Synchronization in complex networks of phase oscillators: A survey

Automatica

Volumn 50, Issue 6, 2014, Pages 1539-1564

  Dörfler, Florian ^a   Bullo, Francesco ^b  

^a ETH ZURICH   (Switzerland)

^b University of California   (United States)

Author keywords

Complex networks; Coupled oscillators; Kuramoto model; Nonlinear analysis; Synchronization

Indexed keywords

COMPLEX NETWORKS; ELECTRIC MACHINE CONTROL; NONLINEAR ANALYSIS; OSCILLATORS (MECHANICAL); PATTERN MATCHING; SURVEYS;

COUPLED OSCILLATORS; ELECTRIC POWER NETWORKS; INTERCONNECTION TOPOLOGIES; KURAMOTO MODELS; MULTI-DISCIPLINARY RESEARCH; PARTIAL SYNCHRONIZATION; PHASE AND FREQUENCY SYNCHRONIZATION; SCIENCE AND ENGINEERING;

SYNCHRONIZATION;

EID: 84902182078     PISSN: 00051098     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.automatica.2014.04.012     Document Type: Article

References (244)

1. A.A. Abidi, and L.O. Chua On the dynamics of Josephson-junction circuits IEE Journal on Electronic Circuits and Systems 3 4 1979 186 200
2. J.A. Acebrón, L.L. Bonilla, C.J.P. Vicente, F. Ritort, and R. Spigler The Kuramoto model: a simple paradigm for synchronization phenomena Reviews of Modern Physics 77 1 2005 137 185
3. R. Adler A study of locking phenomena in oscillators Proceedings of the IRE 34 6 1946 351 357
4. D. Aeyels, and J.A. Rogge Existence of partial entrainment and stability of phase locking behavior of coupled oscillators Progress on Theoretical Physics 112 6 2004 921 942
5. N. Ainsworth, and S. Grijalva A structure-preserving model and sufficient condition for frequency synchronization of lossless droop inverter-based AC networks IEEE Transactions on Power Systems 28 4 2013 4310 4319
6. D. Angeli A Lyapunov approach to incremental stability properties IEEE Transactions on Automatic Control 47 3 2002 410 421
7. S. Aoi, and K. Tsuchiya Locomotion control of a biped robot using nonlinear oscillators Autonomous Robots 19 3 2005 219 232
8. E.V. Appleton Automatic synchronization of triode oscillators Mathematical Proceedings of the Cambridge Philosophical Society 21 pt 111 1922 231
9. A. Araposthatis, S. Sastry, and P. Varaiya Analysis of power-flow equation International Journal of Electrical Power & Energy Systems 3 3 1981 115 126
10. M. Arcak Passivity as a design tool for group coordination IEEE Transactions on Automatic Control 52 8 2007 1380 1390
11. Arcak, M. (2012). Synchronization and pattern formation in diffusively coupled systems. In IEEE conf. on decision and control, Maui, HI, USA, Dec. (pp. 7184-7192).
12. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou Synchronization in complex networks Physics Reports 469 3 2008 93 153
13. B.C. Bag, K.G. Petrosyan, and C.K. Hu Influence of noise on the synchronization of the stochastic Kuramoto model Physical Review E 76 5 2007 056210
14. J. Baillieul, and C.I. Byrnes Geometric critical point analysis of lossless power system models IEEE Transactions on Circuits and Systems 29 11 1982 724 737
15. R. Baldoni, A. Corsaro, L. Querzoni, S. Scipioni, and S.T. Piergiovanni Coupling-based internal clock synchronization for large-scale dynamic distributed systems IEEE Transactions on Parallel and Distributed Systems 21 5 2010 607 619
16. N.J. Balmforth, and R. Sassi A shocking display of synchrony Physica D: Nonlinear Phenomena 143 1 2000 21 55
17. S. Barbarossa, and G. Scutari Decentralized maximum-likelihood estimation for sensor networks composed of nonlinearly coupled dynamical systems IEEE Transactions on Signal Processing 55 7 2007 3456 3470
18. M. Bennett, M.F. Schatz, H. Rockwood, and K. Wiesenfeld Huygens's clocks Proceedings: Mathematical, Physical and Engineering Sciences 458 2019 2002 563 579
19. A.R. Bergen, and D.J. Hill A structure preserving model for power system stability analysis IEEE Transactions on Power Apparatus and Systems 100 1 1981 25 35
20. N. Biggs Algebraic graph theory 2nd ed. 1994 Cambridge University Press
21. N. Biggs Algebraic potential theory on graphs Bulletin of the London Mathematical Society 29 6 1997 641 683
22. I. Blekhman Synchronization in science and technology 1988 American Society of Mechanical Engineers
23. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.U. Hwang Complex networks: structure and dynamics Physics Reports 424 4-5 2006 175 308
24. J.C. Bronski, L. DeVille, and M. Jip Park Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model Chaos: An Interdisciplinary Journal of Nonlinear Science 22 3 2012 033133
25. E. Brown, P. Holmes, and J. Moehlis Globally coupled oscillator networks E. Kaplan, J.E. Marsden, K.R. Sreenivasan, Perspectives and problems in nonlinear science: a celebratory volume in honor of larry sirovich 2003 Springer 183 215
26. E. Brown, J. Moehlis, and P. Holmes On the phase reduction and response dynamics of neural oscillator populations Neural Computation 16 4 2004 673 715
27. J. Buck Synchronous rhythmic flashing of fireflies. II Quarterly Review of Biology 63 3 1988 265 289
28. F. Bullo, J. Cortés, and S. Martínez Distributed control of robotic networks 2009 Princeton University Press
29. M. Bürger, D. Zelazo, and F. Allgöwer Hierarchical clustering of dynamical networks using a saddle-point analysis IEEE Transactions on Automatic Control 58 1 2013 113 124
30. Burylko, O. (2012). Competition and bifurcations in phase oscillator networks with positive and negative couplings. In Proceedings of the nonlinear dynamics of electronic systems, July (pp. 1-4).
