Exploring synchronization in complex oscillator networks

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 real-world 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 self-contained proofs that illustrate a sample of different analysis methods in a tutorial style.

[1]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[2]  Mandel,et al.  Global coupling with time delay in an array of semiconductor lasers , 2000, Physical review letters.

[3]  Michael Chertkov,et al.  Synchronization in complex oscillator networks and smart grids , 2012, Proceedings of the National Academy of Sciences.

[4]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[5]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[6]  G. Ermentrout,et al.  Coupled oscillators and the design of central pattern generators , 1988 .

[7]  J. Martinerie,et al.  The brainweb: Phase synchronization and large-scale integration , 2001, Nature Reviews Neuroscience.

[8]  Peter A. Tass,et al.  A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations , 2003, Biological Cybernetics.

[9]  Seung-Yeal Ha,et al.  Flocking and synchronization of particle models , 2010 .

[10]  L. Moreau,et al.  Stability of continuous-time distributed consensus algorithms , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[11]  A. Mauroy.  On the dichotomic collective behaviors of large populations of pulse-coupled firing oscillators , 2011 .

[12]  Y. Bar-Ness,et al.  Distributed synchronization in wireless networks , 2008, IEEE Signal Processing Magazine.

[13]  Alain Sarlette,et al.  Synchronization and balancing on the N-torus , 2007, Syst. Control. Lett..

[14]  S Post,et al.  Analysis of sigma models via projective structures , 2010 .

[15]  Soon-Jo Chung,et al.  On synchronization of coupled Hopf-Kuramoto oscillators with phase delays , 2010, 49th IEEE Conference on Decision and Control (CDC).

[16]  Marc Timme,et al.  Self-organized synchronization in decentralized power grids. , 2012, Physical review letters.

[17]  Peter W. Sauer,et al.  Power System Dynamics and Stability , 1997 .

[18]  Florian Dörfler,et al.  Kron Reduction of Graphs With Applications to Electrical Networks , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[19]  G. Ermentrout,et al.  Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I. , 1984 .

[20]  Daido,et al.  Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions. , 1992, Physical review letters.

[21]  J. Pantaleone,et al.  Synchronization of metronomes , 2002 .

[22]  Steven H. Strogatz,et al.  Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies , 1988 .

[23]  Chia-Chi Chu,et al.  Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective , 1995, Proc. IEEE.

[24]  Enrique Mallada,et al.  Synchronization of phase-coupled oscillators with arbitrary topology , 2010, Proceedings of the 2010 American Control Conference.

[25]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[26]  Oliver Mason,et al.  Global Phase-Locking in Finite Populations of Phase-Coupled Oscillators , 2007, SIAM J. Appl. Dyn. Syst..

[27]  A. Barabasi,et al.  Physics of the rhythmic applause. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  G. Bard Ermentrout,et al.  Synchronization in a pool of mutually coupled oscillators with random frequencies , 1985 .

[29]  Seung-Yeal Ha,et al.  On the complete synchronization of the Kuramoto phase model , 2010 .

[30]  J. Buck.  Synchronous Rhythmic Flashing of Fireflies. II. , 1938, The Quarterly Review of Biology.

[31]  Francesco Bullo,et al.  Distributed Control of Robotic Networks , 2009 .

[32]  David J. Hill,et al.  Cutset stability criterion for power systems using a structure-preserving model , 1986 .

[33]  G Jongen,et al.  Coupled dynamics of fast spins and slow exchange interactions in the XY spin glass , 2001 .

[34]  Dirk Aeyels,et al.  Partial entrainment in the finite Kuramoto-Sakaguchi model , 2007 .

[35]  Lubos Buzna,et al.  Synchronization in symmetric bipolar population networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  R. Olfati-Saber,et al.  Swarms on Sphere: A Programmable Swarm with Synchronous Behaviors like Oscillator Networks , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[37]  Anna Scaglione,et al.  A scalable synchronization protocol for large scale sensor networks and its applications , 2005, IEEE Journal on Selected Areas in Communications.

