Advertisement

Synchronization and Phase Ordering in Globally Coupled Chaotic Maps

  • O. Alvarez-LlamozaEmail author
  • M. G. Cosenza
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 112)

Abstract

We investigate the processes of synchronization and phase ordering in a system of globally coupled maps possessing bistable, chaotic local dynamics. The stability boundaries of the synchronized states are determined on the space of parameters of the system. The collective properties of the system are characterized by means of the persistence probability of equivalent spin variables that define two phases, and by a magnetization-like order parameter that measures the phase-ordering behavior. As a consequence of the global interaction, the persistence probability saturates for all values of the coupling parameter, in contrast to the transition observed in the temporal behavior of the persistence in coupled maps on regular lattices. A discontinuous transition from a nonordered state to a collective phase-ordered state takes place at a critical value of the coupling. On an interval of the coupling parameter, we find three distinct realizations of the phase-ordered state, which can be discerned by the corresponding values of the saturation persistence. Thus, this statistical quantity can provide information about the transient behaviors that lead to the different phase configurations in the system. The appearance of disordered and phase-ordered states in the globally coupled system can be understood by calculating histograms and the time evolution of local map variables associated to the these collective states.

Keywords

Coupling Strength Coupling Parameter Collective Behavior Spin Variable Chaotic Orbit 
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.

Notes

Acknowledgement

This work is supported by project No. C-1906-14-05-B from CDCHTA, Universidad de Los Andes, Venezuela. O. A. thanks Projet Prometeo, Secretaría de Educación Superior, Ciencia, Tecnología e Innovación, Senescyt, Ecuador. M. G. C. is grateful to the Senior Associates Program of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

