Center manifold reduction for large populations of globally coupled phase oscillators

A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is sin θ, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not sin θ, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.     

Generating macroscopic chaos in a network of globally coupled phase oscillators

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.     

Phase clustering in globally coupled photochemical oscillators

We experimentally investigate the formation of clusters in a population of globally coupled photochemical oscillators. The system consists of catalytic micro-particles in Belousov-Zhabotinsky solution and the coupling exploits the excitatory properties of light; an increase in the light intensity leads to excitation (“firing") of an oscillator. As the coupling strength is increased, a transition occurs from incoherence to clustering, whereby the oscillators split into synchronised groups, to complete synchronisation. Multistability is observed between a one-phase cluster (fully synchronised group) and two-phase clusters (two groups with the same frequency but different phases). The results are reproduced in simulations and we demonstrate that the heterogeneity of the population as well as the relaxational nature of the oscillators is important in the observation of clusters. We also examine the exploitation of the phase model for the prediction of clusters in experiments.     

Finite-size-induced transition in ensemble of globally coupled oscillators

The collective behavior of overdamped nonlinear noise-driven oscillators coupled via mean field is investigated numerically. When a coupling constant is increased, a transition in the dynamics of the mean field is observed. This transition scales with the number of oscillators and disappears when this number tends to infinity. Analytical arguments explaining the observed scaling are presented.     

Clusters and switchers in globally coupled photochemical oscillators

We experimentally investigate the transition to synchronization in a population of photochemical oscillators with weak global coupling. Above a critical coupling strength the oscillators join a one-phase group or two-phase clusters. The number of oscillators in each cluster depends on the initial phase distribution, and irregular switching of oscillators between clusters is observed. The fully synchronized state emerges above a second critical coupling strength. In agreement with earlier theory, the experiments demonstrate the importance of population heterogeneity in cluster multistability.     

Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D～O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D～O(1/N(a)) with a certain constant a>0 in the coherent regime and D～O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.     

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S₃-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z₃-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.     

Diffusion-induced inhomogeneity in globally coupled oscillators: swing-by mechanism

It is shown that for random initial conditions, a large population of identical and sufficiently nonisochronous Stuart-Landau oscillators coupled globally and diffusively exhibits inhomogeneity in a resonant way as the diffusive coupling is intensified, where the diffusive coupling constant is real. A category of inhomogeneous (nonsynchronized) solutions is analytically shown to exist, which is either periodic or quasiperiodic.     

Controlling synchronization in an ensemble of globally coupled oscillators

We propose a technique to control coherent collective oscillations in ensembles of globally coupled units (self-sustained oscillators or maps). We demonstrate numerically and theoretically that a time delayed feedback in the mean field can, depending on the parameters, enhance or suppress the self-synchronization in the population. We discuss possible applications of the technique.     

Clusters in an assembly of globally coupled bistable oscillators

We study the dynamics of an assembly of globally coupled bistable elements. We show that bistability of elements results in some new features of clustering in the assembly when there is global coupling. We provide conditions for the existence of stable amplitude-phase clusters and splay-phase states. Received 12 June 1998 and Received in final form 30 November 1998     

Synchronization in a system of globally coupled oscillators with time delay

We study the synchronization phenomena in a system of globally coupled oscillators with time delay in the coupling. The self-consistency equations for the order parameter are derived, which depend explicitly on the amount of delay. Analysis of these equations reveals that the system in general exhibits discontinuous transitions in addition to the usual continuous transition, between the incoherent state and a multitude of coherent states with different synchronization frequencies. In particular, the phase diagram is obtained on the plane of the coupling strength and the delay time, and ubiquity of multistability as well as suppression of the synchronization frequency is manifested. Numerical simulations are also performed to give consistent results.     

Loss enhanced phase locking in coupled oscillators

Phase locking, which is achieved by transferring some energy from one oscillator to the others, strongly depends on the coupling strength between the oscillators. Typically, the coupling strength must be above a certain threshold in order to achieve phase locking. Here we show how this threshold can be significantly reduced when phase-dependent losses are introduced into the oscillators. Specifically, the coupling strength can be reduced by at least an order of magnitude, thereby substantially decreasing the needed transfer of energy between oscillators. The resulting enhancement of phase locking does not only influence the laser research area, but also affects many other areas that involve coupled ensembles.     

Detecting phase synchronization between coupled non-phase-coherent oscillators

We compare two methods for detecting phase synchronization in coupled non-phase-coherent oscillators. One method is based on the locking of self-sustained oscillators with an irregular signal. The other uses trajectory recurrences in phase space. We identify the pros and cons of both methods and propose guidelines to detect phase synchronization in data series.     

Desynchronization transitions in nonlinearly coupled phase oscillators

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.     

Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering

Experiments on chaotically oscillating arrays of 64 nickel electrodes in sulfuric acid were carried out. External resistors in parallel and series are added to vary the extent of global coupling among the oscillators without changing the other properties of the system. The array is heterogeneous due to small variations in the properties of the electrodes and there is also a small amount of noise. The addition of global coupling transforms a system of independent elements to a state of complete synchronization. At intermediate coupling strengths stable clusters, or condensates of elements, form. All the elements in a cluster follow the same chaotic trajectory but each cluster has its own dynamics; the system is thus temporally chaotic but spatially ordered. Many cluster configurations occur under the same conditions and transitions among them can be produced. For values of the coupling parameter on either side of the stable cluster region a non-stationary behavior occurs in which clustered and synchronized states alternately form and break up. Some statistical properties of the cluster states are determined. (c) 2000 American Institute of Physics.     

Three types of transitions to phase synchronization in coupled chaotic oscillators

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.     

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.     

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.