Exploring Synchronization in Complex Oscillator Networks
Abstract
The emergence of synchronization in a network of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and engineering applications. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among the oscillators. These two ingredients give rise to a rich dynamic behavior that keeps on fascinating the scientific community. In this article, we present a tutorial introduction to coupled oscillator networks, we review the vast literature on theory and applications, and we present a collection of different synchronization notions, conditions, and analysis approaches. We focus on the canonical phase oscillator models occurring in countless realworld synchronization phenomena, and present their rich phenomenology. We review a set of applications relevant to control scientists. We explore different approaches to phase and frequency synchronization, and we present a collection of synchronization conditions and performance estimates. For all results we present selfcontained proofs that illustrate a sample of different analysis methods in a tutorial style.
I Introduction
The scientific interest in synchronization of coupled oscillators can be traced back to the work by Christiaan Huygens on “an odd kind sympathy” between coupled pendulum clocks [1], and it still fascinates the scientific community nowadays [2, 3]. Within the rich modeling phenomenology on synchronization among coupled oscillators, we focus on the canonical model of a continuoustime limitcycle oscillator network with continuous and bidirectional coupling.
A network of coupled phase oscillators: A mechanical analog of a coupled oscillator network is the spring network shown in Figure 1 and consists of a group of kinematic particles constrained to rotate around a circle and assumed to move without colliding.
Each particle is characterized by a phase angle and has a preferred natural rotation frequency . Pairs of interacting particles and are coupled through an elastic spring with stiffness . We refer to the Appendix A for a first principle modeling of the springinterconnected particles depicted in Figure 1.
Formally, each isolated particle is an oscillator with firstorder dynamics . The interaction among such oscillators is modeled by a connected graph with nodes , edges , and positive weights for each undirected edge . Under these assumptions, the overall dynamics of the coupled oscillator network are
(1) 
The rich dynamic behavior of the coupled oscillator model (1) arises from a competition between each oscillator’s tendency to align with its natural frequency and the synchronizationenforcing coupling with its neighbors. Intuitively, a weakly coupled and strongly heterogeneous network does not display any coherent behavior, whereas a strongly coupled and sufficiently homogeneous network is amenable to synchronization, where all frequencies or even all phases become aligned.
History, applications and related literature: The coupled oscillator model (1) has first been proposed by Arthur Winfree [4]. In the case of a complete interaction graph, the coupled oscillator dynamics (1) are nowadays known as the Kuramoto model of coupled oscillators due to Yoshiki Kuramoto [5, 6]. Stephen Strogatz provides an excellent historical account in [7]. We also recommend the survey [8].
Despite its apparent simplicity, the coupled oscillator model (1) gives rise to rich dynamic behavior. This model is encountered in various scientific disciplines ranging from natural sciences over engineering applications to social networks. The model and its variations appear in the study if biological synchronization phenomena such as pacemaker cells in the heart [9], circadian rhythms [10], neuroscience [11, 12, 13], metabolic synchrony in yeast cell populations [14], flashing fireflies [15], chirping crickets [16], biological locomotion [17], animal flocking behavior [18], fish schools [19], and rhythmic applause [20], among others. The coupled oscillator model (1) also appears in physics and chemistry in modeling and analysis of spin glass models [21, 22], flavor evolutions of neutrinos [23], coupled Josephson junctions [24], and in the analysis of chemical oscillations [25].
Some technological applications of the coupled oscillator model (1) include deep brain stimulation [26, 27], vehicle coordination [19, 28, 29, 30, 31], carrier synchronization without phaselocked loops [32], semiconductor lasers [33, 34], microwave oscillators [35], clock synchronization in decentralized computing networks [36, 37, 38, 39, 40, 41], decentralized maximum likelihood estimation [42], and droopcontrolled inverters in microgrids [43]. Finally, the coupled oscillator model (1) also serves as the prototypical example for synchronization in complex networks [44, 45, 46, 47] and its linearization is the wellknown consensus protocol studied in networked control, see the surveys and monographs [48, 49, 50]. Various control scientists explored the coupled oscillator model (1) as a nonlinear generalization of the consensus protocol [51, 52, 53, 54, 55, 56, 57].
Secondorder variations of the coupled oscillator model (1) appear in synchronization phenomena, in population of flashing fireflies [58], in particle models mimicking animal flocking behavior [59, 60], in structurepreserving power system models, [61, 62] in networkreduced power system models [63, 64], in coupled metronomes [65], in pedestrian crowd synchrony on London’s Millennium bridge [66], and in Huygen’s pendulum coupled clocks [67]. Coupled oscillator networks with secondorder dynamics have been theoretically analyzed in [68, 69, 70, 71, 72, 73, 8, 74], among others.