31. L. Buzna, S. Lozano, and A. Diaz-Guilera Synchronization in symmetric bipolar population networks Physical Review E 80 6 2009 66120
32. E. Canale, and P. Monzón Almost global synchronization of symmetric Kuramoto coupled oscillators Systems structure and control 2008 InTech Education and Publishing 167 190
33. Canale, E. A.; Monzón, P.; Robledo, F. (2010a). On the complexity of the classification of synchronizing graphs. In Grid and distributed computing, control and automation, Jeju Island, Korea, Dec. (pp. 186-195).
34. Canale, E. A.; Monzón, P. A.; Robledo, F. (2010b). The wheels: an infinite family of bi-connected planar synchronizing graphs. In IEEE conf. industrial electronics and applications, Taichung, Taiwan, June (pp. 2204-2209).
35. K.S. Chandrashekhar, and D.J. Hill Cutset stability criterion for power systems using a structure-preserving model International Journal of Electrical Power & Energy Systems 8 3 1986 146 157
36. H.-D. Chiang, C.C. Chu, and G. Cauley Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective Proceedings of the IEEE 83 11 1995 1497 1529
37. H. Chiba A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model Ergodic Theory and Dynamical Systems 2014 1 73 in press
38. Y.P. Choi, S.Y. Ha, S. Jung, and Y. Kim Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model Physica D: Nonlinear Phenomena 241 7 2011 735 754
39. Y.-P. Choi, S.-Y. Ha, M. Kang, and M. Kang Exponential synchronization of finite-dimensional Kuramoto model at critical coupling strength Communications in Mathematical Sciences 11 2 2013 385 401
40. Y.-P. Choi, S.-Y. Ha, and S.-B. Yun Complete synchronization of Kuramoto oscillators with finite inertia Physica D 240 1 2011 32 44
41. N. Chopra, and M.W. Spong On exponential synchronization of Kuramoto oscillators IEEE Transactions on Automatic Control 54 2 2009 353 357
42. Chung, S. J.; Slotine, J. J. (2010). On synchronization of coupled Hopf-Kuramoto oscillators with phase delays. In IEEE conf. on decision and control, Atlanta, GA, USA, Dec. (pp. 3181-3187).
43. M.A. Cohen, and S. Grossberg Absolute stability of global pattern formation and parallel memory storage by competitive neural networks IEEE Transactions on Systems, Man and Cybernetics 13 5 1983 815 826
44. Conteville, L.; Panteley, E. (2012). Linear reformulation of the Kuramoto model: asymptotic mapping and stability properties. In Int. conf. on communications, computing and control applications, Hammamet, Tunisia, March.
45. J.D. Crawford Amplitude expansions for instabilities in populations of globally-coupled oscillators Journal of Statistical Physics 74 5 1994 1047 1084
46. J.D. Crawford Scaling and singularities in the entrainment of globally coupled oscillators Physical Review Letters 74 21 1995 4341
47. S.M. Crook, G.B. Ermentrout, M.C. Vanier, and J.M. Bower The role of axonal delay in the synchronization of networks of coupled cortical oscillators Journal of Computational Neuroscience 4 2 1997 161 172
48. M.C. Cross, and P.C. Hohenberg Pattern formation outside of equilibrium Reviews of Modern Physics 65 3 1993 851
49. H. Daido Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions Physical Review Letters 68 7 1992 1073 1076
50. H. Daido Generic scaling at the onset of macroscopic mutual entrainment in limit-cycle oscillators with uniform all-to-all coupling Physical Review Letters 73 5 1994 760 763
51. A.H. Dekker, and R. Taylor Synchronization properties of trees in the Kuramoto model SIAM Journal on Applied Dynamical Systems 12 2 2013 596 617
52. F. De Smet, and D. Aeyels Partial entrainment in the finite Kuramoto-Sakaguchi model Physica D: Nonlinear Phenomena 234 2 2007 81 89
53. L. DeVille Transitions amongst synchronous solutions for the stochastic Kuramoto model Nonlinearity 25 5 2011 1 20
54. A.L. Do, S. Boccaletti, and T. Gross Graphical notation reveals topological stability criteria for collective dynamics in complex networks Physical Review Letters 108 19 2012 194102
55. F. Dörfler, and F. Bullo On the critical coupling for Kuramoto oscillators SIAM Journal on Applied Dynamical Systems 10 3 2011 1070 1099
56. Dörfler, F.; Bullo, F. (2012a). Exploring synchronization in complex oscillator networks. In IEEE conf. on decision and control, Maui, HI, USA, Dec. (pp. 7157-7170).
