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Symmetry and symmetry breaking in coupled oscillator communities

  • Per Sebastian SkardalEmail author
Regular Article
  • 88 Downloads

Abstract

With the recent development of analytical methods for studying the collective dynamics of coupled oscillator systems, the dynamics of communities of coupled oscillators have received a great deal of attention in the nonlinear dynamics community. However, the majority of these works treat systems with a number of symmetries to simplify the analysis. In this work we study the role of symmetry and symmetry-breaking in the collective dynamics of coupled oscillator communities, allowing for a comparison between the macroscopic dynamics of symmetric and asymmetric systems. We begin by treating the symmetric case, deriving the bifurcation diagram as a function of intra- and inter-community coupling strengths. In particular we describe transitions between incoherence, standing wave, and partially synchronized states and reveal bistability regions. When we turn our attention to the asymmetric case we find that the symmetry-breaking complicates the bifurcation diagram. For instance, a pitchfork bifurcation in the symmetric case is broken, giving rise to a Hopf bifurcation. Moreover, an additional partially synchronized state emerges, as well as a new bistability region.

Graphical abstract

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    A.T. Winfree, The Geometry of Biological Time, 2nd edn. (Springer, New York, 2001) Google Scholar
  2. 2.
    S.H. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hypernion, 2003) Google Scholar
  3. 3.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2003) Google Scholar
  4. 4.
    A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Phys. Rep. 469, 93 (2008) MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Buck, Q. Rev. Biol. 63, 265 (1988) CrossRefGoogle Scholar
  6. 6.
    L. Glass, M.C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, 1988) Google Scholar
  7. 7.
    S.H. Strogatz, J. Math. Biol. 25, 327 (1987) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Z. Lu, K. Klein-Cardeña, S. Lee, T.M. Antonsen, M. Girvan, E. Ott, Chaos 26, 094811 (2016) MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Prindle, P. Samayoa, I. Razinkov, T. Danino, L.S. Tsimring, J. Hasty, Nature 481, 39 (2012) CrossRefGoogle Scholar
  10. 10.
    K. Wiesenfeld, P. Colet, S.H. Strogatz, Phys. Rev. E 57, 1563 (1998) CrossRefGoogle Scholar
  11. 11.
    M. Rohen, A. Sorge, M. Timme, D. Witthaut, Phys. Rev. Lett. 109, 064101 (2012) CrossRefGoogle Scholar
  12. 12.
    P.S. Skardal, A. Arenas, Sci. Adv. 1, e1500339 (2015) CrossRefGoogle Scholar
  13. 13.
    Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, New York, 1984) Google Scholar
  14. 14.
    Y. Moreno, A.F. Pacheco, Europhys. Lett. 68, 603 (2004) CrossRefGoogle Scholar
  15. 15.
    T. Ichinomiya, Phys. Rev. E 70, 026116 (2004) CrossRefGoogle Scholar
  16. 16.
    J.G. Restrepo, E. Ott, B.R. Hunt, Phys. Rev. E 71, 036151 (2005) MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Arenas, A. Díaz-Guilera, C.J. Peréz-Vicente, Phys. Rev. Lett. 96, 114102 (2006) CrossRefGoogle Scholar
  18. 18.
    J. Gómez-Gardeñes, Y. Moreno, A. Arenas, Phys. Rev. Lett. 98, 034101 (2007) CrossRefGoogle Scholar
  19. 19.
    J. Gómez-Gardeñes, S. Gómez, A. Arenas, Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011) CrossRefGoogle Scholar
  20. 20.
    J.G. Restrepo, E. Ott, Europhys. Lett. 107, 60006 (2014) CrossRefGoogle Scholar
  21. 21.
    P.S. Skardal, J.G. Restrepo, E. Ott, Phys. Rev. E 91, 060902(R) (2015) MathSciNetCrossRefGoogle Scholar
  22. 22.
    P.S. Skardal, D. Taylor, J. Sun, Phys. Rev. Lett. 113, 144101 (2014) CrossRefGoogle Scholar
  23. 23.
    E. Ott, T.M. Antonsen, Chaos 18, 037113 (2008) MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Ott, T.M. Antonsen, Chaos 19, 023117 (2009) MathSciNetCrossRefGoogle Scholar
  25. 25.
    L.M. Childs, S.H. Strogatz, Chaos 18, 043128 (2008) MathSciNetCrossRefGoogle Scholar
  26. 26.
    W.S. Lee, E. Ott, T.M. Antonsen, Phys. Rev. Lett. 103, 044101 (2009) CrossRefGoogle Scholar
  27. 27.
    P.S. Skardal, E. Ott, J.G. Restrepo, Phys. Rev. E 84, 036208 (2011) CrossRefGoogle Scholar
  28. 28.
    P.S. Skardal, D. Taylor, J.G. Restrepo, Physica D 267, 27 (2014) MathSciNetCrossRefGoogle Scholar
  29. 29.
    H. Hong, S.H. Strogatz, Phys. Rev. Lett. 106, 054102 (2011) CrossRefGoogle Scholar
  30. 30.
    D. Pazó, E. Montbrió, Phys. Rev. X 4, 011009 (2014) Google Scholar
  31. 31.
    C. Laing, Phys. Rev. E 90, 010901(R) (2014) CrossRefGoogle Scholar
  32. 32.
    P.S.S. Skardal, Phys. Rev. E 98, 022207 (2018) MathSciNetCrossRefGoogle Scholar
  33. 33.
    D.M. Abrams, R. Mirollo, S.H. Strogatz, D.A. Wiley, Phys. Rev. Lett. 101, 084103 (2008) CrossRefGoogle Scholar
  34. 34.
    E. Barreto, B. Hunt, E. Ott, P. So, Phys. Rev. E 77, 036107 (2008) MathSciNetCrossRefGoogle Scholar
  35. 35.
    C.R. Laing, Chaos 19, 013113 (2009) MathSciNetCrossRefGoogle Scholar
  36. 36.
    P.S. Skardal, J.G. Restrepo, Phys. Rev. E 85, 016208 (2012) CrossRefGoogle Scholar
  37. 37.
    L.M. Alonso, G.B. Mindlin, Chaos 21, 023102 (2011) MathSciNetCrossRefGoogle Scholar
  38. 38.
    E.A. Martens, C. Bick, M.J. Panaggio, Chaos 26, 094819 (2016) MathSciNetCrossRefGoogle Scholar
  39. 39.
    C. Bick, M.J. Panagio, E.A. Martens, Chaos 28, 071102 (2018) MathSciNetCrossRefGoogle Scholar
  40. 40.
    B. Pietras, N. Deschle, A. Daffertshofer, Phys. Rev. E 94, 052211 (2016) CrossRefGoogle Scholar
  41. 41.
    E.A. Martens, E. Barreto, S.H. Strogatz, E. Ott, P. So, T.M. Antonsen, Phys. Rev. E 79, 026204 (2009) MathSciNetCrossRefGoogle Scholar
  42. 42.
    J.D. Crawford, J. Stat. Phys. 74, 1047 (1994) CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTrinity CollegeHartfordUSA

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