Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> 1 Introduction.- 2 Bifurcations in Neuron Dynamics.- 3 Neural Networks.- 4 Introduction to Canonical Models.- 5 Local Analysis of WCNNs.- 6 Local Analysis of Singularly Perturbed WCNNs.- 7 Local Analysis of Weakly Connected Maps.- 8 Saddle-Node on a Limit Cycle.- 9 Weakly Connected Oscillators.- 10 Multiple Andronov-Hopf Bifurcation.- 11 Multiple Cusp Bifurcation.- 12 Quasi-Static Bifurcations.- 13 Synaptic Organizations of the Brain.- References. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition. In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology. "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties. The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines. Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> Basics on Synchronization and Paradigmatic Models.- Basic Models.- Synchronization Due to External Periodic Forcing.- Synchronization of Two Coupled Systems.- Synchronization in Geometrically Regular Ensembles.- Ensembles of Phase Oscillators.- Chains of Coupled Limit-Cycle Oscillators.- Ensembles of Chaotic Oscillators with a Periodic-Doubling Route to Chaos, R#x00F6 ssler Oscillators.- Intermittent-Like Oscillations in Chains of Coupled Maps.- Regular and Chaotic Phase Synchronization of Coupled Circle Maps.- Controlling Phase Synchronization in Oscillatory Networks.- Chains of Limit-Cycle Oscillators.- Chains and Lattices of Excitable Luo-Rudy Systems.- Synchronization in Complex Networks and Influence of Noise.- Noise-Induced Synchronization in Ensembles of Oscillatory and Excitable Systems.- Networks with Complex Topology. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis frame- work is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small- world networks, Markov processes and gossip-based algo- rithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simu- lation results are presented that demonstrate the role of small- world effects on the speed of consensus algorithms and cooperative control of multivehicle formations. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> In this paper, we study the behavior of a network of N agents, each evolving on the circle. We propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the (possibly time-varying and unidirectional) communication graphs. The global convergence analysis on the N-torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the Euclidean space. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems. Generalizing the work of Moreau, under the assumption that the vector fields satisfy a certain subtangentiality condition, it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time. The proof uses nonsmooth analysis. Second, the rendezvous problem for kinematic point-mass mobile robots is studied when the robots’ fields of view have a fixed radius. The circumcenter control law of Ando e [IEEE Trans. Robotics Automation, 15 (1999), pp. 818-828] is shown to solve the problem. The rendezvous problem is a kind of state agreement problem, but the interaction structure is state dependent. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> Synchronization in complex networks Alex Arenas, 1, 2, 3 Albert D´ iaz-Guilera, 4, 2 Jurgen Kurths, 5 Yamir Moreno, 2, 6 and Changsong Zhou 7 Departament d’Enginyeria Inform` tica i Matem` tiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain a a Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Departament de F´ isica Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain Institute of Physics, University of Potsdam PF 601553, 14415 Potsdam, Germany Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (Dated: February 6, 2008) Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many eﬀorts de- voted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive nu- merical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to diﬀerent disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences. PACS numbers: 05.45.Xt,89.75.Fb,89.75.Hc Contents I. Introduction II. Complex networks in a nutshell III. Coupled phase oscillator models on complex networks A. Phase oscillators 1. The Kuramoto model 2. Kuramoto model on complex networks 3. Onset of synchronization in complex networks 4. Path towards synchronization in complex networks 5. Kuramoto model on structured or modular networks 6. Synchronization by pacemakers B. Pulse-coupled models C. Coupled maps IV. Stability of the synchronized state in complex networks A. Master Stability Function formalism 1. Linear Stability and Master Stability Function 2. Measures of synchronizability 3. Synchronizability of typical network models 4. Synchronizability and structural characteristics of networks 5. Graph theoretical bounds to synchronizability 6. Synchronizability of weighted networks 7. Universal parameters controlling the synchronizability B. Design of synchronizable networks 1. Weighted couplings for enhancing synchronizability 2. Topological