57. F. Dörfler, and F. Bullo Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators SIAM Journal on Control and Optimization 50 3 2012 1616 1642
58. F. Dörfler, and F. Bullo Kron reduction of graphs with applications to electrical networks IEEE Transactions on Circuits and Systems I: Regular Papers 60 1 2013 150 163
59. Dörfler, F.; Bullo, F. (2013b). Novel insights into lossless AC and DC power flow. In IEEE power & energy society general meeting, Vancouver, BC, Canada, July.
60. F. Dörfler, M. Chertkov, and F. Bullo Synchronization in complex oscillator networks and smart grids Proceedings of the National Academy of Sciences 110 6 2013 2005 2010
61. S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes Critical phenomena in complex networks Reviews of Modern Physics 80 4 2008 1275
62. El Ati, A.; Panteley, E. (2013a). Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: a case study. In IEEE conf. on decision and control, Florence, Italy, Dec. (pp. 1319-1324).
63. El Ati, A.; Panteley, E. (2013b). Synchronization of phase oscillators with attractive and repulsive interconnections. In Int. conference on methods and models in automation and robotics (pp. 22-27).
64. G.B. Ermentrout Synchronization in a pool of mutually coupled oscillators with random frequencies Journal of Mathematical Biology 22 1 1985 1 9
65. G.B. Ermentrout An adaptive model for synchrony in the firefly pteroptyx malaccae Journal of Mathematical Biology 29 6 1991 571 585
66. G.B. Ermentrout, and N. Kopell Frequency plateaus in a chain of weakly coupled oscillators, I SIAM Journal on Mathematical Analysis 15 2 1984 215 237
67. G.B. Ermentrout, and N. Kopell Multiple pulse interactions and averaging in systems of coupled neural oscillators Journal of Mathematical Biology 29 3 1991 195 217
68. G. Filatrella, A.H. Nielsen, and N.F. Pedersen Analysis of a power grid using a Kuramoto-like model The European Physical Journal B 61 4 2008 485 491
69. V. Fioriti, S. Ruzzante, E. Castorini, E. Marchei, and V. Rosato Stability of a distributed generation network using the Kuramoto models Critical information infrastructure security Lecture notes in computer science 2009 Springer 14 23
70. F. Forni, and R. Sepulchre A differential Lyapunov framework for contraction analysis IEEE Transactions on Automatic Control 59 3 2014 614 628
71. A. Franci, A. Chaillet, E. Panteley, and F. Lamnabhi-Lagarrigue Desynchronization and inhibition of Kuramoto oscillators by scalar mean-field feedback Mathematics of Control, Signals, and Systems 24 1-2 2012 169 217
72. A. Franci, A. Chaillet, and W. Pasillas-Lépine Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback Automatica 47 6 2011 1193 1202
73. B.A. Francis Oscillator synchronization J. Baillieul, T. Samad, Encyclopedia of systems and control 2015 Springer in press
74. F. Garin, and L. Schenato A survey on distributed estimation and control applications using linear consensus algorithms A. Bemporad, M. Heemels, M. Johansson, Networked control systems LNCIS 2010 Springer 75 107
75. A.K. Ghosh, B. Chance, and E.K. Pye Metabolic coupling and synchronization of NADH oscillations in yeast cell populations Archives of Biochemistry and Biophysics 145 1 1971 319 331
76. J. Gómez-Gardeñes, Y. Moreno, and A. Arenas Paths to synchronization on complex networks Physical Review Letters 98 3 2007 34101
77. Gupta, S.; Campa, A.; Ruffo, S. (2014). Kuramoto model of synchronization: equilibrium and nonequilibrium aspects. arXiv preprint arXiv:1403.2083.
78. S.-Y. Ha, T. Ha, and J.-H. Kim On the complete synchronization of the Kuramoto phase model Physica D: Nonlinear Phenomena 239 17 2010 1692 1700
79. S.Y. Ha, E. Jeong, and M.J. Kang Emergent behaviour of a generalized Viscek-type flocking model Nonlinearity 23 12 2010 3139 3156
80. S.Y. Ha, and M.J. Kang On the basin of attractors for the unidirectionally coupled Kuramoto model in a ring SIAM Journal on Applied Mathematics 72 5 2012 1549 1574
81. S. Ha, C. Lattanzio, B. Rubino, and M. Slemrod Flocking and synchronization of particle models Quarterly Applied Mathematics 69 1 2011 91 103
82. S.-Y. Ha, and Z. Li Complete synchronization of Kuramoto oscillators with hierarchical leadership Communications in Mathematical Sciences 12 3 2014 485 508
83. S.Y. Ha, and M. Slemrod A fast-slow dynamical systems theory for the Kuramoto type phase model Journal of Differential Equations 251 10 2011 2685 2695
84. J.L.V. Hemmen, and W.F. Wreszinski Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators Journal of Statistical Physics 72 1 1993 145 166
85. Hill, D.J.; Chen, G. (2006). Power systems as dynamic networks. In IEEE int. symposium on circuits and systems, Kos, Greece, May (pp. 722-725).