Coupled oscillator models of the form (1) are also studied from a purely theoretic perspective in the physics, dynamical systems, and control communities. At the heart of the coupled oscillator dynamics is the transition from incoherence to synchrony. Here, different notions and degrees of synchronization can be distinguished [74, 75, 76], and the (apparently) incoherent state features rich and largely unexplored dynamics as well [77, 78, 79, 47]. In this article we will be particularly interested in phase and frequency synchronization when all phases become aligned, respectively all frequencies become aligned. We refer to [76, 64, 19, 80, 31, 81, 82, 53, 83, 52, 84, 85, 86, 87, 88, 89, 90, 75, 7, 8, 74, 56, 91, 92, 28, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 95, 109, 110, 111, 112, 113, 114] for an incomplete overview concerning numerous recent research activities. We will review some of literature throughout the paper and refer to the surveys [8, 74, 7, 44, 45, 46] for further applications and numerous additional theoretic results concerning the coupled oscillator model (1).
Contributions and contents: In this paper, we introduce the reader to synchronization in networks of coupled oscillators. We present a sample of important analysis concepts in a tutorial style and from a controltheoretic perspective.
In Section II, we will review a set of selected technological applications which are directly tied to the coupled oscillator model (1) and also relevant to control systems. We will cover vehicle coordination, electric power networks, and clock synchronization in depth, and also justify the importance of the coupled oscillator model (1) as a canonical model. Prompted by these applications, we review the existing results concerning phase synchronization, phase balancing, and frequency synchronization, and we also present some novel results on synchronization in sparselycoupled networks.
In particular, Section III introduces the reader to different synchronization notions, performance metrics, and synchronization conditions. We illustrate these results with a simple yet rich example that nicely explains the basic phenomenology in coupled oscillator networks.
Section IV presents a collection of important results regarding phase synchronization, phase balancing, and frequency synchronization. By now the analysis methods for synchronization have reached a mature level, and we present simple and selfcontained proofs using a sample of different analysis methods. In particular, we present one result on phase synchronization and one result on phase balancing including estimates on the exponential synchronization rate and the region of attraction (see Theorem IV.3 and Theorem IV.4). We also present some implicit and explicit, and necessary and sufficient conditions for frequency synchronization in the classic homogeneous case of a complete and uniformlyweighted coupling graphs (see Theorem IV.5). Concerning frequency synchronization in sparse graphs, we present two partially new synchronization conditions depending on the algebraic connectivity (see Theorem IV.6 and Theorem IV.7).
In our technical presentation, we try to strike a balance between mathematical precision and removing unnecessary technicalities. For this reason some proofs are reported in the appendix and others are only sketched here with references to the detailed proofs elsewhere. Hence, the main technical ideas are conveyed while the tutorial value is maintained.
Finally, Section V concludes the paper. We summarize the limitations of existing analysis methods and suggest some important directions for future research.
Preliminaries and notation: The remainder of this section introduces some notation and recalls some preliminaries.
Vectors and functions: Let and be the dimensional vector of unit and zero entries, and let be the orthogonal complement of in , that is, . Given an tuple , let be the associated vector with maximum and minimum elements and . For an ordered index set of cardinality and an onedimensional array , let be the associated diagonal matrix. Finally, define the continuous function by for .
Geometry on the torus: The set denotes the unit circle, an angle is a point , and an arc is a connected subset of . The geodesic distance between two angles , is the minimum of the counterclockwise and the clockwise arc lengths connecting and . With slight abuse of notation, let denote the geodesic distance between two angles . The torus is the product set is the direct sum of unit circles. For , let be the closed set of angle arrays with the property that there exists an arc of length containing all . Thus, an angle array satisfies . Finally, let be the interior of the set .
Algebraic graph theory: Let be an undirected, connected, and weighted graph without selfloops. Let be its symmetric nonnegative adjacency matrix with zero diagonal, . For each node , define the nodal degree by . Let be the Laplacian matrix defined by . If a number and an arbitrary direction is assigned to each edge , the (oriented) incidence matrix is defined componentwise by if node is the sink node of edge and by if node is the source node of edge ; all other elements are zero. For , the vector has components corresponding to the oriented edge from to , that is, maps node variables , to incremental edge variables . If is the diagonal matrix of edge weights, then . If the graph is connected, then , all nonzero eigenvalues of are strictly positive, and the secondsmallest eigenvalue is called the algebraic connectivity and is a spectral connectivity measure.
Ii Applications of Kuramoto Oscillators Relevant to Control Systems
The mechanical analog in Figure 1 provides an intuitive illustration of the coupled oscillator dynamics (1), and we reviewed a wide range of examples from physics, life sciences, and technology in Section I. Here, we detail a set of selected technological applications which are relevant to control systems scientists.
Iia Flocking, Schooling, and Planar Vehicle Coordination
