On the double sphere model of synchronization

We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist–contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Hong and Strogatz (2011 Phys. Rev. Lett. 106 054102), or inside the sum as in Hong and Strogatz (2012 Phys. Rev. E 85 056210), but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties.

We study fast and slow clustering dynamics of Cucker–Smale ensemble with internal phase dynamics via the Cucker–Smale–Kuramoto (in short, CSK) model. The CSK model describes the emergent dynamics of flocking particles with phase dynamics. It consists of the Cucker–Smale flocking model and the Kuramoto model, and their interplay is registered in the communication weight function between particles. We present a sufficient framework for mono-cluster flocking and complete synchronization in terms of system parameters and initial data. In particular, the mono-cluster flocking will emerge exponentially fast (fast dynamics). On the other hand, when initial spatial-velocity configuration is close to the bi-cluster flocking, we also provide a sufficient framework leading to convergence of bi-cluster flocking algebraically slow depending on the decay rate of the communication weight (slow flocking dynamics). We also provide several numerical examples and compare them with analytical results.

The dynamics of ensembles of phase oscillators are usually described considering their infinite-size limit. In practice, however, this limit is fully accessible only if the Ott–Antonsen theory can be applied, and the heterogeneity is distributed following a rational function. In this work, we demonstrate the usefulness of a moment-based scheme to reproduce the dynamics of infinitely many oscillators. Our analysis is particularized for Gaussian heterogeneities, leading to a Fourier–Hermite decomposition of the oscillator density. The Fourier–Hermite moments obey a set of hierarchical ordinary differential equations. As a preliminary experiment, the effects of truncating the moment system and implementing different closures are tested in the analytically solvable Kuramoto model. The moment-based approach proves to be much more efficient than the direct simulation of a large oscillator ensemble. The convenience of the moment-based approach is exploited in two illustrative examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii) the “enlarged Kuramoto model” (endowed with nonpairwise interactions). In both systems, we obtain new results inaccessible through direct numerical integration of populations.

Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model

Introduction. Recent studies into the properties of spintronic oscillators have led to broadening their scope of practical application as devices for generating and processing signals. The practical implementation of spintronic oscillators is, however, significantly limited by their low power capacity, thus requiring synchronization between devices.Aim. Determination of conditions for the implementation of the synchronous regime of two antiferromagnetic spintronic oscillators coupled by a common current.Materials and methods. To simplify the numerical simulation of a system of coupled resistively antiferromagnetic oscillators, the method of multiple-time-scale analysis was used. This allowed a system of Kuramoto equations to be considered instead of the original system. To determine the locking band of the Kuramoto model, the homoclinic trajectory approximation method was applied.Results. A system of Kuramoto equation for the phases of partial oscillators under the influence of the inertial term and phase shift was obtained. Expressions describing the locking and synchronization band as functions of the system parameters (bias currents and sizes) were derived. The numerically simulated Kuramoto model was used to determine the bands of the synchronous and asynchronous regimes.Conclusion. The results of numerical simulations of the system of Kuramoto equations and the Adler equation for two coupled spintronic oscillators agree well with the theoretically calculated values of locking and synchronization ranges. The scheme for reducing the model of antiferromagnetic oscillators to a Kuramoto model can be further extended to the case of a larger number of coupled oscillators, which will simplify computational experiments and significantly reduce the time required for numerical simulations.

The Kuramoto model serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in large ensembles of coupled dynamical units. In this paper, we present a general framework for analytically capturing the stability and bifurcation of the collective dynamics in oscillator populations by extending the global coupling to depend on an arbitrary function of the Kuramoto order parameter. In this generalized Kuramoto model with rotation and reflection symmetry, we show that all steady states characterizing the long-term macroscopic dynamics can be expressed in a universal profile given by the frequency-dependent version of the Ott-Antonsen reduction, and the introduced empirical stability criterion for each steady state degenerates to a remarkably simple expression described by the self-consistent equation [Iatsenko et al., Phys. Rev. Lett. 110, 064101 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064101]. Here, we provide a detailed description of the spectrum structure in the complex plane by performing a rigorous stability analysis of various steady states in the reduced system. More importantly, we uncover that the empirical stability criterion for each steady state involved in the system is completely equivalent to its linear stability condition that is determined by the nontrivial eigenvalues (discrete spectrum) of the linearization. Our study provides a new and widely applicable approach for exploring the stability properties of collective synchronization, which we believe improves the understanding of the underlying mechanisms of phase transitions and bifurcations in coupled dynamical networks.