86. Y.W. Hong, and A. Scaglione A scalable synchronization protocol for large scale sensor networks and its applications IEEE Journal on Selected Areas in Communications 23 5 2005 1085 1099
87. H. Hong, and S.H. Strogatz Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators Physical Review Letters 106 5 2011 054102
88. F.C. Hoppensteadt, and E.M. Izhikevich Weakly connected neural networks 1997 Springer
89. F.C. Hoppensteadt, and E.M. Izhikevich Synchronization of laser oscillators, associative memory, and optical neurocomputing Physical Review E 62 3 2000 4010 4013
90. F.C. Hoppensteadt, and E.M. Izhikevich Synchronization of MEMS resonators and mechanical neurocomputing IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 48 2 2001 133 138
91. Huepe, C.; Cadiz, R. F.; Colasso, M. (2012). Generating music from flocking dynamics. In American control conference, Montréal, Canada, June (pp. 4339-4344).
92. C. Huygens M. Nijhoff, Oeuvres Complètes De Christiaan Huygens 1893 Societe Hollandaise des Sciences The Hague, Netherlands
93. T. Ichinomiya Frequency synchronization in a random oscillator network Physical Review E 70 2 2004 026116
94. A.J. Ijspeert Central pattern generators for locomotion control in animals and robots: a review Neural Networks 21 4 2008 642 653
95. E.M. Izhikevich Phase models with explicit time delays Physical Review E 58 1 1998 905
96. E.M. Izhikevich Dynamical systems in neuroscience: The geometry of excitability and bursting 2007 MIT Press
97. E.M. Izhikevich, and Y. Kuramoto Weakly coupled oscillators Encyclopedia of mathematical physics 5 2006 448
98. Jadbabaie, A.; Motee, N.; Barahona, M. (2004). On the stability of the Kuramoto model of coupled nonlinear oscillators. In: American control conference, Boston, MA, USA, June (pp. 4296-4301).
99. G. Jongen, J. Anemüller, D. Bollé, A.C.C. Coolen, and C. Perez-Vicente Coupled dynamics of fast spins and slow exchange interactions in the XY spin glass Journal of Physics A: Mathematical and General 34 19 2001 3957 3984
100. E.W. Justh, and P.S. Krishnaprasad Equilibria and steering laws for planar formations Systems & Control Letters 52 1 2004 25 38
101. A.C. Kalloniatis From incoherence to synchronicity in the network Kuramoto model Physical Review E 82 6 2010 066202
102. M. Kapitaniak, K. Czolczynski, P. Perlikowski, A. Stefanski, and T. Kapitaniak Synchronization of clocks Physics Reports 517 2012 1 69
103. H.K. Khalil Nonlinear systems 3rd ed. 2002 Prentice Hall
104. I.Z. Kiss, Y. Zhai, and J.L. Hudson Emerging coherence in a population of chemical oscillators Science 296 5573 2002 1676 1678
105. Klein, D. J. (2008). Coordinated control and estimation for multi-agent systems: theory and practice. Ph.D. Thesis, University of Washington.
106. D.J. Klein, P. Lee, K.A. Morgansen, and T. Javidi Integration of communication and control using discrete time Kuramoto models for multivehicle coordination over broadcast networks IEEE Journal on Selected Areas in Communications 26 4 2008 695 705
107. N. Kopell, and G.B. Ermentrout Coupled oscillators and the design of central pattern generators Mathematical Biosciences 90 1-2 1988 87 109
108. G. Korniss, M.B. Hastings, K.E. Bassler, M.J. Berryman, B. Kozma, and D. Abbott Scaling in small-world resistor networks Physics Letters A 350 5-6 2006 324 330
109. A.J. Korsak On the question of uniqueness of stable load-flow solutions IEEE Transactions on Power Apparatus and Systems 91 3 1972 1093 1100
110. G. Kozyreff, A.G. Vladimirov, and P. Mandel Global coupling with time delay in an array of semiconductor lasers Physical Review Letters 85 18 2000 3809 3812
111. Y. Kuramoto Self-entrainment of a population of coupled non-linear oscillators H. Araki, Int. symposium on mathematical problems in theoretical physics Lecture notes in physics vol. 39 1975 Springer 420 422
112. Y. Kuramoto Chemical oscillations, waves, and turbulence 1984 Springer
113. Y. Kuramoto Cooperative dynamics of oscillator community Progress of Theoretical Physics Supplement 79 1984 223 240
114. Y. Kuramoto Collective synchronization of pulse-coupled oscillators and excitable units Physica D: Nonlinear Phenomena 50 1 1991 15 30
115. C.R. Laing The dynamics of chimera states in heterogeneous Kuramoto networks Physica D: Nonlinear Phenomena 238 16 2009 1569 1588
116. N.E. Leonard, T. Shen, B. Nabet, L. Scardovi, I.D. Couzin, and S.A. Levin Decision versus compromise for animal groups in motion Proceedings of the National Academy of Sciences 109 1 2012 227 232
117. Z. Lin, B. Francis, and M. Maggiore State agreement for continuous-time coupled nonlinear systems SIAM Journal on Control and Optimization 46 1 2007 288 307
118. W.C. Lindsey, F. Ghazvinian, W.C. Hagmann, and K. Dessouky Network synchronization Proceedings of the IEEE 73 10 1985 1445 1467
119. C. Liu, D.R. Weaver, S.H. Strogatz, and S.M. Reppert Cellular construction of a circadian clock: period determination in the suprachiasmatic nuclei Cell 91 6 1997 855 860
120. W. Lohmiller, and J.-J.E. Slotine On contraction analysis for non-linear systems Automatica 34 6 1998 683 696
121. J. Lunze Complete synchronization of Kuramoto oscillators Journal of Physics A: Mathematical and Theoretical 44 2011 425102
122. J. Lunze Synchronization of heterogeneous agents IEEE Transactions on Automatic Control 57 11 2012 2885 2890
123. Y.L. Maistrenko, O.V. Popovych, and P.A. Tass Desynchronization and chaos in the Kuramoto model J.-R. Chazottes, B. Fernandez, Dynamics of coupled map lattices and of related spatially extended systems Lecture notes in physics vol. 671 2005 Springer 285 306
124. Mallada, E.; Tang, A. (2011). Distributed clock synchronization: joint frequency and phase consensus. In IEEE conf. on decision and control and European control conference, Orlando, FL, USA, Dec. (pp. 6742-6747).
