Advertisement

Synchronisation

  • Nicolás RubidoEmail author
Chapter
  • 466 Downloads
Part of the Springer Theses book series (Springer Theses)

Abstract

In this Chapter we derive a general framework to analyse the emergence of collective behaviour in any complex network of interacting phase-oscillators. Collective behaviour is ubiquitous. It appears spontaneously due to the interaction among the dynamical units composing a complex system and corresponds to an ordered state that can be absent in the dynamics of the individual units. Examples where collective behaviour are found include ecosystems, biological systems (as the brain or insect colonies), and even human societies and man-made systems (as the brain or insect colonies), and even human societies and man-made systems.

Keywords

Collective Behaviour Coupling Function Laplacian Matrix Phase Oscillator Dynamical Unit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2003)Google Scholar
  2. 2.
    S. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hyperion, New York, 2003)Google Scholar
  3. 3.
    S.C. Manrubia, A.S. Mikhailov, D.H. Zanette, Emergence of Dynamical Order (Synchronization Phenomena in Complex Systems (World Scientific, Singapore, 2004)Google Scholar
  4. 4.
    L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109 (1998)Google Scholar
  5. 5.
    M. Barahona, L.M. Pecora, Synchronization in Small-World Systems. Phys. Rev. Lett. 89(5), 054101 (2002)Google Scholar
  6. 6.
    J. Sun, E.M. Bollt, T. Nishikawa, Master stability functions for coupled nearly identical dynamical systems. Europhys. Lett. 85, 60011 (2009)Google Scholar
  7. 7.
    S. Acharyya, R. Amritkar, Synchronization of coupled nonidentical dynamical systems. Europhys. Lett. 99(4), 40005 (2012)Google Scholar
  8. 8.
    M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76(11), 1804 (1996)Google Scholar
  9. 9.
    G.V. Osipov, A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Phase synchronization effects in a lattice of nonidentical Rossler oscillators. Phys. Rev. E 55(3), 2353 (1997)Google Scholar
  10. 10.
    Z. Liu, Y.-C. Lai, F.C. Hoppensteadt, Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators. Phys. Rev. E 63, 055201(R) (2001)Google Scholar
  11. 11.
    L. Huang, Q. Chen, Y.-C. Lai, L.M. Pecora, Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Phys. Rev. E 80, 036204 (2009)Google Scholar
  12. 12.
    J. Gómez-Gardeñes, S. Gómez, A. Arenas, Y. Moreno, Explosive synchronization transitions in scale-free networks. Phys. Rev. Lett. 106, 128701 (2011)Google Scholar
  13. 13.
    P. Ji, T.K.DM. Peron, P.J. Menck, F.A. Rodrigues, J. Kurths, Cluster explosive synchronization in complex networks. Phys. Rev. Lett. 110, 218701 (2013)Google Scholar
  14. 14.
    Y. Kuramoto, D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators: a soluble case. Nonlinearity 26, 2469–2498 (2002)Google Scholar
  15. 15.
    D.M. Abrams, S.H. Strogatz, Chimera states for coupled oscillators. Phys. Rev. Lett. 93(17), 174102 (2004)Google Scholar
  16. 16.
    O.E. Omel’chenko, Y.L. Maistrenko, P.A. Tass, Chimera states: the natural link between coherence and incoherence. Phys. Rev. Lett. 100, 044105 (2008)Google Scholar
  17. 17.
    D.M. Abrams, R. Mirollo, S.H. Strogatz, D.A. Wiley, Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101, 084103 (2008)Google Scholar
  18. 18.
    O.E. Omel’chenko, M. Wolfrum, Y.L. Maistrenko, Chimera states as chaotic spatiotemporal patterns. Phys. Rev. E 81, 065201(R) (2010)Google Scholar
  19. 19.
    I. Omelchenko, Y. Maistrenko, P. Hövel, E. Schöll, Loss of coherence in dynamical networks: Spatial chaos and chimera states. Phys. Rev. Lett. 106, 234102 (2011)Google Scholar
  20. 20.
    M. Wolfrum, O.E. Omel’chenko, Chimera states are chaotic transients. Phys. Rev. E 84, 015201(R) (2011)Google Scholar
  21. 21.
    T. Pereira, M.S. Baptista, J. Kurths, General framework for phase synchronization through localized sets. Phys. Rev. E 75, 026216 (2007)Google Scholar