[38]  K. Dessouky,et al.  Network synchronization , 1985, Proceedings of the IEEE.

[39]  Florian Dörfler,et al.  Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators , 2009, Proceedings of the 2010 American Control Conference.

[40]  E. Ott,et al.  Exact results for the Kuramoto model with a bimodal frequency distribution. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  M. Ilić.  Network theoretic conditions for existence and uniqueness of steady state solutions to electric power circuits , 1992, [Proceedings] 1992 IEEE International Symposium on Circuits and Systems.

[42]  S. Strogatz,et al.  The size of the sync basin. , 2006, Chaos.

[43]  Frank Allgöwer,et al.  Consensus reaching in multi-agent packet-switched networks with non-linear coupling , 2009, Int. J. Control.

[44]  D. Aeyels,et al.  Stability of phase locking in a ring of unidirectionally coupled oscillators , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[45]  Shilpa Chakravartula,et al.  Complex Networks: Structure and Dynamics , 2014 .

[46]  A. Jadbabaie,et al.  On the stability of the Kuramoto model of coupled nonlinear oscillators , 2005, Proceedings of the 2004 American Control Conference.

[47]  John L Hudson,et al.  Emerging Coherence in a Population of Chemical Oscillators , 2002, Science.

[48]  S. Strogatz,et al.  Frequency locking in Josephson arrays: Connection with the Kuramoto model , 1998 .

[49]  Hoppensteadt,et al.  Synchronization of laser oscillators, associative memory, and optical neurocomputing , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[50]  Lee DeVille,et al.  Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model. , 2011, Chaos.

[51]  Yongqiang Wang,et al.  Increasing Sync Rate of Pulse-Coupled Oscillators via Phase Response Function Design: Theory and Application to Wireless Networks , 2012, IEEE Transactions on Control Systems Technology.

[52]  Yamir Moreno,et al.  Synchronization of Kuramoto oscillators in scale-free networks , 2004 .

[53]  Florian Dörfler,et al.  Droop-Controlled Inverters are Kuramoto Oscillators , 2012, ArXiv.

[54]  Rodolphe Sepulchre,et al.  Geometry and Symmetries in Coordination Control , 2009 .

[55]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion With Limited Communication , 2008, IEEE Transactions on Automatic Control.

[56]  Frank Allgöwer,et al.  Multi-agent speed consensus via delayed position feedback with application to Kuramoto oscillators , 2009, 2009 European Control Conference (ECC).

[57]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion: All-to-All Communication , 2007, IEEE Transactions on Automatic Control.

[58]  J. Jalife,et al.  Mechanisms of Sinoatrial Pacemaker Synchronization: A New Hypothesis , 1987, Circulation research.

[59]  Hyunsuk Hong,et al.  Inertia effects on periodic synchronization in a system of coupled oscillators , 1999 .

[60]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[61]  Wei Ren,et al.  Information consensus in multivehicle cooperative control , 2007, IEEE Control Systems.

[62]  J. L. Hemmen,et al.  Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators , 1993 .

[63]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[64]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[65]  Raghuraman Mudumbai,et al.  Consensus Based Carrier Synchronization in a Two Node Network , 2011 .

[66]  Eduardo Alberto Canale,et al.  On the Complexity of the Classification of Synchronizing Graphs , 2010, FGIT-GDC/CA.

[67]  A.R. Bergen,et al.  A Structure Preserving Model for Power System Stability Analysis , 1981, IEEE Transactions on Power Apparatus and Systems.

[68]  S. Strogatz,et al.  The spectrum of the locked state for the Kuramoto model of coupled oscillators , 2005 .

[69]  E. Izhikevich.  Weakly Coupled Oscillators , 2006 .

[70]  B. Ermentrout,et al.  An adaptive model for synchrony in the firefly Pteroptyx malaccae , 1991 .

[71]  Jorge Cortes,et al.  Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms , 2009 .

[72]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[73]  Seung‐Yeal Ha,et al.  Complete synchronization of Kuramoto oscillators with finite inertia , 2011 .

