The Kuramoto model is a paradigm for studying oscillator networks with interplay between coupling tending towards synchronization, and heterogeneity in the oscillator population driving away from synchrony. In continuum versions of this model an oscillator population is represented by a probability density on the circle. Ott and Antonsen identified a special class of densities which is invariant under the dynamics and on which the dynamics are low-dimensional and analytically tractable. The reduction to the OA manifold has been used to analyze the dynamics of many variants of the Kuramoto model. To address the fundamental question of whether the OA manifold is attracting, we develop a systematic technique using weighted averages of Poisson measures for analyzing dynamics off the OA manifold. We show that for models with a finite number of populations, the OA manifold is {\it not} attracting in any sense; moreover, the dynamics off the OA manifold is often more complex than on the OA manifold, even at the level of macroscopic order parameters. The OA manifold consists of Poisson densities $ρ_ω$. A simple extension of the OA manifold consists of averages of pairs of Poisson densities; then the hyperbolic distance between the centroids of each Poisson pair is a dynamical invariant (for each $ω$). These conserved quantities, defined on the double Poisson manifold, are a measure of the distance to the OA manifold. This invariance implies that chimera states, which have some but not all populations in sync, can never be stable in the full state space, even if stable in the OA manifold. More broadly, our framework facilitates the analysis of multi-population continuum Kuramoto networks beyond the restrictions of the OA manifold, and has the potential to reveal more intricate dynamical behavior than has previously been observed for these networks.
We are interested in low-dimensional dynamics in an ensemble of coupled nonidentical generalized oscillators on the 3-sphere. The system of governing equations for such an ensemble is referred to as non-Abelian Kuramoto model in the literature. We establish an analogue (or an extension) of the Ott-Antonsen (OA) result for this model.
The Ott--Antonsen ansatz is a powerful tool to extract the behaviors of coupled phase oscillators, but it imposes a strong restriction on the initial condition. Herein, a systematic extension of the Ott--Antonsen ansatz is proposed to relax the restriction, enabling the systematic approximation of the behavior of a globally coupled phase oscillator system with an arbitrary initial condition. The proposed method is validated on the Kuramoto--Sakaguchi model of identical phase oscillators. The method yields cluster and chimera-like solutions that are not obtained by the conventional ansatz.
A novel generalization of the Winfree model of globally coupled phase oscillators, representing phase reduction under finite coupling, is studied analytically. We consider interactions through a non-infinitesimal (or finite) phase-response curve (PRC), in contrast to the infinitesimal PRC of the original model. For a family of non-infinitesimal PRCs, the global dynamics is captured by one complex-valued ordinary differential equation resorting to the Ott-Antonsen ansatz. The phase diagrams are thereupon obtained for four illustrative cases of non-infinitesimal PRC. Bistability between collecti…
We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, CHAOS 18, 037113 (2008)] and appear appropriate for constructing the perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulant…
In a recent paper [Chaos 30, 073139 (2020)] we analyzed an extension of the Winfree model with nonlinear interactions. The nonlinear coupling function Q was mistakenly identified with the non-infinitesimal phase-response curve (PRC). Here, we asses to what extent Q and the actual PRC differ in practice. By means of numerical simulations, we compute the PRCs corresponding to the Q functions previously considered. The results confirm a qualitative similarity between the PRC and the coupling function Q in all cases.
The Watanabe-Strogatz and Ott-Antonsen theories provided a seminal framework for rigorous and comprehensive studies of collective phenomena in a broad class of paradigmatic models for ensembles of coupled oscillators. Recently, a "circular cumulant" approach was suggested for constructing the perturbation theory for the Ott-Antonsen approach. In this paper, we derive the relations between the distribution of Watanabe-Strogatz phases and the circular cumulants of the original phases. These relations are important for the interpretation of the circular cumulant approach in the context of the Wat…
Kuramoto's original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics o…
We analyze accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrins…
Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks. With an additional inertia term, we find a low-dimensional behavior similar to the first-order Kuramoto model, derive a self-consistent equation and seek the time-dependent derivation of the order parameter. Numerical simulations are also conducted to verify our analytical results.
We study a numerical instability of direct simulations with truncated equation chains for the "circular cumulant" representation and two approaches to its suppression. The approaches are tested for a chimera-bearing hierarchical population of coupled oscillators. The stabilization techniques can be efficiently applied without significant effect on the natural system dynamics within a finite vicinity of the Ott-Antonsen manifold for direct numerical simulations with up to 20 cumulants; with increasing deviation from the Ott-Antonsen manifold the stabilization becomes more problematic.
We derive the Kuramoto-Sivashinsky-type phase equation from the Kuramoto-Sakaguchi-type phase model via the Ott-Antonsen-type complex amplitude equation and demonstrate heterogeneity-induced collective-phase turbulence in nonlocally coupled individual-phase oscillators.
