Synchronization in complex networks of phase oscillators: A survey <s> Potential Landscape Analysis <s> In this article we study global stability properties of the Kuramoto model of sinusoidally coupled oscillators. We base our analysis on previous results by the control community that analyze local properties of the consensus point of different kinds of Kuramoto models. We prove that for the complete symmetric case, the consensus point is almost globally stable, that is, the set of trajectories that do not converge to it has zero measure. We present a counter-example of that when the completeness hypothesis is removed. We also show that the general non-symmetric case is more complex and we analyze the particular case of oscillators coupled in a ring structure, where we can establish some global stability properties. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Potential Landscape Analysis <s> A few decades ago, Y. Kuramoto introduced a mathematical model of weakly coupled oscillators that gave a formal framework to some of the works of A.T. Winfree on biological clocks [Kuramoto (1975), Kuramoto (1984), Winfree (1980)]. The model proposes the idea that several oscillators can interact in a way such that the individual oscillation properties change in order to achieve a global behavior for the interconnected system. The Kuramoto model serves as a good representation of many systems in several contexts: biology, engineering, physics, mechanics, etc. [Ermentrout (1985), York (1993), Strogatz (1994), Dussopt et al. (1999), Strogatz (2000), Jadbabaie et a. (2003), Rogge et al. (2004), Marshall et al. (2004), Moshtagh et al. (2005)]. Recently, many works on the control community have focused on the analysis of the Kuramoto model, specially the one with sinusoidal coupling. The consensus or collective synchronization of the individuals is particularly important in many applications representing coordination, cooperation, emerging behavior, etc. Local stability properties of the consensus have been initially explored in [Jadbabaie et al. (2004)]. It must be noted that little attention has been devoted to the influence of the underlying interconnection graph on the stability properties of the system. The reason could be the fact that the local stability does not depend on the interconnection [van Hemmen et al. (1993)]. Global or almost global dynamical properties were studied in [Monzon et al. (2005), Monzon (2006), Monzon et al. (2006)]. In these works, the relevance of the interconnection graph of the system was hinted. In the present chapter, we go deeper on the analysis of the relationships between the dynamical properties of the system and the algebraic properties of the interconnection graph, exploiting the strong algebraic structure that every graph has. We step forward into a classification of the interconnection graphs that ensure almost global attraction of the set of synchronized states. In Section 2 we present the Kuramoto model for sinusoidally coupled oscillators, its general properties and the notion of almost global synchronization; in Section 3 we review some basic facts on algebraic graph theory; the symmetric Kuramoto model and the block analysis are presented in Sections 4 and 5; Section 6 gives some examples and applications of the main results; Section 7 presents the problem of classification of almost global synchronizing topologies. <s> BIB002
A classic analysis approach to oscillator networks with symmetric coupling can be deduced from the potential landscape. The potential energy U : T n → R of the spring network in Fig. 1 is, up to an additive constant, For the complete graph with uniform weights K/n, the magnitude r of the order parameter and the potential energy U (θ) are related by 2 n U (θ) = 1 − r 2 . One can easily verify that the phase-synchronized state is a local minimum of the potential energy. Given the potential energy in equation (25), the coupled oscillator model (1) can be reformulated as the forced gradient systeṁ This rewriting clarifies the competition between the synchronization-enforcing coupling through U (θ) and the synchronization-inhibiting heterogeneous natural frequencies ω i . The unforced system (26) with ω = 0 n is a negative gradient flowθ = −∂U (θ)/∂θ with the potential function U (θ) as natural Lyapunov function. Since the Jacobian J(θ) is the negative Hessian of the potential U (θ), Lemma 4.1 implies that any equilibrium in ∆ G (π/2) is a local minimizer of U (θ). Of particular interest are so-called S 1 -synchronizing graphs for which all critical points of (25) are hyperbolic, the phasesynchronized state is the global minimum of U (θ), and all other critical points are local maxima or saddle points. The class of S 1 -synchronizing graphs includes, among others, complete graphs and acyclic graphs BIB001 BIB002  Canale et al., 2010b,a) . These basic results motivated the study of the critical points and of the curvature of the potential energy U (θ) in the literature on the theory and applications of synchronization, including, the study of transient stability in power systems and the design of motion coordination controllers for planar vehicles, see Subsections 2.1 and 2.2. Some direct consequences of the gradient formulation (26) and of the associated Hessian matrix (21) will be presented in Section 5.