[74]  Dirk Aeyels,et al.  Existence of Partial Entrainment and Stability of Phase Locking Behavior of Coupled Oscillators , 2004 .

[75]  S. Kumagai,et al.  Steady-State Security Regions of Power Systems , 1982 .

[76]  Steven H. Strogatz,et al.  Sync: The Emerging Science of Spontaneous Order , 2003 .

[77]  Peter Saunders.  The geometry of biological time (2nd edn), by Arthur T. Winfree. Pp. 777. £46.50. 2001 ISBN 0 387 98992 7 (Springer). , 2002, The Mathematical Gazette.

[78]  David J. Hill,et al.  Power systems as dynamic networks , 2006, 2006 IEEE International Symposium on Circuits and Systems.

[79]  James M. Bower,et al.  The Role of Axonal Delay in the Synchronization of Networks of Coupled Cortical Oscillators , 1997, Journal of Computational Neuroscience.

[80]  H Hong,et al.  Spontaneous phase oscillation induced by inertia and time delay. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[81]  Peter A Tass,et al.  Phase chaos in coupled oscillators. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[82]  Pablo Monzon,et al.  On the Characterization of Families of Synchronizing Graphs for Kuramoto Coupled Oscillators , 2009 .

[83]  Manfredi Maggiore,et al.  State Agreement for Continuous-Time Coupled Nonlinear Systems , 2007, SIAM J. Control. Optim..

[84]  Edward Ott,et al.  Theoretical mechanics: Crowd synchrony on the Millennium Bridge , 2005, Nature.

[85]  Jan Lunze,et al.  Complete synchronization of Kuramoto oscillators , 2011 .

[86]  S. Sastry,et al.  Analysis of power-flow equation , 1981 .

[87]  Pravin Varaiya,et al.  Hierarchical stability and alert state steering control of interconnected power systems , 1980 .

[88]  T. J. Walker,et al.  Acoustic Synchrony: Two Mechanisms in the Snowy Tree Cricket , 1969, Science.

[89]  Daniel J. Klein.  Coordinated control and estimation for multi-agent systems: Theory and practice , 2008 .

[90]  Vittorio Rosato,et al.  Stability of a Distributed Generation Network Using the Kuramoto Models , 2008, CRITIS.

[91]  A. Lichtenberg,et al.  A First Order Phase Transition Resulting from Finite Inertia in Coupled Oscillator Systems , 1996 .

[92]  R. Spigler,et al.  Synchronization in populations of globally coupled oscillators with inertial effects , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[93]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[94]  Oliver Mason,et al.  A Convergence Result for the Kuramoto Model with All-to-All Coupling , 2011, SIAM J. Appl. Dyn. Syst..

[95]  Alex Arenas,et al.  Paths to synchronization on complex networks. , 2006, Physical review letters.

[96]  Alessio Franci,et al.  Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback , 2011, Autom..

[97]  A. Winfree.  The geometry of biological time , 1991 .

[98]  G. Filatrella,et al.  Analysis of a power grid using a Kuramoto-like model , 2007, 0705.1305.

[99]  Sergio Barbarossa,et al.  Decentralized Maximum-Likelihood Estimation for Sensor Networks Composed of Nonlinearly Coupled Dynamical Systems , 2006, IEEE Transactions on Signal Processing.

[100]  Steven H. Strogatz,et al.  Cellular Construction of a Circadian Clock: Period Determination in the Suprachiasmatic Nuclei , 1997, Cell.

[101]  Peter A. Tass,et al.  Desynchronization and chaos in the kuramoto model , 2005 .

[102]  Franco Robledo,et al.  The wheels: an infinite family of bi-connected planar synchronizing graphs , 2010, 2010 5th IEEE Conference on Industrial Electronics and Applications.

[103]  B Chance,et al.  Metabolic coupling and synchronization of NADH oscillations in yeast cell populations. , 1971, Archives of biochemistry and biophysics.

[104]  M. Hastings,et al.  Scaling in small-world resistor networks , 2005, cond-mat/0508056.

[105]  M. Slemrod,et al.  A fast–slow dynamical systems theory for the Kuramoto type phase model , 2011 .

