Advertisement

Pramana

, Volume 70, Issue 6, pp 1117–1125 | Cite as

Dynamics of delayed-coupled chaotic logistic maps: Influence of network topology, connectivity and delay times

  • Arturo C. Martí
  • Marcelo Ponce
  • Cristina Masoller
Article

Abstract

We review our recent work on the synchronization of a network of delay-coupled maps, focusing on the interplay of the network topology and the delay times that take into account the finite velocity of propagation of interactions. We assume that the elements of the network are identical (N logistic maps in the regime where the individual maps, without coupling, evolve in a chaotic orbit) and that the coupling strengths are uniform throughout the network. We show that if the delay times are sufficiently heterogeneous, for adequate coupling strength the network synchronizes in a spatially homogeneous steady state, which is unstable for the individual maps without coupling. This synchronization behavior is referred to as ‘suppression of chaos by random delays’ and is in contrast with the synchronization when all the interaction delay times are homogeneous, because with homogeneous delays the network synchronizes in a state where the elements display in-phase time-periodic or chaotic oscillations. We analyze the influence of the network topology considering four different types of networks: two regular (a ring-type and a ring-type with a central node) and two random (free-scale Barabasi-Albert and small-world Newman-Watts). We find that when the delay times are sufficiently heterogeneous the synchronization behavior is largely independent of the network topology but depends on the network’s connectivity, i.e., on the average number of neighbors per node.

Keywords

Synchronization coupled map lattices time delays logistic map 

PACS Nos

05.45.Xt 05.65.+b 05.45.Ra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Y Kuramoto, Chemical oscillations, waves, and turbulence (Springer-Verlag, Berlin, 1984)zbMATHGoogle Scholar
  2. [2]
    S Boccaletti, J Kurths, G Osipov, D L Valladares and C S Zhou, Phys. Rep. 1, 366 (2002)MathSciNetGoogle Scholar
  3. [3]
    A Pikovsky, M Rosenblum and J Kurths, Synchronization: A universal concept in nonlinear sciences (Cambridge University Press, Cambridge, England, 2003)Google Scholar
  4. [4]
    S Manrubia, A S Mikhailov and D H Zanette, Emergence of dynamical order (World Scientific, Singapore, 2004)zbMATHGoogle Scholar
  5. [5]
    S Boccaletti, V Latora, Y Moreno, M Chavez and D U Hwang, Phys. Rep. 424, 175 (2006)CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    H G Schuster and P Wagner, Prog. Theor. Phys. 81, 939 (1987)CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    V K Jirsa and M Ding, Phys. Rev. Lett. 93, 070602 (2004)Google Scholar
  8. [8]
    E Niebur et al, Phys. Rev. Lett. 67, 2753 (1991)CrossRefADSGoogle Scholar
  9. [8a]
    Ernst et al, Phys. Rev. Lett. 74, 1570 (1995)CrossRefADSGoogle Scholar
  10. [8b]
    S Kim et al, Phys. Rev. Lett. 79, 2911 (1997)CrossRefADSGoogle Scholar
  11. [8c]
    M K Stephen Yeung and S H Strogatz, Phys. Rev. Lett. 82, 648 (1999)CrossRefADSGoogle Scholar
  12. [8d]
    M G Earl and S H Strogatz, Phys. Rev. E67, 036204 (2003)Google Scholar
  13. [8e]
    M Denker et al, Phys. Rev. Lett. 92, 074103 (2004)Google Scholar
  14. [9]
    D V Ramana Reddy et al, Phys. Rev. Lett. 80, 5109 (1998)CrossRefADSGoogle Scholar
  15. [9a]
    R Herrero et al, Phys. Rev. Lett. 84, 5312 (2000)CrossRefADSGoogle Scholar
  16. [10]
    Y Jiang, Phys. Lett. A267, 342 (2000)ADSGoogle Scholar
  17. [10a]
    C Li et al, Physica A335, 365 (2004)ADSGoogle Scholar
  18. [11]
    F M Atay et al, Phys. Rev. Lett. 92, 144101 (2004)Google Scholar
  19. [12]
    J García-Ojalvo, J Casademont, C R Mirasso, M C Torrent and J M Sancho, Int. J. Bifurcation Chaos 9, 2225 (1999)CrossRefGoogle Scholar
  20. [13]
    G Kozyreff, A G Vladimirov and Paul Mandel, Phys. Rev. Lett. 85, 3809 (2000)CrossRefADSGoogle Scholar
  21. [14]
    J Foss, A Longtin, B Mensour and J Milton, Phys. Rev. Lett. 76, 708 (1996)CrossRefADSGoogle Scholar
  22. [15]
    A Takamatsu et al, Phys. Rev. Lett. 92, 228102 (2004)Google Scholar
  23. [15a]
    M Dhamala et al, Phys. Rev. Lett. 92, 074104 (2004)Google Scholar
  24. [16]
    M Rosemblum and A Pikovsky, Phys. Rev. Lett. 92, 114102 (2004); Phys. Rev. E70, 041904 (2004)Google Scholar
  25. [17]
    K Otsuka and J L Chern, Phys. Rev. A45, 5052 (1992)ADSGoogle Scholar
  26. [18]
    B Doiron et al, Phys. Rev. Lett. 93, 048101 (2004)Google Scholar
  27. [19]
    D H Zanette, Phys. Rev. E62, 3167 (2000)ADSGoogle Scholar
  28. [19a]
    S O Jeong et al, Phys. Rev. Lett. 89, 154104 (2002)Google Scholar
  29. [19b]
    T W Ko et al, Phys. Rev. E69, 056106 (2004)Google Scholar
  30. [20]
    F M Atay, Phys. Rev. Lett. 91, 094101 (2003)Google Scholar
  31. [21]
    A C Martí and C Masoller, Phys. Rev. E67, 056219 (2003)Google Scholar
  32. [22]
    A C Martí and C Masoller, Physica A342, 344 (2004)ADSGoogle Scholar
  33. [23]
    C M Gonzalez, C Masoller, M C Torrent and J García-Ojalvo, Europhys. Lett. 79, 64003 (2007)Google Scholar
  34. [24]
    C Masoller and A C Martí, Phys. Rev. Lett. 94, 134102 (2005)Google Scholar
  35. [25]
    A C Martí, M Ponce and C Masoller, Phys. Rev. E72, 066217 (2005)Google Scholar
  36. [26]
    A C Martí, M Ponce C and C Masoller, Physica A371, 104 (2006)ADSGoogle Scholar
  37. [27]
    R Albert and A-L Barabási, Rev. Mod. Phys. 74, 47 (2002)CrossRefADSGoogle Scholar
  38. [28]
    M E J Newman and D J Watts, Phys. Rev. E60, 7332 (1999); Phys. Lett. A263, 341 (1999)ADSGoogle Scholar
  39. [29]
    A Ahlborn and U Parlitz, Phys. Rev. Lett. 93, 264101 (2004)Google Scholar
  40. [30]
    M Ponce, C Masoller and A C Martí, http://arXiv.org/abs/0805.2420 (2008)

Copyright information

© Indian Academy of Sciences 2008

Authors and Affiliations

  • Arturo C. Martí
    • 1
  • Marcelo Ponce
    • 1
  • Cristina Masoller
    • 2
  1. 1.Instituto de Física, Facultad de CienciasUniversidad de la RepúblicaMontevideoUruguay
  2. 2.Departament de Fisica i Enginyeria NuclearUniversitat Politecnica de CatalunyaTerrassa, BarcelonaSpain

Personalised recommendations