We consider directed networks of Winfree oscillators with power law distributed in- and out-degree distributions. Gaussian and power law distributed intrinsic frequencies are considered, and these frequencies are positively correlated with oscillators' in-degrees. The Ott/Antonsen ansatz is used to derive degree-based mean field equations for the expected dynamics of networks, and these are numerically analysed. In a variety of cases "explosive" transitions between either two different steady states or between a steady state and a periodic solution are found, and these transitions are explaine…
In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the cor…
Kuramoto oscillators are widely used to explain collective phenomena in networks of coupled oscillatory units. We show that simple networks of two populations with a generic coupling scheme, where both coupling strengths and phase lags between and within populations are distinct, can exhibit chaotic dynamics as conjectured by Ott and Antonsen [Chaos, 18, 037113 (2008)]. These chaotic mean-field dynamics arise universally across network size, from the continuum limit of infinitely many oscillators down to very small networks with just two oscillators per population. Hence, complicated dynamics …
We consider large networks of theta neurons and use the Ott/Antonsen ansatz to derive degree-based mean field equations governing the expected dynamics of the networks. Assuming random connectivity we investigate the effects of varying the widths of the in- and out-degree distributions on the dynamics of excitatory or inhibitory synaptically coupled networks, and gap junction coupled networks. For synaptically coupled networks, the dynamics are independent of the out-degree distribution. Broadening the in-degree distribution destroys oscillations in inhibitory networks and decreases the range …
An incorporation of higher-order interactions is known to lead an abrupt first-order transition to synchronization in otherwise smooth second-order one for pair-wise coupled systems. Here, we show that adaptation in higher-order coupling strength may yield completely different phenomena, notably, second-order transition to synchronization and tiered synchronization. Using the Ott-Antonsen approach, we perform rigorous theoretical calculations illustrating the origin of these emerging phenomena along with a complete description of all (un)stable states. Numerical simulations for limited-size ne…
We investigate the effect of preferentially connecting oscillators with similar frequency to each other in networks of coupled phase oscillators (i.e., frequency assortativity). Using the network Kuramoto model as an example, we find that frequency assortativity can induce chaos in the macroscopic dynamics. By applying a mean-field approximation in combination with the dimension reduction method of Ott and Antonsen, we show that the dynamics can be described by a low dimensional system of equations. We use the reduced system to characterize the macroscopic chaos using Lyapunov exponents, bifur…
We analyze a variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements. Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters. We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. Our main result is that when the coupling function is broad, the system can display bistability between steady states of constant high and low activity, whereas when the coupling…
We consider a population of globally coupled oscillators driven by common noise. By applying the Ott-Antonsen ansatz and by averaging over the fast oscillations, we obtain analytically tractable equations for the noisy evolution of the order parameter. While noise always tends to synchronize the oscillators, the coupling can act against synchrony if it is repulsive. For identical oscillators, the fully synchronous state remains stable for small enough repulsive coupling; moreover it is an absorbing state which always wins over the asynchronous regime. For oscillators with a distribution of nat…
Synchronization in a population of oscillators with hyperbolic chaotic phases is studied for two models. One is based on the Kuramoto dynamics of the phase oscillators and on the Bernoulli map applied to these phases. This system possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a map for the evolution of the complex order parameter. Beyond a critical coupling strength, this model demonstrates bistability synchrony-disorder. Another model is based on the coupled autonomous oscillators with hyperbolic chaotic strange attractors of Smale-Williams type. Here a disordered …
We study synchronization of sinusoidally coupled phase oscillators on networks with modular structure and a large number of oscillators in each community. Of particular interest is the hierarchy of local and global synchrony, i.e., synchrony within and between communities, respectively. Using the recent ansatz of Ott and Antonsen, we find that the degree of local synchrony can be determined from a set of coupled low-dimensional equations. If the number of communities in the network is large, a low-dimensional description of global synchrony can be also found. Using these results, we study bifu…
We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically …
We study two groups of nonidentical Kuramoto oscillators with differing frequency distributions. Coupling between the groups is repulsive, while coupling between oscillators of the same group is attractive. This asymmetry of interactions leads to an interesting synchronization behavior. For small coupling strength, the mean-fields of both groups resemble a more weakly coupled Kuramoto model. After increasing the coupling strength beyond a threshold, they reach the Bellerophon state of multiple clusters of averagely entrained oscillators. A further increase in the coupling strength then asympto…
Collective behaviors of populations of coupled oscillators have attracted much attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynam- ical mechanism of collective synchronizations by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe-Strogatz transformation, Ott-Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to diverse col- lective st…
We study the collective dynamics in a network of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the network, whereas the feedback from the resources to the network is comprised of components acting homogeneously or inhomogeneously on individual units of the network. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the network and the resources can give rise to collective activity bursting in the network. To explain the…
We consider ensembles of sine-coupled phase oscillators consisting of subpopulations of identical units, with a general heterogeneous coupling between subpopulations. Using the Watanabe-Strogatz ansatz we reduce the dynamics of the ensemble to a relatively small number of dynamical variables plus microscopic constants of motion. This reduction is independent of the sizes of subpopulations and remains valid in the thermodynamic limits, where these sizes or/and the number of subpopulations are infinite. We demonstrate that the approach to the dynamics of such systems, recently proposed by Ott an…