[106]  Luca Scardovi,et al.  Clustering and synchronization in phase models with state dependent coupling , 2010, 49th IEEE Conference on Decision and Control (CDC).

[107]  Mark W. Spong,et al.  On Exponential Synchronization of Kuramoto Oscillators , 2009, IEEE Transactions on Automatic Control.

[108]  Lee Xavier DeVille,et al.  Transitions amongst synchronous solutions in the stochastic Kuramoto model , 2012 .

[109]  Tara Javidi,et al.  Integration of communication and control using discrete time Kuramoto models for multivehicle coordination over broadcast networks , 2007, 2007 46th IEEE Conference on Decision and Control.

[110]  Oliver Mason,et al.  On Computing the Critical Coupling Coefficient for the Kuramoto Model on a Complete Bipartite Graph , 2009, SIAM J. Appl. Dyn. Syst..

[111]  Pablo Monzón,et al.  Monotone measures and almost global stability of dynamical systems , 2004 .

[112]  Jean‐Claude Sabonnadière,et al.  Stability Analysis of Power Systems , 2013 .

[113]  Seung-Yeal Ha,et al.  Emergent behaviour of a generalized Viscek-type flocking model , 2010 .

[114]  A. Winfree.  Biological rhythms and the behavior of populations of coupled oscillators. , 1967, Journal of theoretical biology.

[115]  J. Pantaleone,et al.  Stability of incoherence in an isotropic gas of oscillating neutrinos , 1998 .

[116]  Pablo Monzon,et al.  Almost Global Synchronization of Symmetric Kuramoto Coupled Oscillators , 2008 .

[117]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[118]  Ali Nabi,et al.  Single input optimal control for globally coupled neuron networks , 2011, Journal of neural engineering.

[119]  P. Holmes,et al.  Globally Coupled Oscillator Networks , 2003 .

[120]  Yongqiang Wang,et al.  On influences of global and local cues on the rate of synchronization of oscillator networks , 2011, Autom..

[121]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[122]  A. Lichtenberg,et al.  Self-synchronization of coupled oscillators with hysteretic responses , 1997 .

[123]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[124]  Enrique Mallada,et al.  Distributed clock synchronization: Joint frequency and phase consensus , 2011, IEEE Conference on Decision and Control and European Control Conference.

[125]  Roberto Baldoni,et al.  Coupling-Based Internal Clock Synchronization for Large-Scale Dynamic Distributed Systems , 2010, IEEE Transactions on Parallel and Distributed Systems.

[126]  P. S. Krishnaprasad,et al.  Equilibria and steering laws for planar formations , 2004, Syst. Control. Lett..

[127]  Naomi Ehrich Leonard,et al.  Decision versus compromise for animal groups in motion , 2011, Proceedings of the National Academy of Sciences.

[128]  Sean P. Meyn,et al.  Synchronization of Coupled Oscillators is a Game , 2010, IEEE Transactions on Automatic Control.

[129]  Heidi M. Rockwood,et al.  Huygens's clocks , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[130]  Carlos J. Tavora,et al.  Equilibrium Analysis of Power Systems , 1972 .

[131]  Johan A K Suykens,et al.  Introduction to Focus Issue: synchronization in complex networks. , 2008, Chaos.

[132]  Florian Dörfler,et al.  On the Critical Coupling for Kuramoto Oscillators , 2010, SIAM J. Appl. Dyn. Syst..

[133]  R. Spigler,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[134]  Alain Sarlette,et al.  Consensus Optimization on Manifolds , 2008, SIAM J. Control. Optim..

[135]  R. C. Compton,et al.  Quasi-optical power combining using mutually synchronized oscillator arrays , 1991 .

[136]  Steven H. Strogatz,et al.  The Spectrum of the Partially Locked State for the Kuramoto Model , 2007, J. Nonlinear Sci..

[137]  R. Sepulchre,et al.  Oscillator Models and Collective Motion , 2007, IEEE Control Systems.

[138]  A. Kalloniatis,et al.  From incoherence to synchronicity in the network Kuramoto model. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.