Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> An exact solution to a model of mutually interacting sinusoidal oscillators is found. Limits on the variation of the native frequencies are determined in order for synchronization to occur. These limits are computed for different distributions of native frequencies. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> A Lyapunov function for the phase-locked state of the Kuramoto model of non-linearly coupled oscillators is presented. It is also valid for finite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kuramoto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscillators,XY spins, is ferromagnetic, whereas the random frequencies induce random fields which try to break the ferromagnetic order, i.e., global phase locking. The ensuing arguments imply asymptotic stability of the phase-locked state (up to degeneracy) and hold for any probability distribution of the frequencies. Special attention is given to discrete distribution functions. We argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results, but that a partial locking of some but not all sublattices is not to be expected. The order parameter of the phase-locked state is shown to have a strictly positive lower bound (r ⩾ 1/2), so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens-Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> Abstract This article explores the Kuramoto model describing the synchronization of a population of coupled oscillators. Two versions of this model are considered: a discrete version suitable for a population with a finite number of oscillators, and a continuum model found in the limit of an infinite population. When the strength of the coupling between the oscillators exceeds a threshold, the oscillators partially synchronize. We explore the transition in the continuum model, which takes the form of a bifurcation of a discrete mode from a continuous spectrum. We use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition, and compare with the synchronization predicted by the discrete model. There are similarities with instabilities in ideal plasmas and inviscid fluids, but these are superficial. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> Abstract 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. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> We solve a longstanding stability problem for the Kuramoto model of coupled oscillators. This system has attracted mathematical attention, in part because of its applications in fields ranging from neuroscience to condensed-matter physics, and also because it provides a beautiful connection between nonlinear dynamics and statistical mechanics. The model consists of a large population of phase oscillators with all-to-all sinusoidal coupling. The oscillators' intrinsic frequencies are randomly distributed across the population according to a prescribed probability density, here taken to be unimodal and symmetric about its mean. As the coupling between the oscillators is increased, the system spontaneously synchronizes: the oscillators near the center of the frequency distribution lock their phases together and run at the same frequency, while those in the tails remain unlocked and drift at different frequencies. Although this ``partially locked'' state has been observed in simulations for decades, its stability has never been analyzed mathematically. Part of the difficulty is in formulating a reasonable infinite-N limit of the model. Here we describe such a continuum limit, and prove that the corresponding partially locked state is, in fact, neutrally stable, contrary to what one might have expected. The possible implications of this result are discussed. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> We prove a convergence result for the standard Kuramoto model with all-to-all coupling. Specifically, we show that the critical coupling strength associated with the onset of completely phase-locked behavior converges in probability as the number of oscillators tends to infinity. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> We consider general heterogeneous ensembles of phase oscillators, sine coupled to arbitrary external fields. Starting with the infinitely large ensembles, we extend the Watanabe-Strogatz theory, valid for identical oscillators, to cover the case of an arbitrary parameter distribution. The obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion. As a particular case of the general setup we consider hierarchically organized ensembles, consisting of a finite number of subpopulations, whereas the number of elements in a subpopulation can be both finite or infinite. Next, we link the Watanabe-Strogatz and Ott-Antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion. The approach is applied to the standard Kuramoto-Sakaguchi model, to its extension for the case of nonlinear coupling, and to the description of two interacting subpopulations, exhibiting a chimera state. With these examples we illustrate that, although the asymptotic dynamics can be found within the framework of the Ott-Antonsen theory, the transients depend on the constants of motion. The most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> The synchronization of coupled oscillators is a fascinating manifestation of self-organization that nature employs to orchestrate essential processes of life, such as the beating of the heart. Although it was long thought that synchrony or disorder were mutually exclusive steady states for a network of identical oscillators, numerous theoretical studies in recent years have revealed the intriguing possibility of `chimera states', in which the symmetry of the oscillator