125. E. Mallada, and A. Tang Synchronization of weakly coupled oscillators: coupling, delay and topology Journal of Physics A: Mathematical and Theoretical 46 50 2013
126. Mangesius, H.; Hirche, S.; Obradovic, D. (2012). Quasi-stationarity of electric power grid dynamics based on a spatially embedded Kuramoto model. In American control conference, Montréal, Canada, June (pp. 2159-2164).
127. E.A. Martens, E. Barreto, S.H. Strogatz, E. Ott, P. So, and T.M. Antonsen Exact results for the Kuramoto model with a bimodal frequency distribution Physical Review E 79 2 2009 26204
128. E.A. Martens, S. Thutupalli, A. Fourrière, and O. Hallatschek Chimera states in mechanical oscillator networks Proceedings of the National Academy of Sciences 110 26 2013 10563 10567
129. O. Mason, and M. Verwoerd Graph theory and networks in biology IET Systems Biology 1 2 2007 89 119
130. Mauroy, A.; Sacré, P.; Sepulchre, R.J. (2012). Kick synchronization versus diffusive synchronization. In IEEE conf. on decision and control, Maui, HI, USA, Dec. (pp. 7171-7183).
131. A. Mauroy, and R. Sepulchre Contraction of monotone phase-coupled oscillators Systems & Control Letters 61 11 2012 1097 1102
132. D. Mehta, and M. Kastner Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian Annals of Physics 326 6 2011 1425 1440
133. M. Mesbahi, and M. Egerstedt Graph theoretic methods in multiagent networks 2010 Princeton University Press
134. D.C. Michaels, E.P. Matyas, and J. Jalife Mechanisms of sinoatrial pacemaker synchronization: a new hypothesis Circulation Research 61 5 1987 704 714
135. R.E. Mirollo, and S.H. Strogatz Synchronization of pulse-coupled biological oscillators SIAM Journal on Applied Mathematics 50 6 1990 1645 1662
136. R.E. Mirollo, and S.H. Strogatz The spectrum of the locked state for the Kuramoto model of coupled oscillators Physica D: Nonlinear Phenomena 205 1-4 2005 249 266
137. R. Mirollo, and S.H. Strogatz The spectrum of the partially locked state for the Kuramoto model Journal of Nonlinear Science 17 4 2007 309 347
138. A. Mirzaei, M.E. Heidari, R. Bagheri, S. Chehrazi, and A.A. Abidi The quadrature LC oscillator: a complete portrait based on injection locking IEEE Journal of Solid-State Circuits 42 9 2007 1916 1932
139. Mizumoto, T.; Otsuka, T.; Nakadai, K.; Takahashi, T.; Komatani, K.; Ogata, T. et al. (2010). Human-robot ensemble between robot thereminist and human percussionist using coupled oscillator model. In: IEEE/RSJ int. conf. on intelligent robots & systems, Taipei, Taiwan, Oct. (pp. 1957-1963).
140. Monzón, P.; Paganini, F. (2005). Global considerations on the Kuramoto model of sinusoidally coupled oscillators. In: IEEE conf. on decision and control, San Diego, CA, USA, Dec. (pp. 3923-3928).
141. Moreau, L. (2004). Stability of continuous-time distributed consensus algorithms. In: IEEE conf. on decision and control, Nassau, Bahamas (pp. 3998-4003).
142. L. Moreau Stability of multiagent systems with time-dependent communication links IEEE Transactions on Automatic Control 50 2 2005 169 182
143. Y. Moreno, and A.F. Pacheco Synchronization of Kuramoto oscillators in scale-free networks EPL (Europhysics Letters) 68 2004 603
144. A.E. Motter, S.A. Myers, M. Anghel, and T. Nishikawa Spontaneous synchrony in power-grid networks Nature Physics 9 2013 191 197
145. U. Münz, A. Papachristodoulou, and F. Allgöwer Consensus reaching in multi-agent packet-switched networks with non-linear coupling International Journal of Control 82 5 2009 953 969
146. A. Nabi, and J. Moehlis Single input optimal control for globally coupled neuron networks Journal of Neural Engineering 8 6 2011 065008
147. Z. Néda, E. Ravasz, T. Vicsek, Y. Brechet, and A.-L. Barabási Physics of the rhythmic applause Physical Review E 61 6 2000 6987 6992
148. T. Nishikawa, A.E. Motter, Y.C. Lai, and F.C. Hoppensteadt Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? Physical Review Letters 91 1 2003 14101
149. J. Ochab, and P.F. Gora Synchronization of coupled oscillators in a local one-dimensional Kuramoto model Acta Physica Polonica. Series B, Proceedings Supplement 3 2 2010 453 462
150. Olfati-Saber, R. (2006). Swarms on sphere: a programmable swarm with synchronous behaviors like oscillator networks. In IEEE conf. on decision and control, San Diego, CA, USA (pp. 5060-5066).
