On the critical coupling for kuramoto oscillators

SIAM Journal on Applied Dynamical Systems

Volumn 10, Issue 3, 2011, Pages 1070-1099

  Dörfler, Florian ^a   Bullo, Francesco ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Coupled oscillators; Kuramoto model; Synchronization

Indexed keywords

ANALYSIS TOOLS; ARBITRARY DISTRIBUTION; COUPLED OSCILLATORS; CRITICAL COUPLING; FIRST-ORDER; INCOHERENCY; INERTIAL COEFFICIENTS; INERTIAL EFFECT; KURAMOTO MODELS; LOCAL FREQUENCY; MULTI RATE; ORDER PARAMETER; PERFORMANCE INDICES; PHASE SYNCHRONIZATION; PRACTICAL STABILITY; SECOND ORDERS; STATISTICAL COMPARISONS; SUFFICIENT CONDITIONS; TIME CONSTANTS; TIME VARYING; VISCOUS DAMPING;

DYNAMICAL SYSTEMS; HETEROGENEOUS NETWORKS; NATURAL FREQUENCIES; OSCILLATORS (MECHANICAL); TIME VARYING NETWORKS; TOPOLOGY;

SYNCHRONIZATION;

EID: 80055017077     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/10081530X     Document Type: Article

References (56)