An emerging research field in control is the coordination of autonomous vehicles based on locally available information and inspired by biological flocking phenomena. Consider a set of particles in the plane , which we identify with the complex plane . Each particle is characterized by its position , its heading angle , and a steering control law depending on the position and heading of itself and other vehicles. For simplicity, we assume that all particles have constant and unit speed. The particle kinematics are then given by [115]
(2) 
where is the imaginary unit. If the control is identically zero, then particle travels in a straight line with orientation , and if is a nonzero constant, then the particle traverses a circle with radius .
The interaction among the particles is modeled by a possibly timevarying interaction graph determined by communication and sensing patterns. Some interesting motion patterns emerge if the controllers use only relative phase information between neighboring particles, that is, for and . For example, the control with gain results in
(3) 
The controlled phase dynamics (3) correspond to the coupled oscillator model (1) with a timevarying interaction graph with weights and identically timevarying natural frequencies for all . The controlled phase dynamics (3) give rise to very interesting coordination patterns that mimic animal flocking behavior [18] and fish schools [19]. Inspired by these biological phenomena, the controlled phase dynamics (3) and its variations have also been studied in the context of tracking and formation controllers in swarms of autonomous vehicles [19, 28, 29, 30, 31]. A few trajectories are illustrated in Figure 2, and we refer to [19, 28, 29, 30, 31] for other control laws and motion patterns.
In the following sections, we will present various tools to analyze the motion patterns in Figure 2, which we will refer to as phase synchronization and phase balancing.
IiB Power Grids with Synchronous Generators and Inverters
Here, we present the structurepreserving power network model introduced in [61] and refer to [62, Chapter 7] for detailed derivation from a higher order first principle model. Additionally, we equip the power network model with a set of inverters and refer to [43] for a detailed modeling.
Consider an alternating current (AC) power network modeled as an undirected, connected, and weighted graph with node set , transmission lines , and admittance matrix . For each node, consider the voltage phasor corresponding to the phase and magnitude of the sinusoidal solution to the circuit equations. If the network is lossless, then the active power flow from node to is , where we used the shorthand .
In the following, we assume that the node set is partitioned as , where are load buses, are conventional synchronous generators, and are gridconnected direct current (DC) power sources, such as solar farms. The active power drawn by a load consists of a constant term and a frequencydependent term with . The resulting power balance equation is
(4) 
If the generator reactances are absorbed into the admittance matrix, then the swing dynamics of generator are
(5) 
where and are the generator rotor angle and frequency, is the mechanical power input, and , and are the inertia and damping coefficients.
We assume that each DC source is connected to the AC grid via an DC/AC inverter, the inverter output impendances are absorbed into the admittance matrix, and each inverter is equipped with a conventional droopcontroller. For a droopcontrolled inverter with droopslope , the deviation of the power output from its nominal value is proportional to the frequency deviation . This gives rise to the inverter dynamics
(6) 
These power network devices are illustrated in Figure 3.
Finally, we remark that different load models such as constant power/current/susceptance loads and synchronous motor loads can be modeled and analyzed by the same set of equations (4)(6), see [62, 116, 117, 63, 64].
Synchronization is pervasive in the operation of power networks. All generating units of an interconnected grid must remain in strict frequency synchronism while continuously following demand and rejecting disturbances. Notice that, with exception of the inertial terms and the possibly nonunit coefficients , the power network dynamics (4)(6) are a perfect electrical analog of the coupled oscillator model (1) with . Thus, it is not surprising that scientists from different disciplines recently advocated coupled oscillator approaches to analyze synchronization in power networks [118, 119, 43, 120, 114, 64, 121, 122, 97, 69].
The theoretic tools presented in the following sections establish how frequency synchronization in power networks depend on the nodal parameters as well as the interconnecting electrical network with weights . Ultimately, this deep understanding of synchrony gives us the correct intuition to design controllers and remedial action schemes preventing the loss of synchrony.
IiC Clock Synchronization in Decentralized Networks
Another emerging technological application of the coupled oscillator model (1) is clock synchronization in decentralized computing networks, such as wireless and distributed software networks. A natural approach to clock synchronization is to treat each clock as a coupled oscillator and follow a diffusionbased protocol to synchronize them, see the historic and recent surveys [36, 37], the landmark paper [38], and the interesting recent results [39, 40, 41].
Consider a set of distributed processors interconnected in a (possibly directed) communication network. Each processor is equipped with an internal software clock, and these clocks need to be synchronized for distributed computing and network routing tasks. For simplicity, we consider only analog clocks with continuous coupling since digital clocks are essentially discretized analog clocks and pulsecoupled clocks can be modeled continuously after a phase reduction and averaging analysis.
For our purposes, the clock of processor is a voltagecontrolled oscillator which outputs a harmonic waveform , where is the accumulated instantaneous phase. For uncoupled nodes, the phase evolves as
