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The European Physical Journal Special Topics

, Volume 222, Issue 10, pp 2417–2428 | Cite as

Sequential activity and multistability in an ensemble of coupled Van der Pol oscillators

  • Tatiana A. LevanovaEmail author
  • Maxim A. Komarov
  • Grigory V. Osipov
Regular Article Nonlinear Dynamics and Synchronization

Abstract

In this paper collective dynamics of an ensemble of inhibitory coupled Van der Pol oscillators are studied. It was found that a stable heteroclinic contour and a stable heteroclinic channel between saddle cycles exist. These heteroclinic structures are responsible for the sequential activity of different oscillations. The corresponding bifurcations leading to the appearance of heteroclinic trajectories are analyzed.

Keywords

Saddle Point European Physical Journal Special Topic Invariant Manifold Stable Manifold Stable Limit Cycle 
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.

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References

  1. 1.
    V.S. Afraimovich, M.I. Rabinovich, P. Varona, Int. J. Bifurcation Chaos 14, 1195 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    M.I. Rabinovich, R. Huerta, P. Varona, V.S. Afraimovich, Biol. Cybern. 95, 519 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. Varona, M.I. Rabinovich, A.I. Selverston, Y.I. Arshavsky, Chaos 12, 672 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  4. 4.
    M.S. Fee, A.A. Kozhevnokov, R.H.R. Hahnloser, Annals NY Acad. Sci. 1016, 153 (2004)ADSCrossRefGoogle Scholar
  5. 5.
    M.A. Komarov, G.V. Osipov, J.A.K. Suykens, M.I. Rabinovich, Chaos 19, 015197 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.I. Rabinovich, R. Huerta, P. Varona, V.S. Afraimovich, PLoS Comput. Biol. 4, e1000072 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Arshwin, P. Chossat, J. Nonlin. Sci. 8, 105 (1998)Google Scholar
  8. 8.
    P. Arshwin, M. Field, Arch. Rat. Mech. Anal. 148, 107 (1999)CrossRefGoogle Scholar
  9. 9.
    V.S. Afraimovich, V.P. Zhigulin, M.I. Rabinovich, Chaos 14, 1123 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  10. 10.
    T. Nowotny, M.I. Rabinovich, Phys. Rev. Lett. 98, 128106 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    W.O. Friesen, G.S. Sent, Biol. Cybern. 28, 27 (1977)CrossRefGoogle Scholar
  12. 12.
    A. Shilnikov, R. Gordon, I. Belykh, Chaos 18, 037120 (2008)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    V.S. Afraimovich, S.-B. Hsu, H.-E. Lin, Int. J. Bifurcation Chaos 11, 435 (2001)MathSciNetADSCrossRefzbMATHGoogle Scholar
  14. 14.
    A.A. Andronov, A.A. Vitt, S.E. Khaikin, Theory of ascillations (Pergamon Press, New York, 1966)Google Scholar
  15. 15.
    D. Zwillinger, Handbook of Differential Equations, 3rd ed. (MA: Academic Press, Boston, 1997)Google Scholar
  16. 16.
    M.A. Komarov, G.V. Osipov, J.A.K. Suykens, EPL 86, 60006 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    Y. Kifer, Israel J. Math. 40, 74 (1981)MathSciNetCrossRefGoogle Scholar
  18. 18.
    E. Stone, P. Holmes, SIAM J. Appl. Math. 50, 726 (1990)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Horita, H. Hata, H. Mori, T. Morita, K. Tomita, S. Kuroki, H. Okamoto, Progr. Theor. Phys. 80, 793 (1988)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    E. Ott, Chaos in Dynamical Systems (Cambridge University Press, New York, 1993)Google Scholar
  21. 21.
    V.S. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stohastic Systems. Tutorial and Modern Developments, 2nd ed. (Springer Verlag, Berlin, 2007)Google Scholar
  22. 22.
    D. Hansel, H. Sompolinsky, J. Comput. Neurosc. 3, 7 (1996)CrossRefGoogle Scholar
  23. 23.
    M.I. Rabinovich, P. Varona, A.I Selverston, H.D. Abarbanel, Rev. Mod. Phys. 78, 1213 (2006)ADSCrossRefGoogle Scholar
  24. 24.
    P. Seliger, L.S. Tsimring, M.I. Rabinovich, Phys. Rev. E 67, 011905 (2003)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    M. Jones, A. Fontanini, B.F. Sadacca, P. Miller, D.B. Katz, Proc. Natl. Acad. Sci. U.S.A. 104, 18772 (2007)ADSCrossRefGoogle Scholar
  26. 26.
    M.A. Komarov, G.V. Osipov, M.S. Burtsev, Chaos 20, 045119 (2010)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  • Tatiana A. Levanova
    • 1
    Email author
  • Maxim A. Komarov
    • 1
  • Grigory V. Osipov
    • 1
  1. 1.Department of Computational Mathematics and CyberneticsNizhny Novgorod State UniversityNizhny NovgorodRussia

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