Synchronization in complex networks of phase oscillators: A survey <s> Theorem 6.3 (Full phase locking thresholds) <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> Theorem 6.3 (Full phase locking thresholds) <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> Theorem 6.3 (Full phase locking thresholds) <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> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Theorem 6.3 (Full phase locking thresholds) <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> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Theorem 6.3 (Full phase locking thresholds) <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> BIB005
Consider the infinite-dimensional Kuramoto model (18)- (19) with coupling gain K and frequency distribution g : R → R ≥0 with bounded support. The following statements hold for the full phase-locking threshold K lock : (i) Lower bound: For any continuous, even, and uni- where A proof of the lower bound (i) can be found in (Ermentrout, 1985, Corollary 2(b)) and BIB003 . Notice that the two thresholds K partial (reported in (37)) and K lock coincide for the uniform distribution: This remarkable identity was also observed by Hemmen and Wreszinski (1993); BIB003 ;  ; BIB005 . The upper bound (ii) on bipolar distributions has been proved by BIB002 and earlier by BIB001 for the symmetric case (p = 1/2 and ω max = −ω min = ω 0 ). Bipolar and more general bimodal frequency distributions g(ω) have attracted tremendous research interest by dynamical system researchers thanks to their rich bifurcation diagram, see  BIB004 ). The uniform and bipolar distributions are shown in Fig. 13 together with the associated stationary phase distributions in the critical case K K lock (explicitly calculated by BIB002 ). For later reference, Fig. 13 also shows the tripolar distribution g trip,n (ω) = 