where is the nominal period, is an offset (frequency offset or skew), and is the initial phase. To synchronize their internal clocks, the processors follow a diffusionbased protocol. In a first step, neighboring oscillators continuously communicate their respective waveforms to another. Second, through a phase detector each node measures a convex combination of phase differences as
where are convex () and detectorspecific weights, and is an odd periodic function. Finally, is fed to a (firstorder and constant) phaselocked loop filter whose output drives the local phase according to
(7) 
The goal of the synchronization protocol (7) is to synchronize the frequencies or even the phases in the processor network. For an undirected communication protocol, symmetric weights , and a sinusoidal coupling function , the synchronization protocol (7) equals again the coupled oscillator model (1).
The tools developed in the next section will enable us to state conditions when the protocol (7) successfully achieves phase or frequency synchronization. Of course, the protocol (7) is merely a starting point, more sophisticated phaselocked loop filters can be constructed to enhance steadystate deviations from synchrony, and communication and phase noise as well as timedelays can be considered in the design.
IiD Canonical Coupled Oscillator Model
The importance of the coupled oscillator model (1) does not stem only from the various examples listed in Sections I and II. Even though model (1) appears to be quite specific (a phase oscillator with constant driving term and continuous, diffusive, and sinusoidal coupling), it is the canonical model of coupled limitcycle oscillators [123]. In the following, we briefly sketch how such general models can be reduced to model (1). We schematically follow the approaches [124, Chapter 10],[125] developed in the computational neuroscience community without aiming at mathematical precision, and we refer to [123, 126] for further details.
Consider an oscillator modeled as a dynamical system with state and nonlinear dynamics , which admit a locally exponentially stable periodic orbit with period . By a change of variables, any trajectory in a local neighborhood of can be characterized by a phase variable with dynamics , where .
Now consider a weakly forced oscillator of the form
(8) 
where is sufficiently small and is a timedependent forcing term. For small forcing , the attractive limit cycle persists, and the phase dynamics are obtained as
where is the infinitesimal phase response curve (or linear response function), and we dropped higher order terms.
Now consider such limit cycle oscillators, where is the state of oscillator with limit cycle and period . We assume that the oscillators are weakly coupled with interaction graph and dynamics
(9) 
where is the coupling function for the pair . The coupling can possibly be impulsive. The weak coupling in (9) can be identified with the weak forcing in (8), and a transformation to phase coordinates yields
where . The local change of variables then yields the coupled phase dynamics
An averaging analysis applied to the dynamics results in
(10) 
where and the averaged coupling functions are
Notice that the averaged coupling functions are periodic and the coupling is diffusive. If all functions are odd, a firstorder Fourier series expansion of yields as first harmonic with some coefficient . In this case, the dynamics (10) in the slow time scale reduce exactly to the coupled oscillator model (1).
This analysis justifies calling the coupled oscillator model (1) the canonical model for coupled limitcycle oscillators.
Iii Synchronization Notions and Metrics
In this section, we introduce different notions of synchronization. Whereas the first four subsections address the commonly studied notions of synchronization associated with a coherent behavior and cohesive phases, Subsection IIIE addresses the converse concept of phase balancing.
Iiia Synchronization Notions
The coupled oscillator model (1) evolves on , and features an important symmetry, namely the rotational invariance of the angular variable . This symmetry gives rise to the rich synchronization dynamics. Different levels of synchronization can be distinguished, and the most commonly studied notions are phase and frequency synchronization.
Phase synchronization: A solution to the coupled oscillator model (1) achieves phase synchronization if all phases become identical as .
Phase cohesiveness: As we will see later, phase synchronization can occur only if all natural frequencies are identical. If the natural frequencies are not identical, then each pairwise distance can converge to a constant but not necessarily zero value. The concept of phase cohesiveness formalizes this possibility. For , let be the closed set of angle arrays with the property for all , that is, each pairwise phase distance is bounded by . Also, let be the interior of . Notice that but the two sets are generally not equal. A solution is then said to be phase cohesive if there exists a length such that for all .
Frequency synchronization: A solution achieves frequency synchronization if all frequencies converge to a common frequency as . The explicit synchronization frequency of the coupled oscillator model (1) can be obtained by summing over all equations in (1) as . In the frequencysynchronized case, this sum simplifies to . In conclusion, if a solution of the coupled oscillator model (1) achieves frequency synchronization, then it does so with synchronization frequency equal to . By transforming to a rotating frame with frequency and by replacing by , we obtain (or equivalently ). In what follows, without loss of generality, we will sometimes assume that so that .
Remark 1 (Terminology)