We consider systems of many spatially distributed phase oscillators that interact with their neighbors. Each oscillator is allowed to have a different natural frequency, as well as a different response time to the signals it receives from other oscillators in its neighborhood. Using the ansatz of Ott and Antonsen (Ref. \cite{OA1}) and adopting a strategy similar to that employed in the recent work of Laing (Ref. \cite{Laing2}), we reduce the microscopic dynamics of these systems to a macroscopic partial-differential-equation description. Using this macroscopic formulation, we numerically find …
We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling-wave and $π$-states, in addition to the classical fully synchronized a…
Janus oscillators have been recently introduced as a remarkably simple phase oscillator model that exhibits non-trivial dynamical patterns -- such as chimeras, explosive transitions, and asymmetry-induced synchronization -- that once were only observed in specifically tailored models. Here we study ensembles of Janus oscillators coupled on large homogeneous and heterogeneous networks. By virtue of the Ott-Antonsen reduction scheme, we find that the rich dynamics of Janus oscillators persists in the thermodynamic limit of random regular, Erdős-Rényi and scale-free random networks. We uncover fo…
We study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher-order modes dominate the coupling between oscillators. For the first time, we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony, cluster asymmetry, and switching. We use a variation of the recent dimensionality-reduction technique of Ott and Antonsen [Chaos {\bf 18}, 037113 (2008)] and find an analytic description of the degree of cluster synchrony valid on a globally attracting …
We generalize the Kuramoto model for the synchronization transition of globally coupled phase oscillators to populations by incorporating an additional heterogeneity with the coupling strength, where each oscillator pair interacts with different coupling strength weighted by a genera; function of their natural frequency. The expression for the critical coupling can be straightforwardly extended to a generalized explicit formula analytically, and s self-consistency approach is developed to predict the stationary states in the thermodynamic limit. The landau damping effect is further revealed by…
We propose a coupled system of fast and slow phase oscillators. We observe two-step transitions to quasi-periodic motions by direct numerical simulations of this coupled oscillator system. A low-dimensional equation for order parameters is derived using the Ott-Antonsen ansatz. The applicability of the ansatz is checked by the comparison of numerical results of the coupled oscillator system and the reduced low-dimensional equation. We investigate further several interesting phenomena in which mutual interactions between the fast and slow oscillators play an essential role. Fast oscillations ap…
A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $ω^T L ω$, where $ω$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matr…
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 FourierHermite decomposition of the oscillator density. The Fourier-Hermite moments obey a set of hierarchical ordinary dif…
In this paper, we propose a framework to investigate the collective dynamics in ensembles of globally coupled phase oscillators when higher-order modes dominate the coupling. The spatiotemporal properties of the attractors in various regions of parameter space are analyzed. Furthermore, a detailed linear stability analysis proves that the stationary symmetric distribution is only neutrally stable in the marginal regime which stems from the generalized time-reversal symmetry. Moreover, the critical parameters of the transition among various regimes are determined analytically by both the Ott-An…
The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a {\it frustrated} version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model, but written in terms of unit vectors. Replacing the coupling constant…
We study the impact of random pinning fields on the emergence of synchrony in the Kuramoto model on complete graphs and uncorrelated random complex networks. We consider random fields with uniformly distributed directions and homogeneous and heterogeneous (Gaussian) field magnitude distribution. In our analysis we apply the Ott-Antonsen method and the annealed-network approximation to find the critical behavior of the order parameter. In the case of homogeneous fields, we find a tricritical point above which a second-order phase transition gives place to a first-order phase transition when the…
We have studied the dynamics of the paradigmatic Kuramoto-Sakaguchi model of identical coupled phase oscilla- tors with various kinds of time-dependent connectivity using Eulerian discretization. We first explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities, namely the strength of incoherence and the discontinity measure. In the quasi-static limit of the changing of coupling range, we have observed how the system relaxes from one state to another and have identified a few interesting collective dynamical states along the wa…
We consider large networks of globally coupled spiking neurons and derive an exact low-dimensional description of their collective dynamics in the thermodynamic limit. Individual neurons are described by the Ermentrout-Kopell canonical model that can be excitable or tonically spiking, and interact with other neurons via pulses. Utilizing the equivalence of the quadratic integrate- and-fire and the theta neuron formulations, we first derive the dynamical equations in terms of the Kuramoto-Daido order parameters (Fourier modes of the phase distribution) and relate them to two biophysically relev…
The Kuramoto model is a paradigmatic tool for studying the dynamics of collective behavior in large ensembles of coupled dynamical systems. Over the past decade a great deal of progress has been made in analytical descriptions of the macroscopic dynamics of the Kuramoto mode, facilitated by the discovery of Ott and Antonsen's dimensionality reduction method. However, the vast majority of these works relies on a critical assumption where the oscillators' natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which allows for a convenient closure of the evolution equations fro…
Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree ---consisting of a large population of all-to-all pulse-coupled oscillators--- is still missing. Here we show that the dynamics of the Winfree model evolves into the so-called Ott-Antonsen manifold. This important property allows for an exact description of this h…
