Advertisement

The European Physical Journal Special Topics

, Volume 227, Issue 10–11, pp 1205–1219 | Cite as

Characteristics of in-out intermittency in delay-coupled FitzHugh–Nagumo oscillators

  • Arindam SahaEmail author
  • Ulrike Feudel
Regular Article
Part of the following topical collections:
  1. Advances in Nonlinear Dynamics of Complex Networks: Adaptivity, Stochasticity, Delays

Abstract

We analyze a pair of delay-coupled FitzHugh–Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the invariant subsets involved – a saddle fixed point and a saddle periodic orbit – neither of which are chaotic as in the previously reported cases of in-out intermittency. Based on the analysis of a periodic attractor possessing in-out dynamics, we can characterize the approach to the invariant synchronization manifold and the spiralling out to the saddle periodic orbit with subsequent ejection from the manifold. Due to the striking similarities, this analysis of in-out dynamics also explains in-out intermittency

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Elaskar, E. del Río, Introduction to Chaotic Intermittency (Springer International Publishing, Cham, 2017) Google Scholar
  2. 2.
    G.K. Batchelor, A.A. Townsend, Proc. R. Soc. Lond. A 199, 238 (1949) ADSCrossRefGoogle Scholar
  3. 3.
    P. de Anna, T. Le Borgne, M. Dentz, A.M. Tartakovsky, D. Bolster, P. Davy, Phys. Rev. Lett. 110, 184502 (2013) ADSCrossRefGoogle Scholar
  4. 4.
    J.R. Sanmartin, O.L. Rebollal, E. del Rio, S. Elaskar, Phys. Plasmas 11, 2026 (2004) ADSCrossRefGoogle Scholar
  5. 5.
    C. Stan, C.P. Cristescu, D.G. Dimitriu, Phys. Plasmas 17, 042115 (2010) ADSCrossRefGoogle Scholar
  6. 6.
    M. Dubois, M.A. Rubio, P. Berge, Phys. Rev. Lett. 51, 1446 (1983) ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    J.M. Malasoma, P. Werny, M.A. Boiron, Chaos Solitons Fractals 15, 487 (2003) ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    S.G. Stavrinides, A.N. Miliou, Th. Laopoulos, A.N. Anagnostopoulos, Int. J. Bifurc. Chaos 18, 1561 (2008) CrossRefGoogle Scholar
  9. 9.
    G. Sanchez-Arriaga, J.R. Sanmartin, S.A. Elaskar, Phys. Plasmas 14, 082108 (2007) ADSCrossRefGoogle Scholar
  10. 10.
    A.C.-L. Chian, in Complex systems approach to economic dynamics (Springer Science & Business Media, New York, USA, 2007), Vol. 592 Google Scholar
  11. 11.
    J.J. Żebrowski, R. Baranowski, Physica A 336, 74 (2004) ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Paradisi, P. Allegrini, A. Gemignani, M. Laurino, D. Menicucci, A. Piarulli, AIP Conf. Proc. 1510, 151 (2013) ADSCrossRefGoogle Scholar
  13. 13.
    P. Manneville, Y. Pomeau, Phys. Lett. A 75, 1 (1979) Google Scholar
  14. 14.
    P. Manneville, J. Phys. France 41, 1235 (1980) CrossRefGoogle Scholar
  15. 15.
    M. Marek, I. Schreiber, in Chaotic behaviour of deterministic dissipative systems (Cambridge University Press, Cambridge, UK, 1995), Vol. 1 Google Scholar
  16. 16.
    A.H. Nayfeh, B. Balachandran, Applied nonlinear dynamics: analytical, computational and experimental methods (John Wiley & Sons, Weinheim, Germany, 2008) Google Scholar
  17. 17.
    S.N. Rasband, Chaotic dynamics of nonlinear systems (Courier Dover Publications, Mineola, NY, USA, 2015) Google Scholar
  18. 18.
    H.G. Schuster, W. Just, Deterministic chaos: an introduction (John Wiley & Sons, Weinheim, Germany, 2006) Google Scholar
  19. 19.
    H. Kaplan, Phys. Rev. Lett. 68, 553 (1992) ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    T.J. Price, T. Mullin, Physica D 48, 29 (1991) ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    K.J. Lee, Y. Kwak, T.K. Lim, Phys. Rev. Lett. 81, 321 (1998) ADSCrossRefGoogle Scholar
  22. 22.
    A.E. Hramov, A.A. Koronovskii, M.K. Kurovskaya, S. Boccaletti, Phys. Rev. Lett. 97, 114101 (2006) ADSCrossRefGoogle Scholar
  23. 23.
    N. Platt, E.A. Spiegel, C. Tresser, Phys. Rev. Lett. 70, 279 (1993) ADSCrossRefGoogle Scholar
  24. 24.
    A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, J. Kurths, Phys. Rev. Lett. 79, 47 (1997) ADSCrossRefGoogle Scholar
  25. 25.
    S.C. Venkataramani, T.M. Antonsen, E. Ott, J.C. Sommerer, Physica D 96, 66 (1996) ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    S.C. Venkataramani, T.M. Antonsen, E. Ott, J.C. Sommerer, Phys. Lett. A 207, 173 (1995) ADSCrossRefGoogle Scholar
  27. 27.
    P. Ashwin, E. Covas, R. Tavakol, Nonlinearity 12, 563 (1999) ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    E. Covas, R. Tavakol, P. Ashwin, A. Tworkowski, J.M. Brooke, Chaos 11, 404 (2001) ADSCrossRefGoogle Scholar
  29. 29.
