Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model

Chaos

Volumn 22, Issue 3, 2012, Pages

  Bronski, Jared C ^a   DeVille, Lee ^a   Jip Park, Moon ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84866952734     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.4745197     Document Type: Article

References (36)

1. Acebrón J.A. Bonilla L.L. Vicente C.J. P. Ritort F. Spigler R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 2005, 77(1):137. 10.1103/RevModPhys.77.137
2. Balmforth N.J. Sassi R. A shocking display of synchrony. Phys. D 2000, 143(1-4):21-55. 10.1016/S0167-2789(00)00095-6
3. Bennett M. Schatz M.F. Rockwood H. Wiesenfeld K. Huygens's Clocks. Philos. R. Soc. London, Ser. A 2002, 458(2019):563-579. 10.1098/rspa.2001.0888
4. Bertin E. Clusel M. Generalized extreme value statistics and sum of correlated variables. J. Phys. A 2006, 39(24):7607-7619. 10.1088/0305-4470/39/24/001
5. Chaiken S. A combinatorial proof of the all-minors matrix tree theorem. SIAM J. Algebraic Discrete Methods 1982, 3:319-329. 10.1137/0603033
6. Conway J.H. Sloane N.J. A. Sphere packings, lattices and groups. Fundamental Principles of Mathematical Sciences 1999, 290. 3rd ed. (Springer-Verlag, New York, ), Vol.
7. Dörfler F. Bullo F. On the critical coupling for Kuramoto oscillators. SIAM J. Appl. Dyn. Syst. 2011, 10:1070-1099. 10.1137/10081530X
8. Ermentrout G.B. Synchronization in a pool of mutually coupled oscillators with random frequencies. J. Math. Biol. 1985, 22(1):1-9. 10.1007/BF00276542
9. Ha S.-Y. Jeong E. Kang M.-J. Emergent behaviour of a generalized Viscek-type flocking model. Nonlinearity 2010, 23(12):3139-3156. 10.1088/0951-7715/23/12/008
10. Ha S.-Y. Lattanzio C. Rubino B. Slemrod M. Flocking and synchronization of particle models. Quart. Appl. Math. 2011, 69(1):91-103. 10.1090/S0033-569X-2010-01200-7
11. Hansel D. Sompolinsky H. Synchronization and computation in a chaotic neural network. Phys. Rev. Lett. 1992, 68(5):718-721. 10.1103/PhysRevLett.68.718
12. Huygens C. Horoloquium Oscilatorium 1673, (Parisiis, Paris).
13. Jadbabaie A. Motee N. Barahona M. On the stability of the Kuramoto model of coupled nonlinear oscillators. Proceedings of the American Control Conference 2004, 5:4296-4301. in Boston, MA, 30 June -2 July Vol. pp.
14. Keener J.P. Principles of Applied Mathematics 2000, revised edition (Westview Press).
15. Knight B.W. Dynamics of encoding in a population of neurons. J. Gen. Physiol. 1972, 59(6):734-766. 10.1085/jgp.59.6.734
16. Kopell N. Ermentrout G.B. Symmetry and phaselocking in chains of weakly coupled oscillators. Commun. Pure Appl. Math. 1986, 39(5):623-660. 10.1002/cpa.3160390504
17. Kuramoto Y. Self-entrainment of a population of coupled non-linear oscillators. International Symposium on Mathematical Problems in Theoretical Physics, Kyoto Univ., Kyoto, 1975 39:420-422. in (Springer, Berlin), Vol. pp.
18. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence 1984, 19. Springer Series in Synergetics (Springer-Verlag, Berlin, ), Vol.
19. Kuramoto Y. Collective synchronization of pulse-coupled oscillators and excitable units. Physica D 1991, 50(1):15-30. 10.1016/0167-2789(91)90075-K
20. Medvedev G.S. Kopell N. Synchronization and transient dynamics in the chains of electrically coupled FitzHugh-Nagumo oscillators. SIAM J. Appl. Math. 2001, 61(5):1762-1801. 10.1137/S0036139900368807
21. Strogatz S.H. Mirollo R.E. Strogatz S.H. Mirollo R.E. Collective synchronisation in lattices of nonlinear oscillators with randomness. Correigendum: J. Phys. A 1988, 21(24):4649. 10.1088/0305-4470/21/13/005, 10.1088/0305-4470/21/24/518, ; and.
22. Strogatz S.H. Mirollo R.E. Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies. Physica D. 1988, 31(2):143-168. 10.1016/0167-2789(88)90074-7
23. Mirollo R.E. Strogatz S.H. Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 1990, 50(6):1645-1662. 10.1137/0150098
24. Mirollo R.E. Strogatz S.H. The spectrum of the locked state for the Kuramoto model of coupled oscillators. Phys. D 2005, 205(1-4):249-266. 10.1016/j.physd.2005.01.017
25. Peskin C.S. Mathematical Aspects of Heart Physiology 1975, (Courant Institute of Mathematical Sciences, New York University, New York, ). Notes based on a course given at New York University during the year 1973/74, see.
26. Pikovsky A. Rosenblum M. Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences 2003, and (Cambridge University Press).
27. Reed M. Simon B. Methods of Modern Mathematical Physics. IV. Analysis of operators 1978, and (Academic, New York).
28. Resnick S.I. Extreme Values, Regular Variation and Point Processes 2008, Springer Series in Operations Research and Financial Engineering (Springer, New York).
29. Simon B. Spectral analysis of rank one perturbations and applications. Lectures on Mathematical Physics 1993, in Vancouver Summer School.
30. Strogatz S. Sync: The Emerging Science of Spontaneous Order 2003, (Hyperion).
31. Strogatz S.H. From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Phys. D 2000, 143(1-4):1-20. 10.1016/S0167-2789(00)00094-4
32. Taylor D. Ott E. Restrepo J.G. Spontaneous synchronization of coupled oscillator systems with frequency adaptation. Phys. Rev. E 2010, 81(4):046214. 10.1103/PhysRevE.81.046214
33. Van Hemmen J.L. Wreszinski W.F. Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators. J. Stat. Phys. 1993, 72(1):145-166. 10.1007/BF01048044
34. Verwoerd M. Mason O. Global phase-locking in finite populations of phase-coupled oscillators. SIAM J. Appl. Dyn. Syst. 2008, 7(1):134-160. 10.1137/070686858
35. Verwoerd M. Mason O. On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph. SIAM J. Appl. Dyn. Syst. 2009, 8(1):417-453. 10.1137/080725726
36. Winfree A.T. The geometry of biological time. Interdisciplinary Applied Mathematics 2001, 12. 2nd ed. (Springer-Verlag, New York, ), Vol.