaviors exhibit strong dependence on local events: addition of new links will improve network synchronizability while rewiring of links will decrease synchronization.     

Synchronization of Coupled Oscillators on Newman-Watts Small-World Networks

We investigate the collection behaviour of coupled phase oscillators on Newman-Watts small-world networks in one and two dimensions. Each component of the network is assumed as an oscillator and each interacts with the others following the Kuramoto model We then study the onset of global synchronization of phases and frequencies based on dynamic simulations and finite-size scaling. Both the phase and frequency synchronization are observed to emerge in the presence of a tiny fraction of shortcuts and enhanced with the increases of nearest neighbours and lattice dimensions.     

The synchronization transition in correlated oscillator populations

The synchronization transition of correlated ensembles of coupled Kuramoto oscillators on sparse random networks is investigated. Extensive numerical simulations show that correlations between the native frequencies of adjacent oscillators on the network systematically shift the critical point as well as the critical exponents characterizing the transition. Negative correlations imply an onset of synchronization for smaller coupling, whereas positive correlations shift the critical coupling towards larger interaction strengths. For negatively correlated oscillators the transition still exhibits critical behaviour similar to that of the all-to-all coupled Kuramoto system, while positive correlations change the universality class of the transition depending on the correlation strength. Crucially, the paper demonstrates that the synchronization behaviour is not only determined by the coupling architecture, but also strongly influenced by the oscillator placement on the coupling network.     

Dynamics of clustering patterns in the Kuramoto model with unidirectional coupling

We study the synchronization transition in the Kuramoto model by considering a unidirectional coupling with a chain structure. The microscopic clustering features are characterized in the system. We identify several clustering patterns for the long-time evolution of the effective frequencies and reveal the phase transition between them. Theoretically, the recursive approach is developed in order to obtain analytical insights; the essential bifurcation schemes of the clustering patterns are clarified and the phase diagram is illustrated in order to depict the various phase transitions of the system. Furthermore, these recursive theories can be extended to a larger system. Our theoretical analysis is in agreement with the numerical simulations and can aid in understanding the clustering patterns in the Kuramoto model with a general structure.     

Contractivity of Transport Distances for the Kinetic Kuramoto Equation

We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of non-identical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein \(p\) -distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous \(L^1\) -contraction properties of the Kuramoto phase model.     

Effects of frustration on explosive synchronization

In this study, we consider the emergence of explosive synchronization in scale-free networks by considering the Kuramoto model of coupled phase oscillators. The natural frequencies of oscillators are assumed to be correlated with their degrees and frustration is included in the system. This assumption can enhance or delay the explosive transition to synchronization. Interestingly, a de-synchronization phenomenon occurs and the type of phase transition is also changed. Furthermore, we provide an analytical treatment based on a star graph, which resembles that obtained in scale-free networks. Finally, a self-consistent approach is implemented to study the de-synchronization regime. Our findings have important implications for controlling synchronization in complex networks because frustration is a controllable parameter in experiments and a discontinuous abrupt phase transition is always dangerous in engineering in the real world.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Synchronization in large directed networks of coupled phase oscillators

We study the emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks. We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider the case of networks with mixed positive-negative coupling strengths. We compare our theory with numerical simulations and find good agreement.     

Explosive synchronization of multi-layer frequency-weighted coupled complex systems

Synchronization is a phenomenon that is ubiquitous in engineering and natural ecosystems.The study of explosive synchronization on a single-layer network gives the critical transition coupling strength that causes explosive synchronization.However, no significant findings have been made on multi-layer complex networks.This paper proposes a frequency-weighted Kuramoto model on a two-layer network and the critical coupling strength of explosive synchronization is obtained by both theoretical analysis and numerical validation.It is found that the critical value is affected by the interaction strength between layers and the number of network oscillators.The explosive synchronization will be hindered by enhancing the interaction and promoted by increasing the number of network oscillators.Our results have importance across a range of engineering and biological research fields.     

