Journal of Statistical Physics

, Volume 174, Issue 1, pp 160–187 | Cite as

Synchronization and Stability for Quantum Kuramoto

  • Lee DeVilleEmail author


We present and analyze a nonabelian version of the Kuramoto system, which we call the Quantum Kuramoto system. We study the stability of several classes of special solutions to this system, and show that for certain connection topologies the system supports multiple attractors. We also present estimates on the maximal possible heterogeneity in this system that can support an attractor, and study the effect of modifications analogous to phase-lag.


Kuramoto model Kuramoto–Sakaguchi model Lohe model Synchronization Quantum synchronization 

Mathematics Subject Classification

82C10 34D06 58C40 15A18 



The author thanks Jared Bronski, Thomas Carty, and Eddie Nijholt for illuminating discussions in the course of writing this manuscript. The author would also like to thank an anonymous referee for suggesting a line of investigation that culminated in the entirely new Theorem 2.21 and in enhancements to the conclusions of Theorem 2.18.


  1. 1.
    Abrams, D.M., Mirollo, R., Strogatz, S.H., Wiley, D.A.: Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101(8), 084103 (2008)ADSGoogle Scholar
  2. 2.
    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. 77(1), 137 (2005)ADSGoogle Scholar
  3. 3.
    Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)ADSMathSciNetGoogle Scholar
  4. 4.
    Axler, S.: Linear Algebra Done Right, vol. 2. Springer, New York (1997)zbMATHGoogle Scholar
  5. 5.
    Balmforth, N.J., Sassi, R.: A shocking display of synchrony. Phys. D 143(1–4), 21–55 (2000). Bifurcations, patterns and symmetryMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bronski, J.C., DeVille, L., Park, M.J.: Fully synchronous solutions and the synchronization phase transition for the finite-\(N\) Kuramoto model. Chaos 22(3), 033133 (2012)ADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bronski, J., Carty, T., DeVille, L.: Configurational stability for the Kuramoto–Sakaguchi model. submitted (2017)Google Scholar
  8. 8.
    Bronski, J.C., DeVille, L., Ferguson, T.: Graph homology and stability of coupled oscillator networks. SIAM J. Appl. Math. 76(3), 1126–1151 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bronski, J.C., Ferguson, T.: Volume bounds for the phase-locking region in the Kuramoto model. SIAM J. Appl. Dyn. Syst. 17(1), 128–156 (2018)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chi, D., Choi, S.-H., Ha, S.-Y.: Emergent behaviors of a holonomic particle system on a sphere. J. Math. Phys. 55(5), 052703 (2014)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Choi, S.-H., Ha, S.-Y.: Quantum synchronization of the Schrödinger–Lohe model. J. Phys. A 47(35), 355104 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Choi, S.-H., Ha, S.-Y.: Large-time dynamics of the asymptotic Lohe model with a small time-delay. J. Phys. A 48(42), 425101 (2015)ADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    Choi, S.-H., Ha, S.-Y.: Time-delayed interactions and synchronization of identical Lohe oscillators. Quart. Appl. Math. 74(2), 297–319 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chopra, N., Spong, M.: On synchronization of Kuramoto oscillators. In: Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05. 44th IEEE Conference on, pp 3916–3922. IEEE (2005)Google Scholar
  15. 15.
    Chopra, N., Spong, M.W.: On exponential synchronization of Kuramoto oscillators. IEEE Trans. Autom. Control 54(2), 353–357 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Collins, J.J., Stewart, I.N.: Coupled nonlinear oscillators and the symmetries of animal gaits. J. Nonlinear Sci. 3((1), 349–392 (1993)ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Davis, P.J.: Circulant Matrices. American Mathematical Society, Providence (2012)zbMATHGoogle Scholar
  18. 18.
