Synchronization in complex networks of phase oscillators: A survey <s> Infinite-Dimensional Kuramoto Models <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> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Infinite-Dimensional Kuramoto Models <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> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Infinite-Dimensional Kuramoto Models <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> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Infinite-Dimensional Kuramoto Models <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> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Infinite-Dimensional Kuramoto Models <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> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Infinite-Dimensional Kuramoto Models <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> BIB006
In his ingenious analysis of the continuum-limit model (18)-(19) Kuramoto considered continuous, even, and unimodal distributions g(ω) of the natural frequencies (achieving their maximum at g(0)), and found that partially-synchronized solutions (if existent) must satisfy the self-consistency equation   One trivial solution to the self-consistency equation (36) is r = 0 corresponding to the uniform incoherent state shown in Fig. 10(a) . The second solution for r > 0 corresponds to the partially-synchronized state illustrated in Fig. 10(b) . When canceling the variable r from both sides of (36) and taking the limit r 0, the self-consistency equation (36) delivers the bifurcation parameter Kuramoto conjectured that the uniform incoherent state would become unstable for K > K partial and concluded famously that "surprisingly enough, this seemingly obvious fact seems difficult to prove." The resolution of this long-standing conjecture and Kuramoto's ingenious yet incomplete analysis inspired generations of scientists, see BIB002 for an historical account. We present the following general result by (Chiba, 2014, Theorem 3.5 ). Theorem 6.1 (Instability of the incoherent state) Consider the infinite-dimensional Kuramoto model (18)- (19) with coupling gain K and frequency distribution g : R → R ≥0 . Assume that the number of roots {y 1 , y 2 , . . . } of the equation is countable, and g(ω) is continuous at {y 1 , y 2 , . . . }. If , then the incoherent state ρ(θ, t, ω) = 1/(2π) is unstable. Notice that Theorem 6.1 is fairly general and includes bimodal distributions. It can be shown that for a continuous, even, and unimodal distribution g(ω), the unique root of (38) is given by y 1 = 0, see (Chiba, 2014, Corollary 3.6 ). This observation leads to the following corollary, which can be found in BIB001 BIB003  BIB004 BIB005 BIB006  , and references therein. Corollary 6.2 (Instability beyond Kuramoto's critical transition point) Consider the infinitedimensional Kuramoto model (18)-(19) with coupling gain K and frequency distribution g : R → R ≥0 . Suppose that g(ω) is continuous at the origin, even, and unimodal. If K is greater than K partial as given in (37), then the incoherent state ρ(θ, t, ω) = 1/(2π) is unstable. A linear stability analysis of the associated partiallysynchronized state illustrated in Fig. 10(b) is discussed by BIB004 and reveals linear neutral stability. To the best of the authors' knowledge, a nonlinear stability analysis of the partially-synchronized state is still outstanding. If the distribution g(ω) is restricted to have bounded support, then the fully phase-locked state (illustrated in Fig. 10(c) ) can be achieved when the coupling K is larger than the second critical threshold K lock ≥ K partial . In this case, two distributions of interest are the uniform distribution and the bipolar distribution given by gtrip,n(ω) gbip(ω) Fig. 13 . Extremal distributions g(ω) of the natural frequencies and their stationary phase distributions in the critical case K K lock : Panels (a) and (b) show the (generally non-symmetric) bipolar distribution g bip (ω) = p · δ(ω − ωmax) + (1 − p) · δ(ω − ωmin) and its associated bipolar phase distribution. Panels (c) and (d) show the uniform distribution g unif (ω) = 1/(2ω0) and its associated uniform phase distribution. Finally, panels (e) and (f) show the tripolar distribution gtrip,n(ω) = 1 n δ(ω − ω0) + n−2 n δ(ω) + 1 n δ(ω + ω0) and its associated tripolar phase distribution for n → ∞. where ω 0 > 0, ω max > ω min , and p ∈ [0, 1]. These two distributions are particularly interesting since they yield the smallest and the largest threshold K lock .