Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <s> We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <s> Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and transient stability of a power network to the underlying system parameters and initial conditions. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <s> Problems of synchronization of coupled nonlinear oscillators and, in particular, problem of synchronization of Kuramoto model, have attracted attention of researchers from diverse fields including physics, biology, neuroscience, and mathematics during several last decades. In this paper for the classical “all-to-all” Kuramoto model we construct an auxiliary linear model that preserves information on the natural frequencies and interconnection gains of the original Kuramoto model and depends on the phase locked soltions of the Kuramoto model. Stability properties of this model are analysed and we show that solutions of the new linear system exponentially converge to a stable periodic limit cycle for almost all initial conditions. Finally we show that asymptotically, in the time limit, this model maps on the original Kuramoto model. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Exact and implicit conditions: <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> BIB011
The articles BIB002 BIB003 ) derive a set of implicit consistency equations for the exact critical coupling strength K critical for which frequency-synchronized solutions exist. The consistency equation can be easily motivated. Each equilibrium solution to the Kuramoto model (16)- (17) is characterized by [θ * ] ∈ T n such that the righthand side of (17) equals zero. We denote the corresponding value of the order parameter (16) by r ∞ ∈ [0, 1] and, without loss of generality, we assume that its phase ψ is zero. Hence, we arrive at the equations The equations (42) are solvable only if Kr ∞ ≥ ω ∞ , and thus necessarily r ∞ > 0 unless ω = 0 n . By eliminating θ * from (42), we arrive at the consistency equation where the ± signs are due to the equality: cos(arcsin(x)) = ± √ 1 − x 2 for x ∈ ] − 1, 1[. In fact, the consistency equation (43) is a set of 2 n equations corresponding to different possible equilibria θ * in (42) and thus different choices of the ± signs, although not all choices yield feasible solutions r ∞ ≥ 0. We refer to for a discussion of the consistency equation (43) and its infinite-dimensional counterpart (36). The implicit consistency equation (43) marks the starting point for the analyses in BIB002 BIB003 . By collecting various results in these three references, we arrive at the following statement, which has not been presented in this complete and self-contained form so far. Theorem 6.5 (Implicit formulae for the exact critical coupling) Consider the Kuramoto model (3) with n ≥ 2 oscillators, natural frequencies ω ∈ 1 ⊥ n \{0 n }, and coupling strength K. Compute u * ∈ [ ω ∞ , 2 ω ∞ ] as unique solution to the equation The following statements are equivalent: (i) Critical coupling: the coupling strength K is larger than a critical value, that is, (45) (ii) Stable frequency synchronization: there exists a locally exponentially stable equilibrium manifold The implicit formulae (44)- (45) have been established by Verwoerd and Mason (2008, Theorem 3) , who showed that K critical is the smallest nonnegative value of the coupling strength for which the Kuramoto model (3) admits a frequency-synchronized solution. We remark that Verwoerd and Mason also extended the implicit formulae (44)- (45) to complete bipartite graphs  , Theorem 3) and infinite-dimensional networks (Verwoerd and Mason, 2011, Theorem 4) . Moreover, they provided bisection algorithms to compute K critical with predefined precision in a finite number of iterations. BIB001 and BIB002 found similar implicit formulae and carried out a local stability analysis , Theorems 1 and 3) and BIB002 , Sections 3 and 4) showing a saddle-node bifurcation for K = K critical : for K < K critical no frequency-synchronized solution (i.e., equilibrium manifolds) exists and for K > K critical a locally exponentially stable (corresponding to all + signs in (43)) and multiple unstable phase-locked solutions co-exist. As shown by  , the Kuramoto model (3) can be embedded into a higher-dimensional, linear, and complex-valued system, 9 and the above stability results can also be elegantly established via linear systems theory; see also the recent work by BIB009 ; El Ati and Panteley (2013a,b). Sufficient, explicit, and tight conditions: For the purpose of analyzing and selecting a sufficiently strong coupling in applications, Theorem 6.5 has three drawbacks. The stability results are local and the region of attraction of a synchronized solution is unknown. Second, the exact formulae (44)-(45) are implicit and thus not suited for performance estimates. For example, it is unclear which value of asymptotic arc invariance can be achieved if K > c · K critical for some c > 1. Third and finally, the natural frequencies ω i are often time-varying or uncertain in most applications. In this case, the exact value of K critical needs to be estimated in continuous time, or a conservatively strong coupling K K critical has to be chosen. The following theorem provides an explicit upper bound on the critical coupling together with performance