modiﬁcation for enhancing synchronizability 3. Optimization of synchronizability C. Beyond the Master Stability Function formalism V. Applications A. Biological systems and neuroscience 1. Genetic networks 2. Circadian rhythms 3. Ecology 4. Neuronal networks 5. Cortical networks B. Computer science and engineering 1. Parallel/Distributed computation 2. Data mining 3. Consensus problems 4. Communication networks 5. Material and traﬃc ﬂow 6. Power-grids C. Social sciences and economy 1. Opinion formation 2. Finance 3. World Trade Web D. Perspectives VI. Conclusions Acknowledgments References I. INTRODUCTION Synchronization, as an emerging phenomenon of a pop- ulation of dynamically interacting units, has fascinated humans from ancestral times. No matter whether the phenomenon is spontaneous or induced, synchronization captivates our minds and becomes one of the most in- teresting scientiﬁc problems. Synchronization processes are ubiquitous in nature and play a very important role in many diﬀerent contexts as biology, ecology, climatol- ogy, sociology, technology, or even in arts ((Osipov et al., 2007; Pikovsky et al., 2001). It is known that synchrony <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Gr(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle. <s> BIB012 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> We derive a master stability function (MSF) for synchronization in networks of coupled dynamical systems with small but arbitrary parametric variations. Analogous to the MSF for identical systems, our generalized MSF simultaneously solves the linear-stability problem for near-synchronous states (NSS) for all possible connectivity structures. We also derive a general sufficient condition for stable near-synchronization and show that the synchronization error scales linearly with the magnitude of parameter variations. Our analysis underlines the significant role played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems. <s> BIB013 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> In this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators, the Kuramoto model. We consider the special case of finite oscillators with distinct, bounded natural frequencies. Our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization. This bound improves the one derived by Jadbabaie . We then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group. We also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time. <s> BIB014 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical model and prototypical example: <s> Abstract Consensus problems have attracted significant attention in the control community over the last decade. They act as a rich source of new mathematical problems pertaining to the growing field of cooperative and distributed control. This paper is an introduction to consensus problems whose underlying state-space is not a linear space, but instead a highly symmetric nonlinear space such as the circle and other relevant generalizations. A geometric approach is shown to highlight the connection between several fundamental models of consensus, synchronization, and coordination, to raise significant global convergence issues not present in linear models, and to be relevant for a number of engineering applications, including the design of planar or spatial coordinated motions. <s> BIB015
Diffusively-coupled phase oscillators appear to be quite specific at first glance, but they are locally canonical models for weakly coupled and periodic limit-cycle oscillators BIB001 . This fact is established in work by the computational neuroscience community which has developed different approaches Kopell, 1984, 1991; BIB001  BIB005 to reduce general periodic limit-cycle oscillators and weak interaction models to diffusivelycoupled phase oscillator networks of the forṁ where h ij : S 1 → R are 2π-periodic coupling functions. Among such phase oscillators networks, the often encountered and most thoroughly studied case is that of anti-symmetric coupling without higher-order harmonics, that is, the oscillator network (1) with sinusoidal coupling. Moreover, the coupled oscillator model (1) serves as the prototypical example for synchronization in complex networks BIB002 BIB006 BIB007 BIB013 BIB011 , and its linearization is the well-known consensus protocol studied in networked control, see the surveys and monographs BIB008     . Indeed, numerous control scientists explored the coupled oscillator model (1) as a nonlinear generalization of the consensus protocol BIB003 BIB004 BIB009 Olfati-Saber, 2006; BIB010 BIB014 BIB012 BIB015 . The importance of phase oscillator networks does not stem only from their importance as local canonical models. Often they are naturally encountered in applications by first-principle modeling, as phenomenological models, or as a result of control design. In the following, we review a set of selected applications in sciences and technology.