References

  1. 1.
    Wiesenfeld, K., Hadley, P.: Attractor crowding in oscillator arrays. Phys. Rev. Lett. 62, 1335–1338 (1989)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Wiesenfeld, K., Bracikowski, C., James, G., Roy, R.: Observation of antiphase states in a multimode laser. Phys. Rev. Lett. 65, 1749–1752 (1990)CrossRefGoogle Scholar
  3. 3.
    Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    Nakagawa N., Kuramoto, Y.: From collective oscillations to collective chaos in a globally coupled oscillator system. Phys. D 75, 74–80 (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    G. Grüner: The dynamics of charge-density waves. Rev. Mod. Phys. 60, 1129–1181 (1988)CrossRefGoogle Scholar
  6. 6.
    Kaneko K., Tsuda, I.: Complex Systems: Chaos and Beyond. Springer, Berlin (2001)CrossRefGoogle Scholar
  7. 7.
    Yakovenko, V.M.: Encyclopedia of Complexity and System Science, edited by Meyers, R.A. Springer, New York (2009)Google Scholar
  8. 8.
    Newman, M., Barabási, A.L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  9. 9.
    González-Avella, J.C., Eguiluz, V.M., Cosenza, M.G., Klemm, K., Herrera, J.L., San Miguel, M.: Local versus global interactions in nonequilibrium transitions: a model of social dynamics. Phys. Rev. E 73, 04611–9 (2006)CrossRefGoogle Scholar
  10. 10.
    González-Avella, J.C., Cosenza, M.G., San Miguel, M.: A model for cross-cultural reciprocal interactions through mass media. PLoS ONE 7(12), e5103–5 (2012)CrossRefGoogle Scholar
  11. 11.
    Manrubia, S.C., Mikhailov, A.S., Zanette, D.H.: Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems. World Scientific, Singapore (2004)Google Scholar
  12. 12.
    Garcia-Ojalvo, J., Elowitz, M.B., Strogatz, S.H.: Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc. Natl. Acad. Sci. USA 101, 10955–10960 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Wang, W., Kiss, I.Z., Hudson, J.L.: Experiments on arrays of globally coupled chaotic electrochemical oscillators: synchronization and clustering. Chaos 10, 248–256 (2000)CrossRefGoogle Scholar
  14. 14.
    Miyakawa, K., Yamada, K.: Synchronization and clustering in globally coupled salt-water oscillators. Phys. D 151, 217–227 (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    De Monte, S., dOvidio, F., Danø, S., Sørensen, P.G.: Dynamical quorum sensing. Proc. Natl. Acad. Sci. USA 104, 18377–18381 (2007)CrossRefGoogle Scholar
  16. 16.
    Taylor, A.F., Tinsley, M.R., Wang, F., Huang, Z., Showalter, K.: Dynamical quorum sensing and synchronization in large populations of chemical oscillators. Science 323, 614–617 (2009)CrossRefGoogle Scholar
  17. 17.
    Hagerstrom, A.M., Murphy, T.E., Roy, R., Hövel, P., Omelchenko, I., Schöll, E.: Experimental observation of chimeras in coupled-map lattices. Nat. Phys. 8, 658–661 (2012)CrossRefGoogle Scholar
  18. 18.
    Kaneko, K.: Clustering, coding, switching, hierarchical ordering, and control in networks of chaotic elements. Phys. D 41, 137–172 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Miller, J., Huse, D.A.: Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled map lattice. Phys. Rev. E 48, 2528–2535 (1993)CrossRefGoogle Scholar
  20. 20.
    O’Hern, C., Egolf, D., Greenside, H.S.: Lyapunov spectral analysis of a nonequilibrium Ising-like transition. Phys. Rev. E 53, 3374–3386 (1996)CrossRefGoogle Scholar
  21. 21.
    Lemaître, A., Chaté, H.: Phase ordering and onset of collective behavior in chaotic coupled map lattices. Phys. Rev. Lett. 82, 1140–1143 (1999)CrossRefGoogle Scholar
  22. 22.
    Kockelkoren, J., Lemaître, J., Chaté, H.: Phase-ordering and persistence: relative effects of space-discretization, chaos, and anisotropy. Physica A 288, 326–337 (2000)CrossRefGoogle Scholar
  23. 23.
    Wang, W., Liu, Z., Hu, B.: Phase order in chaotic maps and coupled map lattices. Phys. Rev. Lett. 84, 2610–2613 (2000)CrossRefGoogle Scholar
  24. 24.
    Schmüser, F., Just, M., Kantz, H.: On the relation between coupled map lattices and kinetic Ising models. Phys. Rev. E 61, 675–3684 (2000)CrossRefGoogle Scholar
  25. 25.
    Angelini, L., Pellicoro, M., Stramaglia, S.: Phase ordering in chaotic map lattices with additive noise. Phys. Lett. A 285, 293–300 (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    Angelini, L.: Antiferromagnetic effects in chaotic map lattices with a conservation law. Phys. Lett. A 307, 41–49 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Tucci, K., Cosenza, M.G., Alvarez-Llamoza, O.: Phase separation in coupled chaotic maps on fractal networks. Phys. Rev. E 68, 02720–2 (2003)CrossRefGoogle Scholar
  28. 28.
    Echeverria, C., Tucci, K., Cosenza, M.G.: Phase growth in bistable systems with impurities. Phys. Rev. E 77, 01620–4 (2008)CrossRefGoogle Scholar
  29. 29.
    Waller, I. Kapral, R: Spatial and temporal structure in systems of coupled nonlinear oscillators. Phys. Rev. A 30, 2047–2055 (1984)CrossRefGoogle Scholar
  30. 30.
    Derrida, B., Bray, A.J., Godrèche, C.: Non-trivial exponents in the zero temperature dynamics of the 1d Ising and Potts model. J. Phys. A 27, L357–L361 (1994)CrossRefzbMATHGoogle Scholar
  31. 31.
    Herrera, J.L., Cosenza, M.G., Tucci, K., González-Avella, J.C.: General coevolution of topology and dynamics in networks. Europhys. Lett. 95, 5800–6 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Departamento de FísicaFACYT, Universidad de CaraboboValenciaVenezuela
  2. 2.Centro de Física FundamentalUniversidad de Los AndesMéridaVenezuela
  3. 3.Universidad Católica de CuencaCuencaEcuado

Personalised recommendations