estimates, convergence rates, and a guaranteed semi-global region of attraction. This bound is tight and thus necessary and sufficient when considering arbitrary distributions with compact support of the natural frequencies. The result has been originally presented in (Dörfler and Bullo, 2011, Theorem 4 .1). Theorem 6.6 (Explicit, sufficient, and tight critical coupling and practical phase sync) Consider the Kuramoto model (3) with n ≥ 2 oscillators, natural frequencies ω ∈ 1 ⊥ n and coupling strength K. The following statements are equivalent: (i) Critical coupling: the coupling strength K is larger than a critical value, that is, (ii) Admissible initial arc invariance: there exists γ max ∈ ]π/2, π] such that the Kuramoto model (3) achieves exponential frequency synchronization for all possible distributions of the natural frequencies ω i supported on the compact interval [ω min , ω max ] and for all initial phases θ(0) ∈ Arc n (γ max ); and (iii) Arc invariance of sync manifold: there exists γ min ∈ [0, π/2[ such that the Kuramoto model (3) has a locally exponentially stable synchronization manifold in Arc n (γ min ) for all possible distributions of the natural frequencies ω i supported on the compact interval [ω min , ω max ]. If the equivalent conditions (i), (ii), and (iii) hold, then the ratio K critical /K and the arc lengths γ min ∈ [0, π/2[ and γ max ∈ ]π/2, π] are related uniquely via sin(γ min ) = sin(γ max ) = K critical /K. Moreover, the Kuramoto model (3) achieves practical phase synchronization, that is, the set Arc n (γ) is positively invariant for every γ ∈ [γ min , γ max ], and each trajectory originating in Arc n (γ max ) approaches asymptotically Arc n (γ min ). The proof of Theorem 6.6 relies on the Jacobian and contraction properties developed in Subsections 4.1 and 5.2. If all angles at time t ≥ 0 belong to a closed arc of length γ ∈ [0, π[, that is, θ(t) ∈ Arc n (γ), then the arc length t → V (θ(t)) given in (32) is non-increasing if The above inequality holds true for γ ∈ [γ min , γ max ] if and only if condition (46) holds true. Additionally, t → V (θ(t)) is strictly decreasing for γ ∈ ]γ min , γ max [, the angles θ(t) reach the set Arc n (γ min ) ∈ ∆ G (π/2), and frequency synchronization and stability follow from the results developed in Subsection 4.1. Hence, condition (46) implies properties (ii) and (iii) of Theorem 6.6. The converse implications follow since condition (46) is also necessary for synchronization with bipolar natural frequencies ω = ω bip ω 0 ·(−p·1 n−p , +(n−p)·1 p ) with ω 0 ∈ R, p ∈ {1, . . . , n − 1}, and for any of its permutations. In recent work,  show that the bipolar distribution ω bip is the unique worst-case distribution, where synchronization fails for K = K critical . Besides establishing a tight condition for K critical , Theorem 6.6 establishes some properties of the transient evolution of the Kuramoto dynamics (3) and shows that the asymptotic synchronization behavior of the Kuramoto model (3) is best described by the terminology practical phase synchronization, see also BIB008 BIB005 . Notice also that Theorem 6.6 reduces to Theorem 5.2 for identical natural frequencies. We remark that similar analysis results are reported in BIB004 BIB006 BIB010 BIB007 BIB011 , and the bound γ min on the ultimate phase distances can be improved for particular pairs of oscillators, see (Choi et al., 2011a, Theorem 5.2) . Finally, we remark that the proof strategy via the contraction Lyapunov function (32) can be adapted to more general cases, for example, the conclusions of Theorem 6.6 can be extended to time-varying natural frequencies, see BIB005 and the illustration in Fig. 14. Comparison and statistical analysis: Theorem 6.6 states the tight and explicit upper bound (46) on the critical coupling strength K critical . Likewise, Lemma 6.4 states the tight and explicit lower bound (41) on K critical . The exact critical coupling lies somewhere in-between and can be obtained from the implicit formulae (44)- (45). By collecting these results, we can state the following corollary, which improves upon the explicit bounds proposed by Verwoerd and Mason (2008, Corollary 7) . Corollary 6.7 (Tight explicit bounds) Consider the Kuramoto model (3) with n ≥ 2 oscillators, natural frequencies ω ∈ 1 ⊥ n \ {0 n }, and coupling strength K. Compute the exact critical coupling K critical according to (44)-(45). The explicit necessary condition (41) and sufficient condition (46) provide tight upper and lower bounds on the exact critical coupling K critical , that is, where γ ∈ [π/2, π] is defined in (40). Moreover, the lower bound is tight for ω = ω trip ω 0 ·(+1, −1, 0 n−2 ), and the upper bound is tight for ω = ω bip ω 0 ·(−p·1 n−p , +(n− p) · 1 p ), where ω 0 ∈ R, p ∈ {1, . . . , n − 1}, and both ω trip  (a) The simulations illustrate the phase cohesiveness of the angles θ(t) in Arcn(γmin), the boundedness and convergence of the frequency variations (between consecutive switching instances)θ(t) − ωavg(t)1n, as well as the monotonicity of the Lyapunov function V (θ(t)) in Arcn(γ) for γ ∈ [γmin, γmax]. and ω bip are defined modulo index permutations. Corollary 6.7 is the finite-dimensional counterpart to Theorem 6.3 and identifies bipolar and tripolar frequency distributions ω bip and ω trip as the extreme choices for the resulting critical coupling K critical .