1. J. A. Acebrón, L. L. Bonilla, and R. Spigler, Synchronization in populations of globally coupled oscillators with inertial effects, Phys. Rev. E, 62 (2000), pp. 3437-3454.
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, Rev. Modern Phys., 77 (2005), pp. 137-185. (Pubitemid 41490657)
3. D. Aeyels and J. A. Rogge, Existence of partial entrainment and stability of phase locking behavior of coupled oscillators, Progress Theoret. Phys., 112 (2004), pp. 921-942. (Pubitemid 40366573)
4. F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38 (2000), pp. 1102-1119.
5. H. Attouch and P. E. Mainǵe, Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 1 (2009).
6. J. Baillieul and C. I. Byrnes, Geometric critical point analysis of lossless power system models, IEEE Trans. Circuits Syst., 29 (1982), pp. 724-737. (Pubitemid 13491290)
7. A. R. Bergen and D. J. Hill, A structure preserving model for power system stability analysis, IEEE Trans. Power Apparatus Systems, 100 (1981), pp. 25-35.
8. L. Buzna, S. Lozano, and A. Diaz-Guilera, Synchronization in symmetric bipolar population networks, Phys. Rev. E, 80 (2009), 66120.
9. E. Canale and P. Monzón, Almost global synchronization of symmetric Kuramoto coupled oscillators, in Systems Structure and Control, InTech Education and Publishing, 2008, pp. 167-190.
10. E. Canale and P. Monzón, On the characterization of families of synchronizing graphs for Kuramoto coupled oscillators, in IFAC Workshop on Distributed Estimation and Control in Networked Systems, Venice, Italy, 2009, pp. 42-47
11. D. Carlson and H. Schneider, Inertia theorems for matrices: The semidefinite case, J. Math. Anal. Appl., 6 (1963), pp. 430-446.
12. H.-D. Chiang and C. C. Chu, Theoretical foundation of the BCU method for direct stability analysis of network-reduction power system models with small transfer conductances, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), pp. 252-265.
13. H. D. Chiang and F. F. Wu, Stability of nonlinear systems described by a second-order vector differential equa tio n, IEEE Trans. Circuits Syst., 35 (2002), pp. 703-711.
14. Y.-P. Choi, S.-Y. Ha, and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Phys. D, 240 (2011), pp. 32-44.
15. N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control, 54 (2009), pp. 353-357.
16. C. C. Chu, Transient Dynamics of Electric Power Systems: Direct Stability Assessment and Chaotic Motions, Ph.D. thesis, Cornell University, 1996.
17. S. J. Chung and J. J. Slotine, On synchroniz ation of coupled Hopf-Kuramoto oscillators with phase delays, in Proceedings of the IEEE Conference on Decision and Control, Atlanta, GA, 2010, pp. 3181-3187.
18. E. A. Coayla-Teran, S. E. A. Mohammed, and P. R. C. Ruffino, Hartman-Grobman theorems along hyperbolic stationary trajectories, Dyn. Syst., 17 (2007), pp. 281-292.
19. D. Cumin and C. P. Unsworth, Generalising the Kuramoto model for the study of neuronal synchronisation in the brain, Phys. D, 226 (2007), pp. 181-196. (Pubitemid 46185616)
20. F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Phys. D, 234 (2007), pp. 81-89. (Pubitemid 47374836)
21. F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators, in American Control Conference, Baltimore, MD, 2010, pp. 930-937. Extended version available at http://arxiv.org/abs/0910.5673.
22. F. Dörfler and F. Bullo, On the critical coupling strength for Kuramoto oscillators, in Proceedings of the American Control Conference, San Francisco, CA, 2011, pp. 3239-3244.
23. F. Dörfler and F. Bullo, Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011, to appear.
24. G. B. Ermentrout, Synchronization in a pool o f mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), pp. 1-9. (Pubitemid 15057330)
25. G. B. Ermentrout, An adaptive model for synchrony in the firefly pteroptyx malaccae, J. Math. Biol., 29 (1991), pp. 571-585.
26. A. Franci, A. Chaillet, and W. Pasillas-Ĺepine, Phase-locking between Kuramoto oscillators: Robustness to time-varying natural frequencies, in Proceedings of the IEEE Conference on Decision and Control, Atlanta, GA, 2010, pp. 1587-1592.
27. X. Goudou and J. Munier, The gradient and heavy ball with friction dynamical systems: The quasi-co nvex case, Math. Prog., 116 (2009), pp. 173-191.
28. S.-Y. Ha, T. Ha, and J.-H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D, 239 (2010), pp. 1692-1700.
29. H. Hong, M. Y. Choi, J. Yi, and K. S. Soh, Inertia effects on periodic synchronization in a system of coupled oscillators, Phys. Rev. E, 59 (1999), pp. 353-363. (Pubitemid 129693908)
30. H. Hong, G. S. Jeon, and M. Y. Choi, Spontaneous phase oscillation induced by inertia and time delay, Phys. Rev. E, 65 (2002), 026208.
31. A. Jadbabaie, N. Motee, and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscil lators, in Proceedings of the American Control Conference, Boston, MA, 2004, pp. 4296-4301.
32. H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, Upper Saddle River, NJ, 2002.
33. D. E. Koditschek, Strict global Lyapunov functions for mechanical systems, in Proceedings of the 7th American Control Conference, 1988, pp. 1770-1775.
34. D. E. Koditschek, The application of total energy as a Lyapunov function for mechanical control systems, in Dynamics and Control of Multibody Systems, J. E. Marsden, P. S. Krishnaprasad, and J. C. Simo, eds., Comtemp. Math. 97, AMS, Providence, RI, 1989, pp. 131-157.
35. Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, H. Araki, ed., Lecture Notes in Phys. 39, Springer, Berlin, 1975, pp. 420-422.
36. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984.
37. Z. Lin, B. Francis, and M. Maggiore, State agreement for continuous-time coupled nonlinear systems, SIAM J. Control Optim., 46 (2007), pp. 288-307. (Pubitemid 351271298)
38. Y. L. Maistrenko, O. V. Popovych, and P. A. Tass, Desynchronization and chaos in the Kuramoto model, in Dynamics of Coupled Map Lattic es and of Related Spatially Extended Systems, J.-R. Chazottes and B. Fernandez, eds., Lecture No tes in Phys. 671, Springer, Berlin, 2005, pp. 285-306.
39. 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, Phys. Rev. E, 79 (2009), 26204.
40. D. Mehta and M. Kastner, Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian, Ann. Phys., 326 (2011), pp. 1425-1440.
41. R. E. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Phys. D, 205 (2005), pp. 249-266. (Pubitemid 40753192)
42. P. Monzón, Almost Global Stability of Dynamical Systems, Ph.D. thesis, Universidad de la República, Montevideo, Uruguay, 2006.
43. C. Robinson, Dynamical Systems: Stability, Sym bolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1999.
44. A. Sarlette, Geometry and Symmetries in Co ordination Control, Ph.D. thesis, University of Lìege, Belgium, 2009.
45. L. Scardovi, Clustering and synchronization in phase models with state dependent coupling, in Proceedings of the IEEE Conference on Decision and Control, Atlanta, GA, 2010, pp. 627-632.
46. G. S. Schmidt, U. Münz, and F. Allgöwer, Multi-agent speed consensus via delayed position feedback with application to Kuramoto oscillators, in Proceedings of the European Control Conference, Budapest, Hungary, 2009, pp. 2464-2469.
47. R. Sepulchre, D. A. Paley, and N. E. Leonard, Stabilization of planar collective motion: All-to-all communication, IEEE Trans. Automat. Control, 52 (2007), pp. 811-824. (Pubitemid 46796067)
48. S. H. Strogatz, From Kuramoto to Crawford: E xploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), pp. 1-20.
49. H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems, Phys. Re v. Lett., 78 (1997), pp. 2104-2107. (Pubitemid 127651817)
50. H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Phys. D, 100 (1997), pp. 279-300. (Pubitemid 127701343)
51. P. A. Tass, A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations, Biol. Cybernet., 89 (2003), pp. 81-88.
52. J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Statist. Phys., 72 (1993), pp. 145-166.
53. M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 134-160.
54. M. Verwoerd and O. Mason, On computing t he critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 417-453.
55. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), pp. 1563-1569. (Pubitemid 128627146)
56. F. Zhang, The Schur Complement and Its Applications, Springer, Berlin, 2005.