    De Smet, F., Aeyels, D.: Partial entrainment in the finite Kuramoto–Sakaguchi model. Phys. D 234(2), 81–89 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Delabays, R., Coletta, T., Jacquod, P.: Multistability of phase-locking and topological winding numbers in locally coupled kuramoto models on single-loop networks. J. Math. Phys. 57(3), 032701 (2016)ADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Delabays, R., Coletta, T., Jacquod, P.: Multistability of phase-locking in equal-frequency Kuramoto models on planar graphs. J. Math. Phys. 58(3), 032703 (2017)ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    DeVille, L.: Transitions amongst synchronous solutions in the stochastic Kuramoto model. Nonlinearity 25(5), 1473 (2012)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    DeVille, L., Ermentrout, B.: Phase-locked patterns of the Kuramoto model on 3-regular graphs. Chaos 26(9), 094820 (2016)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Dörfler, F., Chertkov, M., Bullo, F.: Synchronization in complex oscillator networks and smart grids. Proc. Nat. Acad. Sci. 110(6), 2005–2010 (2013)ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    Dörfler, F., Bullo, F.: On the critical coupling for Kuramoto oscillators. SIAM J. Appl. Dyn. Syst. 10(3), 1070–1099 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Dörfler, F., Bullo, F.: Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators. SIAM J. Control Optim. 50(3), 1616–1642 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Dörfler, F., Bullo, F.: Synchronization in complex networks of phase oscillators: a survey. Automatica 50(6), 1539–1564 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. Rev. Modern Phys. 80(4), 1275 (2008)ADSGoogle Scholar
  28. 28.
    Ermentrout, G.B.: Synchronization in a pool of mutually coupled oscillators with random frequencies. J. Math. Biol. 22(1), 1–9 (1985)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ermentrout, G.B.: Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 52(6), 1665–1687 (1992)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Ferguson, T.: Topological states in the Kuramoto model. SIAM J. Appl. Dyn. Syst. 17(1), 484–499 (2018)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Galán, R.F., Ermentrout, G., Urban, N.N.: Efficient estimation of phase-resetting curves in real neurons and its significance for neural-network modeling. Physi. Rev. Lett. 94(15), 158101 (2005)ADSGoogle Scholar
  32. 32.
    Gray, R.M., et al.: Toeplitz and circulant matrices: a review. Found. Trends® Commun. Inform. Theory 2(3), 155–239 (2006)zbMATHGoogle Scholar
  33. 33.
    Ha, S.-Y., Jeong, E., Kang, M.-J.: Emergent behaviour of a generalized Viscek-type flocking model. Nonlinearity 23(12), 3139–3156 (2010)ADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Ha, S.-Y., Ko, D., Park, J., Zhang, X.: Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3(2), 209–267 (2016)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Ha, S.-Y., Ko, D., Ryoo, S .W.: Emergent dynamics of a generalized Lohe model on some class of Lie groups. J. Stat. Phys. 168(1), 171–207 (2017)ADSMathSciNetzbMATHGoogle Scholar
  36. 36.
    Ha, S.-Y., Lattanzio, C., Rubino, B., Slemrod, M.: Flocking and synchronization of particle models. Quart. Appl. Math. 69(1), 91–103 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Ha, S.-Y., Ryoo, S.W.: On the emergence and orbital stability of phase-locked states for the Lohe model. J. Stat. Phys. 163(2), 411–439 (2016)ADSMathSciNetzbMATHGoogle Scholar
  38. 38.
    Ha, S.-Y., Xiao, Q.: Remarks on the nonlinear stability of the Kuramoto–Sakaguchi equation. J. Differ. Equ. 259(6), 2430–2457 (2015)ADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Hansel, D., Sompolinsky, H.: Synchronization and computation in a chaotic neural network. Phys. Rev. Lett. 68(5), 718–721 (1992)ADSGoogle Scholar
  40. 40.
    Kirkland, S., Severini, S.: \(\alpha \)-kuramoto partitions from the frustrated Kuramoto model generalise equitable partitions. Appl. Anal. Discret. Math. 9, 29–38 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics (Kyoto University, Kyoto, 1975), pp 420–422. Lecture Notes in Physics, vol 39. Springer, Berlin (1975)Google Scholar
  42. 42.
    Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)zbMATHGoogle Scholar
  43. 43.
    Kuramoto, Y.: Collective synchronization of pulse-coupled oscillators and excitable units. Phys. D 50(1), 15–30 (1991)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. arXiv:cond-mat/0210694 (2002)
  45. 45.
    Lohe, M.A.: Non-abelian Kuramoto models and synchronization. J. Phys. A 42(39), 395101 (2009)ADSMathSciNetzbMATHGoogle Scholar
  46. 46.
    Lohe, M.A.: Quantum synchronization over quantum networks. J. Phys. A 43(46), 465301 (2010)ADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    Mehta, D., Daleo, N.S., Dörfler, F., Hauenstein, J.D.: Algebraic geometrization of the Kuramoto model: equilibria and stability analysis. Chaos 25(5), 053103 (2015)ADSMathSciNetzbMATHGoogle Scholar
  48. 48.
    Mehta, D., Hughes, C., Kastner, M., Wales, D.J.: Potential energy landscape of the two-dimensional XY model: higher-index stationary points. J. Chem. Phys. 140(22), 224503 (2014)ADSGoogle Scholar
  49. 49.
    Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645–1662 (1990)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Mirollo, R.E., Strogatz, S.H.: The spectrum of the locked state for the Kuramoto model of coupled oscillators. Phys. D 205(1–4), 249–266 (2005)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Omelchenko, E., Wolfrum, M.: Bifurcations in the Sakaguchi–Kuramoto model. Phys. D 263, 74–85 (2013)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  53. 53.
    Sakaguchi, H., Shinomoto, S., Kuramoto, Y.: Local and global self-entrainments in oscillator lattices. Prog. Theor. Phys. 77(5), 1005–1010 (1987)ADSGoogle Scholar
  54. 54.
    Sakaguchi, H., Shinomoto, S., Kuramoto, Y.: Mutual entrainment in oscillator lattices with nonvariational type interaction. Prog. Theor. Phys. 79(5), 1069–1079 (1988)ADSGoogle Scholar
  55. 55.
    Sastry, S., Varaiya, P.: Hierarchical stability and alert state steering control of interconnected power systems. IEEE Trans. Circuits Syst. 27(11), 1102–1112 (1980)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Sastry, S., Varaiya, P.: Coherency for interconnected power systems. IEEE Trans. Autom. Control 26(1), 218–226 (1981)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Progress Theor. Phys. 75(5), 1105–1110 (1986)ADSGoogle Scholar
  58. 58.
    Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143(1–4), 1–20 (2000). Bifurcations, patterns and symmetryMathSciNetzbMATHGoogle Scholar
  59. 59.
    Strogatz, S.H.: Sync: The Emerging Science of Spontaneous Order. Hyperion (2003)Google Scholar
  60. 60.
    Strogatz, S.H., Stewart, I.: Coupled oscillators and biological synchronization. Sci. Am. 269(6), 102–109 (1993)Google Scholar
  61. 61.
    Taylor, D., Ott, E., Restrepo, J.G.: Spontaneous synchronization of coupled oscillator systems with frequency adaptation. Phys. Rev. E 81(4), 046214–8 (2010)ADSMathSciNetGoogle Scholar
  62. 62.
    Tee, G.J.: Eigenvectors of block circulant and alternating circulant matrices. N. Z. J. Math. 36(8), 195–211 (2007)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Verwoerd, M., Mason, O.: Global phase-locking in finite populations of phase-coupled oscillators. SIAM J. Appl. Dyn. Syst. 7(1), 134–160 (2008)ADSMathSciNetzbMATHGoogle Scholar
  64. 64.
    Verwoerd, M., Mason, O.: On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph. SIAM J. Appl. Dyn. Syst. 8(1), 417–453 (2009)ADSMathSciNetzbMATHGoogle Scholar
  65. 65.
    Weiner, J.L., Wilkens, G.R.: Quaternions and rotations in ê4. Am. Math. Mon. 112(1), 69–76 (2005)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16, 015103 (2006)ADSMathSciNetzbMATHGoogle Scholar
  67. 67.
    Winfree, A.T.: The Geometry of Biological Time. Interdisciplinary Applied Mathematics, vol. 12. Springer, New York (2001)zbMATHGoogle Scholar
  68. 68.
    Witthaut, D., Wimberger, S., Burioni, R., Timme, M.: Classical synchronization indicates persistent entanglement in isolated quantum systems. Nat. Commun. 8, 14829 (2017)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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