Alternative terminologies for phase synchronization include full, exact, or perfect synchronization. For a frequencysynchronized solution all phase distances are constant in a rotating coordinate frame with frequency , and the terminology phase locking is sometimes used instead of frequency synchronization. Other commonly used terms include frequency locking, frequency entrainment, or also partial synchronization.
Synchronization: The main object under study in most applications and theoretic analyses are phase cohesive and frequencysynchronized solutions, that is, all oscillators rotate with the same synchronization frequency, and all their pairwise phase distances are bounded. In the following, we restrict our attention to synchronized solutions with sufficiently small phase distances for . Of course, there may exist other possible solutions, but these are not necessarily stable (see our analysis in Section IV) or not relevant in most applications^{1}^{1}1For example, in power network applications the coupling terms are power flows along transmission lines , and the phase distances are bounded well below due to thermal constraints. In Subsection IIIE, we present a converse synchronization notion, where the goal is to maximize phase distances.. We say that a solution to the coupled oscillator model (1) is synchronized if there exists for some and (identically zero for ) such that for all .
Synchronization manifold: The geometric object under study in synchronization is the synchronization manifold. Given a point and an angle , let be the rotation of counterclockwise by the angle . For , define the equivalence class
Clearly, if for some , then . Given a synchronized solution characterized by for some , the set is a synchronization manifold of the coupledoscillator model (1). Note that a synchronized solution takes value in a synchronization manifold due to rotational symmetry, and for (implying ) a synchronization manifold is also an equilibrium manifold of the coupled oscillator model (1). These geometric concepts are illustrated in Figure 4 for the twodimensional case.
IiiB A Simple yet Illustrative Example
The following example illustrates the different notions of synchronization introduced above and points out various important geometric subtleties occurring on the compact state space . Consider oscillators with . We restrict our attention to angles contained in an open halfcircle: for angles , with , the angular difference is the number in with magnitude equal to the geodesic distance and with positive sign if and only if the counterclockwise path length from to on is smaller than the clockwise path length. With this definition the twodimensional oscillator dynamics can be reduced to the scalar difference dynamics . After scaling time as and introducing the difference dynamics are
(11) 
The scalar dynamics (11) can be analyzed graphically by plotting the vector field over the difference variable , as in Figure 5(a). Figure 5(a) displays a saddlenode bifurcation at . For no equilibrium of (11) exists, and for an asymptotically stable equilibrium together with a saddle point exists.
For all trajectories converge exponentially to , that is, the oscillators synchronize exponentially. Additionally, the oscillators are phase cohesive if an only if , where all trajectories remain bounded. For the difference will increase beyond , and by definition will change its sign since the oscillators change orientation. Ultimately, converges to the equilibrium in the branch where . In the configuration space this implies that the distance increases to its maximum value and shrinks again, that is, the oscillators are not phase cohesive and revolve once around the circle before converging to the equilibrium manifold. Since , strongly coupled oscillators with practically achieve phase synchronization from every initial condition in an open semicircle. In the critical case, , the saddle equilibrium manifold at is globally attractive but not stable. An representative trajectory is illustrated in Figure 5(b).
In conclusion, the simple but already rich dimensional case shows that two oscillators are phase cohesive and synchronize if and only if , that is, if and only if the coupling dominates the nonuniformity as . The ratio determines the ultimate phase cohesiveness as well as the set of admissible initial conditions. For , practical phase synchronization is achieved for all angles in an open semicircle. More general coupled oscillator networks display the same phenomenology, but the threshold from incoherence to synchrony is generally unknown.
IiiC Synchronization Metrics
The notion of phase cohesiveness can be understood as a performance measure for synchronization and phase synchronization is simply the extreme case of phase cohesiveness with . An alternative performance measure is the magnitude of the socalled order parameter introduced by Kuramoto [5, 6]:
The order parameter is the centroid of all oscillators represented as points on the unit circle in . The magnitude of the order parameter is a synchronization measure: if all oscillators are phasesynchronized, then , and if all oscillators are spaced equally on the unit circle, then . The latter case is characterized in Subsection IIIE. For a complete graph, the magnitude of the order parameter serves as an average performance index for synchronization, and phase cohesiveness can be understood as a worstcase performance index. Extensions of the order parameter tailored to noncomplete graphs have been proposed in [52, 19, 56].
For a complete graph and for sufficiently small, the set reduces to , the arc of length containing all oscillators. The order parameter is contained within the convex hull of this arc since it is the centroid of all oscillators, see Figure 6. In this case, the magnitude of the order parameter can be related to the arc length .
Lemma III.1
(Shortest arc length and order parameter, [74, Lemma 2.1]) Given an angle array with , let be the magnitude of the order parameter, and let be the length of the shortest arc containing all angles, that is, . The following statements hold:

if , then ; and

if , then .
IiiD Synchronization Conditions
The coupled oscillator dynamics (1) feature (i) the synchronizing coupling described by the graph and (ii) the desynchronizing effect of the nonuniform natural frequencies . Loosely speaking, synchronization occurs when the coupling dominates the nonuniformity. Various conditions have been proposed in the synchronization and power systems literature to quantify this tradeoff.
The coupling is typically quantified by the algebraic connectivity [127, 64, 52, 128, 44, 45] or the weighted nodal degree [129, 117, 130, 64, 97], and the nonuniformity is quantified by either absolute norms or incremental norms , where typically . Sometimes, these conditions can be evaluated only numerically since they are statedependent [127, 129] or arise from a nontrivial linearization process, such as the Master stability function formalism [44, 45, 131]. In general, concise and accurate results are known only for specific topologies such as complete graphs [74], linear chains [108], and bipartite graphs [82] with uniform weights.
For arbitrary coupling topologies only sufficient conditions are known [127, 64, 52, 129] as well as numerical investigations for random networks [132, 89, 128, 98, 99]. Simulation studies indicate that these conditions are conservative estimates on the threshold from incoherence to synchrony. Literally, every review article on synchronization draws attention to the problem of finding sharp synchronization conditions [46, 8, 74, 7, 44, 45, 114].
IiiE Phase Balancing and Splay State
In certain applications in neuroscience [11, 12, 13], deepbrain stimulation [26, 27], and vehicle coordination [19, 28, 29, 30, 31], one is not interested in the coherent behavior with synchronized (or nearly synchronized) phases, but rather in the phenomenon of synchronized frequencies and desychronized phases.
Whereas the phasesynchronized state is characterized by the order parameter achieving its maximal (unit) magnitude, we say that a solution to the coupled oscillator model (1) achieves phase balancing if all phases converge to as , that is, the oscillators are distributed over the unit circle , such that their centroid vanishes. We refer to [28] for a geometric characterization of the balanced state.
One balanced state of particular interest in neuroscience applications [11, 12, 13, 26, 27] is the socalled splay state corresponding to phases uniformly distributed around the unit circle with distances . Other highly symmetric balanced states consist of multiple clusters of collocated phases, where the clusters themselves are arranged in splay state, see [28, 29].
Iv Analysis of Synchronization
In this section we present several analysis approaches to synchronization in the coupled oscillator model (1). We begin with a few basic ideas to provide important intuition as well as the analytic basis for further analysis.
Iva Some Simple Yet Important Insights
The potential energy of the elastic spring network in Figure 1 is, up to an additive constant, given by
(12) 
By means of the potential energy, the coupled oscillator model (1) can reformulated as the forced gradient system
(13) 
where denotes the partial derivative. It can be easily verified that the phasesynchronized state for all is a local minimum of the potential energy (12). The gradient formulation (13) clearly emphasizes the competition between the synchronizationenforcing coupling through the potential and the synchronizationinhibiting heterogeneous natural frequencies .
We next note that has to satisfy certain bounds, relative to the weighted nodal degree, in order for a synchronized solution to exist.
Lemma IV.1
(Necessary sync conditions) Consider the coupled oscillator model (1) with graph , frequencies , and nodal degree for each oscillator . If there exists a synchronized solution for some , then the following conditions hold:

Absolute bound: For each node ,
(14) 
Incremental bound: For all distinct ,
(15)
Since , the synchronization frequency is zero, and phase and frequency synchronized solutions are equilibrium solutions determined by the equations
(16) 
Since for , the equilibrium equations (16) have no solution if condition (14) is not satisfied. Since , an incremental bound on seems to be more appropriate than an absolute bound. The subtraction of the th and th equation (16) yields
Again, since the coupling is bounded, the above equation has no solution in if condition (15) is not satisfied.
The following result is fundamental for various approaches to phase and frequency synchronization. To the best of the authors’ knowledge this result has been first established in [133], and it has been reproved numerous times.
Lemma IV.2
(Stable synchronization in ) Consider the coupled oscillator model (1) with a connected graph and frequencies . The following statements hold:

Jacobian: The Jacobian of the coupled oscillator model (1) evaluated at is given by

Local stability and uniqueness: If there exists an equilibrium , then

is a Laplacian matrix;

the equilibrium manifold is locally exponentially stable; and

this equilibrium manifold is unique in .

Since and , we obtain that the Jacobian is equal to minus the Laplacian matrix of the connected graph with the (possibly negative) weights . Equivalently, in compact notation . This completes the proof of statement 1).
The Jacobian evaluated for an equilibrium is minus the Laplacian matrix of the graph with strictly positive weights for every . Hence, is negative semidefinite with the nullspace arising from the rotational symmetry, see Figure 4. Consequently, the equilibrium point is locally (transversally) exponentially stable, or equivalently, the corresponding equilibrium manifold is locally exponentially stable.
The uniqueness statement follows since the righthand side of the coupled oscillator model (1) is a onetoone function (modulo rotational symmetry) for , see [134, Corollary 1]. This completes the proof of statement 2).
By Lemma IV.2, any equilibrium in is stable which supports the notion of phase cohesiveness as a performance metric. Since the Jacobian is the negative Hessian of the potential defined in (12), Lemma IV.2 also implies that any equilibrium in is a local minimizer of . Of particular interest are socalled synchronizing graphs for which all critical points of (12) are hyperbolic, the phasesynchronized state is the global minimum of , and all other critical points are local maxima or saddle points. The class of synchronizing graphs includes, among others, complete graphs and acyclic graphs [100, 101, 102, 103].
IvB Phase Synchronization
If all natural frequencies are identical, for all , then a transformation of the coupled oscillator model (1) to a rotating frame with frequency leads to
(17) 
The analysis of the coupled oscillator model (17) is particularly simple and local phase synchronization can be concluded by various analysis methods. A sample of different analysis schemes (by far not complete) includes the contraction property [54, 100, 92, 64, 138], quadratic Lyapunov functions [52, 64], linearization [81, 103], or order parameter and potential function arguments [56, 28, 80].
The following theorem on phase synchronization summarizes a collection of results originally presented in [56, 54, 103, 100, 28, 74], and it can be easily proved given the insights developed in Subsection IVA.
Theorem IV.3
(Phase synchronization) Consider the coupled oscillator model (1) with a connected graph and with frequency (not necessarily zero mean). The following statements are equivalent:

Stable phase sync: there exists a locally exponentially stable phasesynchronized solution (or a synchronization manifold ); and

Uniformity: there exists a constant such that for all .
If the two equivalent cases (i) and (ii) are true, the following statements hold:

Global convergence: For all initial angles all frequencies converge to and all phases converge to the critical points ;

Semiglobal stability: The region of attraction of the phasesynchronized solution contains the open semicircle , and each arc is positively invariant for every arc length ;

Explicit phase: For initial angles in an open semicircle , the asymptotic synchronization phase is given by^{2}^{2}2This “average” of angles (points on ) is welldefined in an open semicircle. If the parametrization of