151. R. Olfati-Saber, J.A. Fax, and R.M. Murray Consensus and cooperation in networked multi-agent systems Proceedings of the IEEE 95 1 2007 215 233
152. G.V. Osipov, J. Kurths, and C. Zhou Synchronization in oscillatory networks 2007 Springer
153. E. Ott, and T.M. Antonsen Low dimensional behavior of large systems of globally coupled oscillators Chaos: An Interdisciplinary Journal of Nonlinear Science 18 3 2008 037113
154. E. Ott, and T.M. Antonsen Long time evolution of phase oscillator systems Chaos: An Interdisciplinary Journal of Nonlinear Science 19 2 2009 023117
155. F. Paganini, and B.C. Lesieutre Generic properties, one-parameter deformations, and the BCU method IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 46 6 1999 760 763
156. D.A. Paley, N.E. Leonard, R. Sepulchre, D. Grunbaum, and J.K. Parrish Oscillator models and collective motion IEEE Control Systems Magazine 27 4 2007 89 105
157. J. Pantaleone Stability of incoherence in an isotropic gas of oscillating neutrinos Physical Review D 58 7 1998 073002
158. J. Pantaleone Synchronization of metronomes American Journal of Physics 70 2002 992
159. L.M. Pecora, and T.L. Carroll Synchronization in chaotic systems Physical Review Letters 64 8 1990 821 824
160. L.M. Pecora, and T.L. Carroll Master stability functions for synchronized coupled systems Physical Review Letters 80 10 1998 2109 2112
161. C.S. Peskin Mathematical aspects of heart physiology 1975 Courant Institute of Mathematical Sciences
162. A. Pikovsky, and M. Rosenblum Dynamics of heterogeneous oscillator ensembles in terms of collective variables Physica D: Nonlinear Phenomena 240 9 2011 872 881
163. A. Pikovsky, M. Rosenblum, and J. Kurths Synchronization: a universal concept in nonlinear sciences 2003 Cambridge University Press
164. A. Pluchino, S. Boccaletti, V. Latora, and A. Rapisarda Opinion dynamics and synchronization in a network of scientific collaborations Physica A. Statistical Mechanics and its Applications 372 2 2006 316 325
165. A. Pluchino, V. Latora, and A. Rapisarda Compromise and synchronization in opinion dynamics The European Physical Journal B. Condensed Matter and Complex Systems 50 1 2006 169 176
166. O.V. Popovych, Y.L. Maistrenko, and P.A. Tass Phase chaos in coupled oscillators Physical Review E 71 6 2005 065201
167. Rahman, M. M. U.; Mudumbai, R.; Dasgupta, S. (2011). Consensus based carrier synchronization in a two node network. In IFAC world congress, Milan, Italy, Aug. (pp. 10038-10043).
168. R.H. Rand, and P.J. Holmes Bifurcation of periodic motions in two weakly coupled van der Pol oscillators International Journal of Non-Linear Mechanics 15 4 1980 387 399
169. J.W.S.B. Rayleigh The theory of sound. vol. 2 1896 Macmillan
170. W. Ren, R.W. Beard, and E.M. Atkins Information consensus in multivehicle cooperative control: collective group behavior through local interaction IEEE Control Systems Magazine 27 2 2007 71 82
171. J.G. Restrepo, E. Ott, and B.R. Hunt Desynchronization waves and localized instabilities in oscillator arrays Physical Review Letters 93 11 2004 114101
172. J.G. Restrepo, E. Ott, and B.R. Hunt Onset of synchronization in large networks of coupled oscillators Physical Review E 71 3 2005 036151
173. J.G. Restrepo, E. Ott, and B.R. Hunt Synchronization in large directed networks of coupled phase oscillators Chaos: An Interdisciplinary Journal of Nonlinear Science 16 1 2006 015107-1 015107-10
174. Righetti, L.; Ijspeert, A. J. (2006). Programmable central pattern generators: an application to biped locomotion control. In IEEE int. conf. on robotics and automation, Orlando, FL, USA, May (pp. 1585-1590).
175. D.C. Roberts Linear reformulation of the Kuramoto model of self-synchronizing coupled oscillators Physical Review E 77 3 2008 031114
176. J.A. Rogge, and D. Aeyels Stability of phase locking in a ring of unidirectionally coupled oscillators Journal of Physics A: Mathematical and General 37 46 2004 11135 11148
177. M. Rohden, A. Sorge, M. Timme, and D. Witthaut Self-organized synchronization in decentralized power grids Physical Review Letters 109 6 2012 064101
178. G. Russo, and M. Di Bernardo Contraction theory and master stability function: linking two approaches to study synchronization of complex networks IEEE Transactions on Circuits and Systems II: Express Briefs 56 2 2009 177 181
179. Russo, G.; Di Bernardo, M.; Sontag, E. D. (2010). Stability of networked systems: a multi-scale approach using contraction. In IEEE conf. on decision and control, Atlanta, GA, USA (pp. 6559-6564).