The standard Kuramoto model has been instrumental in explaining synchronization and desynchronization, two emergent phenomena often observed in biological, neuronal, and physical systems. While the Kuramoto model has turned out effective with one-dimensional oscillators, real-world systems often involve high-dimensional interacting units, such as biological swarms, necessitating a model of multidimensional oscillators. However, existing high-dimensional generalizations of the Kuramoto model commonly rely on a scalar-valued coupling strength, which limits their ability to capture the full complexity of high-dimensional interactions. This work introduces a matrix, A, to couple the interconnected components of the oscillators in a d-dimensional space, leading to a matrix-coupled multidimensional Kuramoto model that approximates a prototypical swarm dynamics by its first-order Fourier harmonics. Moreover, the matrix A introduces an inter-dimensional higher-order interaction that partly accounts for the emergence of 2 d system modes in a d-dimensional population, where each dimension can either be synchronized or desynchronized, represented by a set of almost binary order parameters. The binary system modes capture characteristic swarm behaviors such as fish milling or polarized schooling. Additionally, our findings provides a theoretical analogy to cerebral activity, where the resting state and the activated state coexist unihemispherically. It also suggests a new possibility for information storage in oscillatory neural networks.

The Kuramoto model in complex networks

The present paper is a brief survey of the history and development of the famous Kuramoto model of coupled phase oscillators. We consider several systems generalizing the classical Kuramoto model and given on symmetric oscillatory networks with different functions of interaction between the elements. We describe the collective dynamics and bifurcations of transitions between different modes of interacting elements, namely: partial and complete synchronization, global antiphase mode, slow switching, and chimera states. We show the relationship between the symmetries of networks and the existence of invariant manifolds of the system, cluster states, and more complicated collective behaviors. In part II, we plan to consider several models with nonglobal symmetric coupling.

Carpets of beating cilia represent a paradigmatic example of self-organized synchronization of noisy biological oscillators, characterized by traveling waves of cilia phase. We present a multi-scale model of a cilia carpet that comprises realistic hydrodynamic interactions between cilia computed for a chiral cilia beat pattern from unicellular Paramecium and active noise of the cilia beat. We demonstrate an abrupt loss of global synchronization beyond a characteristic noise strength. We characterize stochastic transitions between synchronized and disordered dynamics, which generalize the notion of phase slips in pairs of coupled noisy phase oscillators. Our theoretical work establishes a link between the two-dimensional Kuramoto model of phase oscillators with mirror-symmetric oscillator coupling and detailed models of biological oscillators with asymmetric, chiral interactions.

Inferring the connectivity of coupled oscillators and anticipating their transition to synchrony through lag-time analysis

Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood. Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored. In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2]. Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.

The aim of this paper is to investigate complex dynamic networks which can model high-voltage power grids with renewable, fluctuating energy sources. For this purpose we use the Kuramoto model with inertia to model the network of power plants and consumers. In particular, we analyse the synchronization transition of networks of N phase oscillators with inertia (rotators) whose natural frequencies are bimodally distributed, corresponding to the distribution of generator and consumer power. First, we start from globally coupled networks whose links are successively diluted, resulting in a random Erdös-Renyi network. We focus on the changes in the hysteretic loop while varying inertial mass and dilution. Second, we implement Gaussian white noise describing the randomly fluctuating input power, and investigate its role in shaping the dynamics. Finally, we briefly discuss power grid networks under the impact of both topological disorder and external noise sources.

We describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties. These models may be defined on any complex network where the variable at each node is an element of the unitary group U(n), or a subgroup of U(n). The nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of U(n). Synchronization of trajectories with phase locking occurs as for the Kuramoto model, for values of the coupling constant larger than a critical value, and may be measured by various order and disorder parameters. Limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables. We perform numerical computations for n = 2, for which the SU(2) group manifold is S3, for a range of natural frequencies and all-to-all coupling, in order to confirm synchronization properties. We also describe a second generalization of the Kuramoto model which is formulated in terms of real m-vectors confined to the (m − 1)-sphere for any positive integer m, and investigate trajectories numerically for the S2 model. This model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase-locked, even for large values of the coupling constant.

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.