Synchronization in Complex Networks of Phase Oscillators: A Survey Florian D¨orfler a , Francesco Bullo b a Department of Electrical Engineering, University of California Los Angeles, USA b Department of Mechanical Engineering, University of California Santa Barbara, USA Abstract The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research. 1 Introduction Synchronization in networks of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and technological applications. A coupled oscillator net- work 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. Within the rich modeling phenomenology on synchro- nization among coupled oscillators, this article focuses on the widely adapted model of a continuous-time and periodic limit-cycle oscillator network with continuous, bidirectional, and antisymmetric coupling. We consider a system of n oscillators, each characterized by a phase angle θ i ∈ S 1 and a natural rotation frequency ω i ∈ R. The dynamics of each isolated oscillator are thus ˙ θ i = ω i ? This material is based in part upon work supported by UCLA startup funds and NSF grants IIS-0904501 and CPS- 1135819. A preliminary short version of this document ap- peared as (D¨ orfler and Bullo, 2012a). Email addresses: dorfler@seas.ucla.edu (Florian D¨ orfler), bullo@engineering.ucsb.edu (Francesco Bullo). for i ∈{1,...,n}. The interaction topology and coupling strength among the oscillators are modeled by a con- nected, undirected, and weighted graph G =(V , E ,A) with nodes V = {1,...,n}, edges E ⊂ V×V , and positive weights a ij = a ji > 0 for each undirected edge {i, j }∈E . The interaction between neighboring oscillators is as- sumed to be additive, anti-symmetric, diffusive, 1 and proportional to the coupling strengths a ij . In this case, the simplest 2π-periodic interaction function between neighboring oscillators {i, j }∈E is a ij sin(θ i - θ j ), and the overall model of coupled phase oscillators reads as ˙ θ i = ω i - X n j=1 a ij sin(θ i - θ j ) , i ∈{1,...,n} . (1) Despite its apparent simplicity, this coupled oscillator model gives rise to rich dynamic behavior, and it is en- countered in many scientific disciplines ranging from natural and life sciences to engineering. This article sur- veys recent results and applications of the coupled oscil- lator model (1) and of its variations. The motivations for this survey are manifold. Recent 1 The interaction between two oscillators is diffusive if its strength depends on the corresponding phase difference; such interactions arise for example in the discretization of the Laplace operator in diffusive partial differential equations. Preprint submitted to Automatica 23 March 2014