180. Sacré, P. (2013). Systems analysis of oscillator models in the space of phase response curves. Ph.D. Thesis, University of Liège, Sep.
181. P. Sacré, and R. Sepulchre Sensitivity analysis of oscillator models in the space of phase-response curves: oscillators as open systems IEEE Control Systems Magazine 34 2 2014 50 74
182. Sarlette, A. (2009). Geometry and symmetries in coordination control. Ph.D. Thesis, University of Liège, Belgium, Jan.
183. A. Sarlette, and R. Sepulchre Consensus optimization on manifolds SIAM Journal on Control and Optimization 48 1 2009 56 76
184. A. Sarlette, and R. Sepulchre Synchronization on the circle J. Dubbeldam, K. Green, D. Lenstra, The complexity of dynamical systems 2011 Wiley
185. S. Sastry, and P. Varaiya Hierarchical stability and alert state steering control of interconnected power systems IEEE Transactions on Circuits and Systems 27 11 1980 1102 1112
186. P.W. Sauer, and M.A. Pai Power system dynamics and stability 1998 Prentice Hall
187. Scardovi, L. (2010). Clustering and synchronization in phase models with state dependent coupling. In IEEE conf. on decision and control, Atlanta, GA, USA, Dec. (pp. 627-632).
188. L. Scardovi, A. Sarlette, and R. Sepulchre Synchronization and balancing on the N-torus Systems & Control Letters 56 5 2007 335 341
189. G.S. Schmidt, A. Papachristodoulou, U. Münz, and F. Allgöwer Frequency synchronization and phase agreement in Kuramoto oscillator networks with delays Automatica 48 12 2012 3008 3017
190. R. Sepulchre Consensus on nonlinear spaces Annual Reviews in Control 35 1 2011 56 64
191. R. Sepulchre, D.A. Paley, and N.E. Leonard Stabilization of planar collective motion: all-to-all communication IEEE Transactions on Automatic Control 52 5 2007 811 824
192. R. Sepulchre, D.A. Paley, and N.E. Leonard Stabilization of planar collective motion with limited communication IEEE Transactions on Automatic Control 53 3 2008 706 719
193. S.Y. Shafi, M. Arcak, M. Jovanović, and A.K. Packard Synchronization of diffusively-coupled limit cycle oscillators Automatica 49 12 2013 3613 3622
194. S.B. Shim, M. Imboden, and P. Mohanty Synchronized oscillation in coupled nanomechanical oscillators Science 316 5821 2007 95 99
195. O. Simeone, U. Spagnolini, Y. Bar-Ness, and S.H. Strogatz Distributed synchronization in wireless networks IEEE Signal Processing Magazine 25 5 2008 81 97
196. J.W. Simpson-Porco, F. Dörfler, and F. Bullo Synchronization and power sharing for droop-controlled inverters in islanded microgrids Automatica 49 9 2013 2603 2611
197. P.S. Skardal, J. Sun, D. Taylor, and J.G. Restrepo Effects of degree-frequency correlations on network synchronization: universality and full phase-locking EPL (Europhysics Letters) 101 2 2013 20001
198. P. So, T.B. Luke, and E. Barreto Networks of theta neurons with time-varying excitability: macroscopic chaos, multistability, and final-state uncertainty Physica D: Nonlinear Phenomena 267 2014 16 26
199. B. Sonnenschein, and L. Schimansky-Geier Onset of synchronization in complex networks of noisy oscillators Physical Review E 85 5 2012 051116
200. E.D. Sontag Contractive systems with inputs J.C. Willems, S. Hara, Y. Ohta, H. Fujioka, Perspectives in mathematical system theory, control, and signal processing 2010 Springer 217 228
201. F. Sorrentino, and M. Porfiri Analysis of parameter mismatches in the master stability function for network synchronization EPL (Europhysics Letters) 93 5 2011 50002
202. S.H. Strogatz From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators Physica D: Nonlinear Phenomena 143 1 2000 1 20
203. S.H. Strogatz Exploring complex networks Nature 410 6825 2001 268 276
204. S.H. Strogatz SYNC: The emerging science of spontaneous order 2003 Hyperion
205. S.H. Strogatz, D.M. Abrams, A. McRobie, B. Eckhardt, and E. Ott Theoretical mechanics: crowd synchrony on the millennium bridge Nature 438 7064 2005 43 44
206. S.H. Strogatz, and R.E. Mirollo Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies Physica D: Nonlinear Phenomena 31 2 1988 143 168
207. D. Subbarao, R. Uma, B. Saha, and M.V.R. Phanendra Self-organization on a power system IEEE Power Engineering Review 21 12 2001 59 61
208. J. Sun, E.M. Bollt, and T. Nishikawa Master stability functions for coupled nearly identical dynamical systems EPL (Europhysics Letters) 85 6 2009 60011
209. J.A.K. Suykens, and G.V. Osipov Introduction to focus issue: synchronization in complex networks Chaos: An Interdisciplinary Journal of Nonlinear Science 18 3 2008 037101-1 037101-4
210. H.A. Tanaka, A.J. Lichtenberg, and S. Oishi Self-synchronization of coupled oscillators with hysteretic responses Physica D: Nonlinear Phenomena 100 3-4 1997 279 300
211. P.A. Tass A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations Biological Cybernetics 89 2 2003 81 88
212. C.J. Tavora, and O.J.M. Smith Equilibrium analysis of power systems IEEE Transactions on Power Apparatus and Systems 91 3 1972 1131 1137
213. C.J. Tavora, and O.J.M. Smith Stability analysis of power systems IEEE Transactions on Power Apparatus and Systems 91 3 1972 1138 1144
214. R. Taylor There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model Journal of Physics A: Mathematical and Theoretical 45 5 2012 1 15
215. Tilton, A. K.; Hsiao-Wecksler, E. T.; Mehta, P. G. (2012). Filtering with rhythms: Application to estimation of gait cycle. In American control conference, Montréal, Canada, June (pp. 3433-3438).