  22. 22.
    T. Pereira, M.S. Baptista, J. Kurths, Phase and average period of chaotic oscillators. Phys. Lett. A 362, 159–165 (2007)Google Scholar
  23. 23.
    K. Wiesenfeld, J.W. Swift, Averaged equations for Josephson junction series arrays. Phys. Rev. E 51(2), 1020 (1995)Google Scholar
  24. 24.
    K. Wiesenfeld, P. Colet, S.H. Strogatz, Synchronization transitions in a disordered Josephson series array. Phys. Rev. Lett. 76(3), 404 (1996)Google Scholar
  25. 25.
    K. Wiesenfeld, P. Colet, S.H. Strogatz, Frequency locking in Josephson arrays: connection with the Kuramoto model. Phys. Rev. E 57(2), 1563 (1998)Google Scholar
  26. 26.
    M. Perrin, G.L. Lippi, A. Politi, Phase transition in a radiation-matter interaction with recoil and collisions. Phys. Rev. Lett. 86(20), 4520 (2001)Google Scholar
  27. 27.
    J. Javaloyes, M. Perrin, A. Politi, Collective atomic recoil laser as a synchronization transition. Phys. Rev. E 78, 011108 (2008)Google Scholar
  28. 28.
    F.M. Orsatti, R. Carareto, J.R.C. Piqueira, Multiple synchronous states in static delay-free mutually connected PLL networks. Signal Process. 90, 2072–2082 (2010)CrossRefzbMATHGoogle Scholar
  29. 29.
    J. Grollier, V. Cros, A. Fert, Synchronization of spin-transfer oscillators driven by stimulated microwave currents. Phys. Rev. B 73, 060409(R) (2006)Google Scholar
  30. 30.
    B. Georges, J. Grollier, V. Cros, A. Fert, Impact of the electrical connection of spin transfer nano-oscillators on their synchronization: an analytical study. Appl. Phys. Lett. 92, 232504 (2008)Google Scholar
  31. 31.
    Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators. Int. Symp. Math. Probab. Theo. Phys. 420–422 (1975)Google Scholar
  32. 32.
    J.A. Acebrón, L.L. Bonilla, C.J. Pérez, Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)Google Scholar
  33. 33.
    A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008)CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    L. Pecora, T. Carroll, G. Johnson, D. Mar, K.S. Fink, Synchronization stability in coupled oscillator arrays: solution for arbitrary configurations. Int. J. Bifurc. Chaos 10(2), 273–290 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    P.N. McGraw, M. Menzinger, Clustering and the synchronization of oscillator networks. Phys. Rev. E 72, 015101(R) (2005)Google Scholar
  36. 36.
    E.A. Martens, E. Barreto, S.H. Strogatz, E. Ott, P. So, T.M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distribution. Phys. Rev. E 79, 026204 (2009)Google Scholar
  37. 37.
    M. Brede, Locals vs. global synchronization in networks of non-identical Kuramoto oscillators. Eur. Phys. J. B 62, 87–94 (2008)CrossRefADSGoogle Scholar
  38. 38.
    R. Carareto, F.M. Orsatti, J.R.C. Piqueira, Optimized network structure for full-synchronization. Commun. Nonlinear Sci. Numer. Simu. 14, 2536–2541 (2009)CrossRefADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    C. Bick, M. Timme, D. Paulikat, D. Rathlev, P. Ashwin, Chaos in symmetric phase oscillator networks. Phys. Rev. Lett. 107, 244101 (2011)Google Scholar
  40. 40.
    M. Komarov, A. Pikovsky, Multiplicity of singular synchronous states in the Kuramoto model of coupled oscillators. Phys. Rev. Lett. 111, 204101 (2013)Google Scholar
  41. 41.
    Z. Zheng, G. Hu, B. Hu, Phase slips and phase synchronization of coupled oscillators. Phys. Rev. Lett. 81(24), 5318 (1998)Google Scholar
  42. 42.
    E. Canale, P. Monzón, Global properties of Kuramoto bidirectionally coupled oscillators in a ring structure, IEEE Control Appl. (CCA) IEEE Int. Sym. Intell. Control (ISIC) (2009), 183–188 (2009)Google Scholar
  43. 43.
    H.F. El-Nashar, P. Muruganandam, F.F. Ferreira, H.A. Cerdeira, Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling. Chaos 19, 013103 (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Universidad de la RepúblicaMontevideoUruguay

Personalised recommendations