    P. Ashwin, E. Covas, R. Tavakol, Phys. Rev. E 64, 066204 (2001) ADSCrossRefGoogle Scholar
  30. 30.
    J.M. Gac, J.J. Żebrowski, Physica D 232, 136 (2007) ADSCrossRefGoogle Scholar
  31. 31.
    Y.-C. Lai, C. Grebogi, Phys. Rev. Lett. 83, 2926 (1999) ADSCrossRefGoogle Scholar
  32. 32.
    V. Dronov, E. Ott, Chaos 10, 291 (2000) ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    S.R. Ujjwal, N. Punetha, R. Ramaswamy, M. Agrawal, A. Prasad, Chaos 26, 063111 (2016) ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Saha, U. Feudel, Chaos 28, 033610 (2018) ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    A. Saha, U. Feudel, Phys. Rev. E 95, 062219 (2017) ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    G. Ansmann, R. Karnatak, K. Lehnertz, U. Feudel, Phys. Rev. E 88, 052911 (2013) ADSCrossRefGoogle Scholar
  37. 37.
    R. Karnatak, G. Ansmann, U. Feudel, K. Lehnertz, Phys. Rev. E 90, 022917 (2014) ADSCrossRefGoogle Scholar
  38. 38.
    S. Albeverio, V. Jentsch, H. Kantz, Extreme Events in Nature and Society, the Frontiers Collection (Springer, Berlin, 2006) Google Scholar
  39. 39.
    D. Helbing, Rev. Mod. Phys. 73, 1067 (2001) ADSCrossRefGoogle Scholar
  40. 40.
    N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373, 675 (2009) ADSCrossRefGoogle Scholar
  41. 41.
    A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Phys. Rev. Lett. 106, 204502 (2011) ADSCrossRefGoogle Scholar
  42. 42.
    A. Chabchoub, N. Hoffmann, M. Onorato, N. Akhmediev, Phys. Rev. X 2, 011015 (2012) Google Scholar
  43. 43.
    M. Onorato, A.R. Osborne, M. Serio, S. Bertone, Phys. Rev. Lett. 86, 5831 (2001) ADSCrossRefGoogle Scholar
  44. 44.
    V. Zakharov, A. Dyachenko, A. Prokofiev, Eur. J. Mech. B: Fluids 25, 677 (2006) ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    C. Bonatto, M. Feyereisen, S. Barland, M. Giudici, C. Masoller, J.R.R. Leite, J.R. Tredicce, Phys. Rev. Lett. 107, 053901 (2011) ADSCrossRefGoogle Scholar
  46. 46.
    N. Akhmediev et al., J. Opt. 18, 063001 (2016) ADSCrossRefGoogle Scholar
  47. 47.
    A.N. Pisarchik, R. Jaimes-Reátegui, R. Sevilla-Escoboza, G. Huerta-Cuellar, Phys. Rev. E 86, 056219 (2012) ADSCrossRefGoogle Scholar
  48. 48.
    D. Sornette, Critical Phenomena in Natural Sciences: Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer Series in Synergetics (Springer, Berlin, 2003) Google Scholar
  49. 49.
    A. Bunde, J. Kropp, H. Schellnhuber, The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes (Springer, Berlin, 2002) Google Scholar
  50. 50.
    J. Feigenbaum, Quant. Financ. 1, 346 (2001) CrossRefGoogle Scholar
  51. 51.
    I. Dobson, B.A. Carreras, V.E. Lynch, D.E. Newman, Chaos 17, 026103 (2007) ADSCrossRefGoogle Scholar
  52. 52.
    S. Bialonski, A.D. Caron, J. Schloen, U. Feudel, H. Kantz, S.D. Moorthi, J. Plankton Res. 38, 1077 (2016) CrossRefGoogle Scholar
  53. 53.
    J.E. Truscott, J. Brindley, Bull. Math. Biol. 56, 981 (1994) CrossRefGoogle Scholar
  54. 54.
    S. Chakraborty, U. Feudel, Theor. Ecol. 7, 221 (2014) CrossRefGoogle Scholar
  55. 55.
    F. Mormann, T. Kreuz, R.G. Andrzejak, P. David, K. Lehnertz, C.E. Elger, Theor. Ecol. 7, 221 (2014) CrossRefGoogle Scholar
  56. 56.
    I. Omelchenko, O.E. Omel’chenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 110, 224101 (2013) ADSCrossRefGoogle Scholar
  57. 57.
    I. Omelchenko, A. Provata, J. Hizanidis, E. Schöll, P. Hövel, Phys. Rev. E 91, 022917 (2015) ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    A.S. Pikovsky, Z. Phys. B 55, 149 (1984) ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    P. Ashwin, J. Buescu, I. Stewart, Nonlinearity 9, 703 (1996) ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    A. Saha, U. Feudel, Indian Acad. Sci. Conf. Ser. 1, 163 (2017) Google Scholar
  61. 61.
    P. Ashwin, J. Buescu, I. Stewart, Phys. Lett. A 193, 126 (1994) ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    E. Ott, J.C. Sommerer, Phys. Lett. A 188, 38 (1994) ADSCrossRefGoogle Scholar
  63. 63.
    P. Ashwin, A.M. Rucklidge, R. Sturman, Physica D 194, 30 (2004) ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    N. Blackbeard, S. Wieczorek, H. Erzgräber, P.S. Dutta, Physica D 286–287, 43 (2014) CrossRefGoogle Scholar
  65. 65.
    W. Michiels, S.-I. Niculescu, Stability and stabilization of time-delay systems: an eigenvalue-based approach (SIAM, Philadelphia, PA, USA, 2007) Google Scholar
  66. 66.
    C. Simmendinger, A. Wunderlin, A. Pelster, Phys. Rev. E 59, 5344 (1999) ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Theoretical Physics/Complex Systems, ICBM, University of OldenburgOldenburgGermany

Personalised recommendations