216. Tönjes, R. (2007). Pattern formation through synchronization in systems of nonidentical autonomous oscillators. Ph.D. Thesis, Universitäts Potsdam, Germany.
217. E.S. Tuna Synchronization analysis of coupled Lienard-type oscillators by averaging Automatica 48 8 2012 1885 1891
218. B. Van Der Pol Forced oscillations in a circuit with non-linear resistance The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 3 13 1927 65 80
219. F. Varela, J.P. Lachaux, E. Rodriguez, and J. Martinerie The brainweb: phase synchronization and large-scale integration Nature Reviews Neuroscience 2 4 2001 229 239
220. E. Vassilieva, G. Pinto, J. Acacio de Barros, and P. Suppes Learning pattern recognition through quasi-synchronization of phase oscillators IEEE Transactions on Neural Networks 22 1 2011 84 95
221. M. Verwoerd, and O. Mason Global phase-locking in finite populations of phase-coupled oscillators SIAM Journal on Applied Dynamical Systems 7 1 2008 134 160
222. M. Verwoerd, and O. Mason On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph SIAM Journal on Applied Dynamical Systems 8 1 2009 417 453
223. M. Verwoerd, and O. Mason A convergence result for the Kuramoto model with all-to-all coupling SIAM Journal on Applied Dynamical Systems 10 3 2011 906 920
224. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet Novel type of phase transition in a system of self-driven particles Physical Review Letters 75 6-7 1995 1226 1229
225. T.J. Walker Acoustic synchrony: two mechanisms in the snowy tree cricket Science 166 3907 1969 891 894
226. Y. Wang, and F.J. Doyle Optimal phase response functions for fast pulse-coupled synchronization in wireless sensor networks IEEE Transactions on Signal Processing 60 10 2012 5583 5588
227. Y. Wang, and F.J. Doyle Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker IEEE Transactions on Automatic Control 58 4 2013 989 994
228. Wang, W.; Ghosh, B. (2007). Kuramoto models, coupled oscillations and laser networks. In SICE annual conference, Kagawa, Japan, Sep. (pp. 130-135).
229. Y. Wang, F. Núñez, and F.J. Doyle Increasing sync rate of pulse-coupled oscillators via phase response function design: theory and application to wireless networks IEEE Transactions on Control Systems Technology 21 4 2013 1455 1462
230. W. Wang, and J.-J.E. Slotine On partial contraction analysis for coupled nonlinear oscillators Biological Cybernetics 92 1 2005 38 53
231. Wieland, P. (2010). From static to dynamic couplings in consensus and synchronization among identical and non-identical systems. Ph.D. Thesis, Universität Stuttgart.
232. N. Wiener Cybernetics; or control and communication in the animal and the machine 1948 Wiley
233. N. Wiener Nonlinear problems in random theory 1958 MIT Press
234. K. Wiesenfeld, P. Colet, and S.H. Strogatz Frequency locking in Josephson arrays: connection with the Kuramoto model Physical Review E 57 2 1998 1563 1569
235. D.A. Wiley, S.H. Strogatz, and M. Girvan The size of the sync basin Chaos: An Interdisciplinary Journal of Nonlinear Science 16 1 2006 015103
236. A.T. Winfree Biological rhythms and the behavior of populations of coupled oscillators Journal of Theoretical Biology 16 1 1967 15 42
237. A.T. Winfree The geometry of biological time 2nd ed. 2001 Springer
238. Witthaut, D.; Timme, M. (2013). Kuramoto dynamics in Hamiltonian systems. arXiv preprint arXiv:1305.1742.
239. F.F. Wu, and S. Kumagai Limits on power injections for power flow equations to have secure solutions 1980 Electronics Research Laboratory College of Engineering, University of California
240. F. Wu, and S. Kumagai Steady-state security regions of power systems IEEE Transactions on Circuits and Systems 29 11 1982 703 711
241. H. Yin, P.G. Mehta, S.P. Meyn, and U.V. Shanbhag Synchronization of coupled oscillators is a game IEEE Transactions on Automatic Control 57 4 2012 920 935
242. R.A. York, and R.C. Compton Quasi-optical power combining using mutually synchronized oscillator arrays IEEE Transactions on Microwave Theory and Techniques 39 6 2002 1000 1009
243. M. Zhang, G.S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson Synchronization of micromechanical oscillators using light Physical Review Letters 109 23 2012 233906
244. J. Zhu Synchronization of Kuramoto model in a high-dimensional linear space Physics Letters A 377 41 2013 2939 2943