Explosive synchronization in clustered scale-free networks: Revealing the existence of chimera state

The collective dynamics of Kuramoto oscillators with a positive correlation between the incoherent and fully coherent domains in clustered scale-free networks is studied. Emergence of chimera states for the onsets of explosive synchronization transition is observed during an intermediate coupling regime when degree-frequency correlation is established for the hubs with the highest degrees. Diagnostic of the abrupt synchronization is revealed by the intrinsic spectral properties of the network graph Laplacian encoded in the heterogeneous phase space manifold, through extensive analytical investigation, presenting realistic MC simulations of nonlocal interactions in discrete time dynamics evolving on the network.     

The role of degree-weighted couplings in the synchronous onset of Kuramoto oscillator networks

This paper investigates the role of asymmetrical degree-dependent weighted couplings in synchronization of a network of Kuramoto oscillators, where the conditions of coupling criticality for the onset of phase synchronization in degree-weighted complex networks are arrived at. The numerical simulations visualize that for networks having power-law or exponential degree distributions, asymmetrical degree-weighted couplings (with increasing weighting exponent β) increases the critical coupling to achieve the onset of phase synchronization in the networks.     

Effects of Correlation between Network Structure and Dynamics of Oscillators on Synchronization Transition in a Kuramoto Model on Scale-Free Networks

A recent study has found an explosive synchronization in a Kurammoto model on scale-free networks when the natural frequencies of oscillators are equal to their degrees. In this work, we introduce a quantity to characterize the correlation between the structural and the dynamical properties and investigate the impacts of the correlation on the synchronization transition in the Kuramoto model on scale-free networks. We find that the synchronization transition may be either a continuous one or a discontinuous one depending on the correlation and that strong correlation always postpones both the transitions from the incoherent state to a synchronous one and the transition from a synchronous state to the incoherent one. We find that the dependence of the synchronization transition on the correlation is also valid for other types of distributions of natural frequency.     

Determining anomalous dynamic patterns in price indexes of the London Metal Exchange by data synchronization

Data synchronization based on the Kuramoto model for collective synchronization and hypothesis testing based on the rank test combined with the random shuffling surrogate method are applied to finding major feature patterns of weekly nonferrous metal returns from the time series of daily spot and futures price indexes in the London Metal Exchange since 1989. Our results suggest the existence of day-of-the-week anomalies in the metal returns. We conjecture that such anomalies are large-scale manifestations of synchronously accumulated risk-aversive actions of individual market players.     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.     

The Influence of Network Properties on the Synchronization of Kuramoto Oscillators Quantified by a Bayesian Regression Analysis

The influence of the network structure on the emergence of collective dynamical behavior is an important topic of research that has not been fully understood yet. In the current work, it is shown how statistical regression analysis can be considered to address this issue. The regression model proposed suggests that the average shortest path length is the network property most influencing the degree of synchronization of Kuramoto oscillators. Moreover, this model revealed to be very accurate, being the predicted and measured values of synchronization highly correlated. Therefore, the regression modeling allows predicting the values of the dynamic variable in terms of network structure.     

Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc. We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N→∞ limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N→∞ limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K _c . We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K _c .     

Partial entrainment in the finite Kuramoto-Sakaguchi model

Although modifications of the Kuramoto model have been the subject of extensive research, the model itself is not yet fully understood. We offer several results and observations, some analytic, others through simulations. We derive a sufficient condition for the existence of a solution exhibiting partial entrainment with respect to a given subset of oscillators; the result also implies persistence of the entrainment behavior under perturbations.The critical values of the coupling strength, defining the transitions between different forms of partial entrainment, are predicted by an analytical approximation, based on the fact that oscillators with large differences in their natural frequencies have little influence on each other’s entrainment behavior; the predictions agree with the actual values, obtained by simulations.We indicate (by simulations) that entrainment can disappear with increasing coupling strength, and that, in arrays of Josephson junctions, a similar phenomenon can be observed, where it is also possible that a junction leaving one entrained subset joins another entrained subset.     

Modelling microwave scattering from long curved leaves

This paper describes the theoretical simulations carried out with a model of the backscattering coefficient of crops, where the leaf geometry is represented by a curved rectangular dielectric sheet. A general formulation is introduced for the bistatic scattering cross section of the curved sheet, and numerical results based on this approach are compared with those obtained by considering disc shaped leaves.