Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> Abstract The qualitative properties of the power-flow equation for a transmission network are studied in terms of global and local aspects, as well as stability. Global aspects include estimates of the number of solutions and topological properties of the stable region. Local aspects are examined through bifurcations of the flow equation. Examples are given to counter some intuitive conjectures, and other conjectures are made. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> A new model for the study of power system stability via Lyapunov functions is proposed. The key feature of the model is an assumption of frequency-dependent load power, rather than the usual impedance loads which are subsequently absorbed into a reduced network. The original network topology is explicitly represented. This approach has the important advantage of rigorously accounting for real power loads in the Lyapunov functions. This compares favorably with existing methods involving approximations to allow for the significant transfer conductances in reduced network models. The preservation of network topology can be exploited in stability analysis, with the concepts of critical and vulnerable cutsets playing central roles in dynamic and transient stability evaluation respectively. Of fundamental importance is the feature that the Lyapunov functions give a true representation of the spatial distribution of stored energy in the system <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> Abstract The direct assessment of large-disturbance stability using the model given by Bergen and Hill is studied. this model differs from the usual classical model with constant-impedance loads in that no network reduction is used; the system structure is preserved with the loads modelled as first-order dynamical systems. A new style of stability criterion is suggested which is based on local transient energy flow across network cutsets. This is found to give a more accurate assessment of stability than the usual approach based on total transient energy in the system. The criterion is applied to 5-bus and 14-bus example systems. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> This paper derives new results in non-linear system analysis using methods inspired from fluid mechanics and differential geometry. Based on a differential analysis of convergence, these results may be viewed as generalizing the classical Krasovskii theorem, and, more loosely, linear eigenvalue analysis. A central feature is that convergence and limit behavior are in a sense treated separately, leading to significant conceptual simplifications. The approach is illustrated by controller and observer designs for simple physical examples. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in "flocks" of oscillators or dynamic elements. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <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> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <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> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <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> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> This paper is concerned with the stability of networked control systems. Using contraction theory, a result is established on the network structure and the properties of the individual component subsystems and their couplings to ensure the overall contractivity of the entire network. Specifically, it is shown that a contraction property on a reduced-order matrix that quantifies the interconnection structure, coupled with contractivity/expansion estimates on the individual component subsystems, suffices to ensure that nearby trajectories of the overall system converge towards each other. The paper also describes a relative contraction result of interest in synchronization problems. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> We present a detailed analysis of the stability of phase-locked solutions to the Kuramoto system of oscillators. We derive an analytical expression counting the dimension of the unstable manifold associated to a given stationary solution. From this we are able to derive a number of consequences, including analytic expressions for the first and last frequency vectors to phase-lock, upper and lower bounds on the probability that a randomly chosen frequency vector will phase-lock, and very sharp results on the large N limit of this model. One of the surprises in this calculation is that for frequencies that are Gaussian distributed, the correct scaling for full synchrony is not the one commonly studied in the literature; rather, there is a logarithmic correction to the scaling which is related to the extremal value statistics of the random frequency vector. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> We propose a graphical notation by which certain spectral properties of complex systems can be rewritten concisely and interpreted topologically. Applying this notation to analyze the stability of a class of networks of coupled dynamical units, we reveal stability criteria on all scales. In particular, we show that in systems such as the Kuramoto model the Coates graph of the Jacobian matrix must contain a spanning tree of positive elements for the system to be locally stable. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Jacobian Analysis <s> There are three key factors of a system of coupled oscillators that characterize the interaction among them: coupling (how to affect), delay (when to affect) and topology (whom to affect). For each of them, the existing work has mainly focused on special cases. With new angles and tools, this paper makes progress in relaxing some assumptions of these factors. There are three main results in this paper. First, by using results from algebraic graph theory, a sufficient condition is obtained which can be used to check equilibrium stability. This condition works for arbitrary topology. It generalizes existing results and also leads to a sufficient condition on the coupling function with which the system is guaranteed to reach synchronization. Second, it is known that identical oscillators with sin() coupling functions are guaranteed to synchronize in phase on a complete graph. Using our results, we demonstrate that for many cases certain structures instead of exact shape of the coupling function such as symmetry and concavity are the keys for global synchronization. Finally, the effect of heterogenous delays is investigated. We develop a new framework by constructing a non-delayed phase model that approximates the original one in the continuum limit. We further derive how its stability properties depend on the delay distribution. In particular, we show that heterogeneity, i.e. wider delay distribution, can help reach in-phase synchronization. <s> BIB012
We begin by drawing some insights from a Jacobian analysis. The right-hand side of the oscillator network (1) defines the vector field f : T n → R n with components , the Jacobian J(θ) of the coupled oscillator model (1) satisfies where B is the incidence matrix of the graph. Notice that for phase cohesive angles θ ∈ ∆ G (π/2), the Jacobian J(θ) equals minus the Laplacian matrix of the graph G(V, E,Ã) with strictly positive weightsã ij = a ij cos(θ i − θ j ) > 0, for {i, j} ∈ E. Hence, J(θ) is negative semidefinite, it inherits the sparsity of the graph G(V, E, A), and its nullspace Span(1 n ) arises from the rotational symmetry of the vector field (20). These basic results are fundamental to various analysis approaches. To the best of the authors' knowledge this set of results was first established by Tavora and Smith (1972b,a) , and it has since been rediscovered several times. Some consequences are collected in the following lemma, whose proof can be found in (Dörfler et al., 2013, Lemma 3) . Consider the coupled oscillator model (1) with a connected graph G(V, E, A) and frequencies ω ∈ 1 ⊥ n . If there exists an equilibrium θ * ∈ ∆ G (π/2), then is a negative semidefinite Laplacian matrix; (ii) the equilibrium manifold [θ * ] ⊂ ∆ G (π/2) is locally exponentially stable (modulo rotational symmetry). Some consequences of the particular form of the Jacobian (21) evaluated in ∆ G (π/2) are collected below. Frequency dynamics: The frequency dynamics obtained by differentiating the phase dynamics (1) are whereã ij (t) = a ij cos(θ i (t) − θ j (t)). The frequency dynamics (22) evolve on the tangent space of T n , that is, the Euclidean space R n . If the set ∆ G (γ) is forward invariant and θ(0) ∈ ∆ G (γ) for some γ ∈ [0, π/2[, then a ij (t) ≥ a ij cos(γ) > 0, for {i, j} ∈ E. Thus, the frequency dynamics (22) can be regarded as linear consensus protocol (5) with time-varying strictly-positive weights. Based on this observation, it can be shown that all frequenciesθ i (t) synchronize exponentially, that is, where λ fe = −λ 2 (L) cos(γ). We refer to (Chopra and Spong, 2009, Theorem 3.1) and BIB008 , Theorem 4.1) for a formal proof, and to (Schmidt et al., 2012, Lemma 3.5) , BIB008 , Theorem 4.1), and (Wang and Doyle, 2013, Theorem 4) for extensions to more general coupling functions, time-delays, and extensions to models with pacemakers. Contraction and incremental stability: Assume again that ∆ G (γ) is forward invariant for some γ ∈ [0, π/2[. Since −J(θ) is negative semidefinite in ∆ G (γ), it follows that the coupled oscillator dynamics (1) are contracting 5 relative to the nullspace 1 n . Consequently, the dynamics (1) are incrementally exponentially stable (modulo symmetry), that is, given any two initial values The application of contraction analysis to the coupled oscillator model (1) yields the incremental exponential stability (24) in 2 -type metrics (Chung and Slotine, 2010, Theorem 7) or in ∞ -type metrics  , Example 6). Choi et al. (2011a, Theorem 4 .1) report the incremental stability (24) in an 1 -metric. Finally, for discontinuous and monotone coupling functions and complete interaction graphs the total variation distance provides yet another 1 -type contraction metric . The results discussed so far in Subsection 4.1 are applicable only to angles inside the phase cohesive set ∆ G (π/2), where all weightsã ij = a ij cos(θ i − θ j ) are strictly positive for {i, j} ∈ E, and the Laplacian properties of the Jacobian J(θ) can be exploited. Outside the set ∆ G (π/2), the associated state-dependent graph G(V, E,Ã) may at times be disconnected and/or have negative weights a ij = a ij cos(θ i − θ j ) < 0. In this more general setting, the standard methods from algebraic and spectral graph theory cannot be applied and many puzzling examples are known BIB001 . A necessary condition for stability of arbitrary equilibrium manifolds [θ * ] ⊂ T n is that the graph induced by the Jacobian J(θ * ) possess a spanning tree with strictly positive weightsã ij > 0 along its edges BIB011 . Sufficient stability and instability conditions can be derived if the graph induced by J(θ * ) admits certain cutsets 5 We refer the reader to BIB004 ) for a treatment of contraction analysis and to BIB005 BIB009  for its extension to systems with symmetries. ( BIB001 BIB002 BIB003 BIB012 . Finally, for the complete graph with uniform weights (see the Kuramoto model (3)), additional insights and identities related to the Jacobian (21) can be found in BIB006 BIB007 BIB010 .