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The European Physical Journal B

, Volume 70, Issue 4, pp 535–541 | Cite as

Synchronization, stickiness effects and intermittent oscillations in coupled nonlinear stochastic networks

Statistical and Nonlinear Physics

Abstract

Long distance reactive and diffusive coupling is introduced in a spatially extended nonlinear stochastic network of interacting particles. The network serves as a substrate for Lotka-Volterra dynamics with 3rd order nonlinearities. If the network includes only local nearest neighbour interactions, the system organizes into a number of local asynchronous oscillators. It is shown that (a) Introduction of all-to-all coupling in the network leads the system into global, center-type, conservative oscillations as dictated by the mean-field dynamics. (b) Introduction of reactive coupling to the network leads the system to intermittent oscillations where the trajectories stick for short times in bounded regions of space, with subsequent jumps between different bounded regions. (c) Introduction of diffusive coupling to the system does not alter the dynamics for small values of the diffusive coupling pdiff, while after a critical value pdiff c the system synchronizes into a limit cycle with specific frequency, deviating non-trivially from the mean-field center-type behaviour. The frequency of the limit cycle oscillations depends on the reaction rates and on the diffusion coupling. The amplitude σ of the limit cycle depends on the control parameter pdiff.

PACS

05.45.Xt Synchronization; coupled oscillators 02.50.Ey Stochastic processes 02.50.Ng Distribution theory and Monte Carlo studies 64.60.Ht Dynamic critical phenomena 82.40.Bj Oscillations, chaos, and bifurcations 

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References

  1. G. Nicolis, I. Prigogine, Exploring Complexity (New York, Freeman, 1989) Google Scholar
  2. J.D. Murray, Mathematical Biology (Springer, 1993) Google Scholar
  3. T. Antal, M. Droz, A. Lipowski, G. Odor, Phys. Rev. E 64, 036118 (2001)Google Scholar
  4. V.S. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems (Berlin, Spinger, 2007) Google Scholar
  5. S.H. Strogatz, Non-linear Dynamics and Chaos (New York, West-View, 1994); A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization (Cambridge University Press, Cambridge, 2001) Google Scholar
  6. A. Provata, G. Nicolis, F. Baras, J. Chem. Phys. 110, 8361 (1999); A. Provata, G.A. Tsekouras, Phys. Rev. E 67, 056602 (2003) Google Scholar
  7. V.P. Zhdanov, Phys. Rev. E 59, 6292 (1999) Google Scholar
  8. A.J. Acebron, L.L. Bonilla, C.J. Perez Vicente, F. Ritort, R. Spigler, Rev. Mod. Phys. 77, 137 (2005) Google Scholar
  9. K. Wood, C. Van De Broeck, R. Kawai, K. Lindenberg, Phys. Rev. Lett. 96, 145701 (2006); K. Wood, C. Van De Broeck, R. Kawai, K. Lindenberg, Phys. Rev. E 74, 031113 (2006); K. Wood, C. Van De Broeck, R. Kawai, K. Lindenberg, Phys. Rev. E 75, 041107 (2007) Google Scholar
  10. P.S. Landa, A.A. Zaikin, V.G. Ushakov, J. Kurths, Phys. Rev. E 61, 4809 (2000) Google Scholar
  11. J. García-Ojalvo, A. Hernández-Machado, J.M. Sancho, Phys. Rev. Lett. 71, 1542 (1993) Google Scholar
  12. B. Blasius, L. Stone, J. Bif. Chaos 10, 2361 (2000) Google Scholar
  13. W.G. Wilson, E. Mccauley, A.M. De Roos, Bull. Math. Biol. 57, 507 (1995); A.M. De Roos, E. Mccauley, W.G. Wilson, Theor. Pop. Biol. 53, 108 (1998) Google Scholar
  14. K. Tainaka, Phys. Rev. E 50, 3401 (1994) Google Scholar
  15. A. Efimov, A.V. Shabunin, A. Provata, Phys. Rev. E 78, 056201 (2008) Google Scholar
  16. M. Kuperman, G. Abramson, Phys. Rev. Lett. 86, 2909 (2001); D.H. Zanette, M. Kuperman, Physica A 309, 445 (2002); D.H. Zanette Phys. Rev. E 64, 050901 (2001) Google Scholar
  17. G. Szabó, A. Szolnoki, R. Izsák, J. Phys. A 37, 2599 (2004); G. Szabó, G. Fáth, Phys. Rep. 446, 97 (2007) Google Scholar
  18. J. Miyazaki, S. Yoshioka, S. Kinoshita, Chem. Phys. Lett. 387, 471 (2004) Google Scholar
  19. N. Kouvaris, A. Provata, Eur. Phys. J. B 66, 97 (2008); N. Kouvaris, A. Provata, Nonlinear Phenomena in Complex Systems 11, 259 (2008) Google Scholar
  20. T. Risler, J. Prost, F. Julicher, Phys. Rev. Lett. 93, 175702 (2004); T. Risler, J. Prost, F. Julicher, Phys. Rev. E 72, 016130 (2005) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”AthensGreece
  2. 2.Department of Mathematical, Physical and Computational Science, Faculty of EngineeringAristotle University of ThessalonikiThessalonikiGreece

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