Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <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> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> We study a network of all-to-all interconnected phase oscillators as modeled by the Kuramoto model. For coupling strengths larger than a critical value, we show the existence of a collective behavior called phase locking: the phase differences between all oscillators are constant in time. As the coupling strength increases, the distance between each pair of phases decreases. Stability of each phase locking solution is proven for general frequency distributions. There exist one unique asymptotically stable phase locking solution. Furthermore a description is given of partial entrainment, which can be regarded as the finite number analogon of partial synchronization in the infinite number case. When the network is partially entraining some phase differences possess an upper and lower bound. Partial entrainment of the three-cell network is analyzed: an estimate of the onset of partial entrainment is given and the existence of partial entrainment is proven. Furthermore, local stability of partial entrainment is proven for the three-cell network with two identical oscillators. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> We analyze the linear stability of the phase-locked state in the Kuramoto model of coupled oscillators. The main result is the first rigorous characterization of the spectrum and its associated eigenvectors, for any finite number of oscillators. All but two of the eigenvalues are negative, and merge into a continuous spectrum as the number of oscillators tends to infinity. One eigenvalue is always zero, by rotational invariance. The final eigenvalue, corresponding to a collective mode, determines the stability of the locked state. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <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> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> In this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators, the Kuramoto model. We consider the special case of finite oscillators with distinct, bounded natural frequencies. Our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization. This bound improves the one derived by Jadbabaie . We then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group. We also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> This paper presents new methods and results on synchronization of coupled Hopf nonlinear oscillators, which are commonly used as the dynamic model of engineered central pattern generators (CPGs). On balanced graphs, any positive coupling gain is proven to induce almost global asymptotic synchronization, and a threshold value for truly global exponential synchronization is also computed. Furthermore, a hierarchical connection between coupled Hopf oscillators and Kuramoto oscillators is identified. Finally, a new result on the synchronization of Kuramoto oscillators with arbitrary time-varying heterogeneous frequencies and delays is derived. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> Abstract We discuss the asymptotic complete phase–frequency synchronization for the Kuramoto phase model with a finite size N . We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N , hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> The Kuramoto model captures various synchronization phenomena in biological and man-made systems of coupled oscillators. It is well-known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength for finite and infinite-dimensional, and for first and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that the multi-rate Kuramoto model is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies, and we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization. Interestingly, these conditions do not depend on the inertiae which contradicts prior observations on the role of inertiae in synchronization of second-order Kuramoto models. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> Motivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phase-locked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). This general result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> Abstract We discuss the asymptotic formation and nonlinear orbital stability of phase-locked states arising from the ensemble of non-identical Kuramoto oscillators. We provide an explicit lower bound for a coupling strength on the formation of phase-locked states, which only depends on the diameters of natural frequencies and initial phase configurations. We show that, when the phases of non-identical oscillators are distributed over the half circle and the coupling strength is sufficiently large, the dynamics of Kuramoto oscillators exhibits two stages (transition and relaxation stages). In a transition stage, initial configurations shrink to configurations whose diameters are strictly less than π 2 in a finite-time, and then the configurations tend to phase-locked states asymptotically. This improves previous results on the formation of phase-locked states by Chopra–Spong (2009) [26] and Ha–Ha–Kim (2010) [27] where their attention were focused only on the latter relaxation stage. We also show that the Kuramoto model is l 1 -contractive in the sense that the l 1 -distance along two smooth Kuramoto flows is less than or equal to that of initial configurations. In particular, when two initial configurations have the same averaged phases, the l 1 -distance between them decays to zero exponentially fast. For the configurations with different phase averages, we use the method of average adjustment and translation-invariant of the Kuramoto model to show that one solution converges to the translation of the other solution exponentially fast. This establishes the orbital stability of the phase-locked states. Our stability analysis does not employ any standard linearization technique around the given phase-locked states, but instead, we use a robust l 1 -metric functional as a Lyapunov functional. In the formation process of phase-locked states, we estimate the number of collisions between oscillators, and lower–upper bounds of the transversal phase differences. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Finite-Dimensional Kuramoto Models <s> We investigate frequency synchronization and phase agreement in networks of non-identical Kuramoto oscillators with delayed coupling. We provide delay-dependent lower bounds for the coupling gain so that a network with all-to-all coupling and constant heterogeneous delays achieves synchronization. The given bounds are determined explicitly by the system parameters. Furthermore, we show that frequency synchronization and phase agreement are possible in general directed networks of non-identical oscillators which contain a spanning tree if the delays satisfy a self-consistency condition. Networks with delays differ at this point notably from networks without delays, because phase agreement is not possible in undelayed non-identical Kuramoto oscillator networks. In addition to the sufficient conditions for synchronization, we investigate the connection between phase agreement and the existence of solutions of a necessary self-consistency condition. <s> BIB012
In the finite-dimensional case, various necessary, sufficient, implicit, and explicit estimates of the critical coupling strength K critical have been proposed BIB001 BIB002 BIB003  BIB004 BIB005 BIB006 Mason, 2008, 2009; BIB007 BIB008 BIB011 BIB010 Bullo, 2011, 2012b; BIB012 . We refer to BIB009 for a comprehensive historical overview and present only the best known results here. Necessary, explicit, and tight conditions: The necessary condition (28) evaluated for γ π/2 gives the following lower bound for the critical coupling: Of course, this often-reported lower bound (39) is generally conservative. The following tighter lower bound has been constructed by BIB006 . Here, we report a refined formulation of their necessary condition. Lemma 6.4 (Explicit, necessary, and tight critical coupling) Consider the Kuramoto model (3) with n ≥ 2 oscillators, natural frequencies ω ∈ 1 ⊥ n , and coupling strength K. The Kuramoto model has a frequency-synchronized solution only if the coupling strength K is larger than a critical value, that is, . (41) Moreover, condition (41) is tight: for ω = ω trip ω 0 · (+1, −1, 0 n−2 ) with ω 0 ∈ R and for any of its permutations, there exists a synchronous solution if and only if Notice that the bound (41) equals the bound (39) for n = 2 and for n → ∞, and it is a strict improvement otherwise. The bound (41) is reported in (Chopra and Spong, 2009, Eqs. (8) and (11)) and is computed using optimization techniques. Though not explicitly stated by BIB006 , it can be verified from their proof that the lower bound (40)- (41) is tight for ω = ω trip . In the critical case K = K critical , the associated arc-invariant equilibrium manifold is given by [θ * ] = [(+γ/2, −γ/2, 0 n−2 )]. In the limit n → ∞ this choice of natural frequencies ω corresponds to the tripolar distribution in Fig. 13(e) , and the associated phases [θ * ] are shown in Fig. 13(f) .