population is broken into a synchronous and an asynchronous part. However, a striking lack of empirical evidence raises the question of whether chimeras are indeed characteristic to natural systems. This calls for a palpable realization of chimera states without any fine-tuning, from which physical mechanisms underlying their emergence can be uncovered. Here, we devise a simple experiment with mechanical oscillators coupled in a hierarchical network to show that chimeras emerge naturally from a competition between two antagonistic synchronization patterns. We identify a wide spectrum of complex states, encompassing and extending the set of previously described chimeras. Our mathematical model shows that the self-organization observed in our experiments is controlled by elementary dynamical equations from mechanics that are ubiquitous in many natural and technological systems. The symmetry breaking mechanism revealed by our experiments may thus be prevalent in systems exhibiting collective behaviour, such as power grids, opto-mechanical crystals or cells communicating via quorum sensing in microbial populations. <s> BIB012 </s> Synchronization in complex networks of phase oscillators: A survey <s> Continuum-limit model: <s> Abstract Using recently developed analytical techniques, we study the macroscopic dynamics of a large heterogeneous network of theta neurons in which the neurons’ excitability parameter varies in time. We demonstrate that such periodic variation can lead to the emergence of macroscopic chaos, multistability, and final-state uncertainty in the collective behavior of the network. Finite-size network effects and rudimentary control via an accessible macroscopic network parameter is also investigated. <s> BIB013
Since infinite-dimensional oscillator networks are surveyed in detail in BIB004  BIB005 , this paper discusses them only very briefly. In what follows, we present an informal Eulerian derivation of the continuum-limit model. We also remark that a treatment of (17) as a stochastic differential equation (in the limit of zero additive white noise) results in a FokkerPlanck equation analogous to the continuum-limit model BIB003 BIB004  . Consider an infinite population of oscillators, and let ρ : S 1 ×R ≥0 ×R → R ≥0 be the probability density function of the oscillators, that is, denotes the fraction of oscillators in Arc n (γ) ⊆ S 1 , at time t, and with frequencies ω ∈ [ω, ω]. Hence, the order parameter is given by Notice that in the discrete (finite-dimensional) case we have ρ(θ, t, ω) = 1 n n j=1 δ(θ − θ j ) (where δ(·) is the Dirac δ-distribution), and the two order parameters (16) and (18) coincide. According to (17), the instantaneous velocity of an oscillator at position θ, at time t, and with natural frequency ω is given by v(θ, t, ω) = ω − Kr(t) sin(θ−ψ(t)). The evolution of the probability density function is then governed by the continuity equation subject to the conservation of the oscillators at time t and with frequency ω, that is, 2π 0 ρ(θ, t, ω) dθ = 1. 3 Synchronization in the continuum-limit model: Similar to the finite-dimensional model (16)-(17), the continuum-limit model (18)-(19) displays a rich set of symmetries BIB008 and dynamics BIB005 BIB009 ). The saddle-node bifurcation from incoherence to synchrony in the finite-dimensional model (16)-(17) (see Subsection 3.2) manifests itself in the infinite-dimensional model (18)-(19) as a phase transition from the uniform incoherent state with density ρ(θ, t, ω) = 1 2π to the so-called partially-synchronized state. The partiallysynchronized state is characterized by a subset of phaselocked oscillators rotating in unison whereas the remaining oscillators are incoherent. 4 The synchronized 3 In some articles, the continuum-limit model (18)- (19) is presented with the density functionρ(θ, t, ω) = ρ(θ, t, ω)g(ω), which satisfies 2π 0ρ (θ, t, ω)dθ = g(ω). 4 An analogous partially-synchronized state can be defined in the finite-dimensional Kuramoto model (16)- (17), where a subset of oscillators is contained in an arc; see BIB006 . A related dynamic phenomenon are chimera states with frequency-synchronized and incoherent oscillators  BIB012 . set of oscillators are those satisfying Kr > |ω| such that v(θ, t, ω) = 0, and the incoherent ones are uniformly spread over the circle, see Fig. 10(a) and Fig. 10(b) for a schematic illustration. This phase transition occurs when K exceeds some critical value K partial . When K is further increased, more and more oscillators become entrained by the mean field (18) and join the set of phase-locked oscillators. For a frequency distribution g(ω) with bounded support, there exists a second critical parameter K lock ≥ K partial , such that for K > K lock all oscillators are phase-locked. This final stage of synchronization is illustrated in Fig. 10(c) . It is often referred to as the fully phase-locked state, and it is reminiscent of frequency synchronization and arc invariance as displayed in the finite-dimensional model (16),(17). Whereas the majority of the literature on the continuumlimit model (18)- (19) focuses on the first phase transition and the calculation of K partial , see  BIB003 BIB004  BIB007 BIB005 BIB008  and references therein, the articles BIB001 BIB002 Strogatz, 2005, 2007;  BIB010 ) discuss the fully phase-locked state and the calculation of K lock . The influential works Antonsen, 2008, 2009) exploit the extensive symmetries of the continuum-limit model (18)-(19) to construct a simple solution ansatz obeying low-dimensional ordinary differential equations. This reduction approach has triggered an extensive line of recent research, e.g., see BIB009  BIB011 BIB013 and references therein.