New Formulation of the Chaotic Intermittency

  • Sergio Elaskar
  • Ezequiel del Río
Chapter
First Online:

Abstract

As we have seen in the previous chapters, in the classical theory of intermittency the uniform density of points reinjected from the chaotic to laminar region is a usual hypothesis. In this chapter we reported on how the reinjection probability density (RPD) can be generalized. Estimation of the universal RPD is based on fitting a linear function to experimental data and it does not require a priori knowledge on the dynamical model behind. We provide special fitting procedure that enables robust estimation of the RPD from relatively short data sets. Thus, the method is applicable for a wide variety of data sets including numerical simulations and real-life experiments. Also an analytical method providing the RPD is explained. It is based on the number of null derivatives of the map at the extreme point. Estimated RPD enables analytic evaluation of the length of the laminar phase of intermittent behaviors. The new characteristic exponent is developed, that now is not a single number but is a function depending on the whole map, not on the only the local region. In conclusion, a generalization of the classical intermittency theory is present.

Keywords

Extreme Point Characteristic Exponent Chaotic Region Laminar Region Unstable Fixed Point 
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.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Schuster H., Just W.: Deterministic Chaos. An Introduction. Wiley VCH Verlag GmbH & Co. KGaA, Weinheim (2005)CrossRefMATHGoogle Scholar
  2. 2.
    Dubois, M., Rubio, M., Berge, P.: Experimental evidence of intermittencies associated with a subharmonic bifurcation. Phys. Rev. Lett. 51, 1446–1449 (1983)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Manneville, P.: Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems. J. Phys. 41, 1235–1243 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Pikovsky, A.: A new type of intermittent transition to chaos. J. Phys. A 16, L109–L112 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kim, C., Kwon, O., Lee, E., Lee, H.: New characteristic relations in type-I intermittency. Phys. Rev. Lett. 73, 525–528 (1994)CrossRefGoogle Scholar
  6. 6.
    Kim, Ch., Yim, G., Ryu, J., Park, Y.: Characteristic relations of type-III intermittency in an electronic circuit. Phys. Rev. Lett. 80, 5317–5320 (1998)CrossRefGoogle Scholar
  7. 7.
    Kim, C., Yim, G., Kim, Y., Kim, J., Lee, H.: Experimental evidence of characteristic relations of type-I intermittency in an electronic circuit. Phys. Rev. E 56, 2573–2577 (1997)CrossRefGoogle Scholar
  8. 8.
    Cho, J., Ko, M., Park, Y., Kim, C.: Experimental observation of the characteristic relations of type-I intermittency in the presence of noise. Phys. Rev. E 65, 036222 (2002)CrossRefGoogle Scholar
  9. 9.
    Kye, W., Rim, S., Kim, Ch.: Experimental observation of characteristic relations of type-III intermittency in the presence of noise in a simple electronic circuit. Phys. Rev. E 68, 036203 (2003)CrossRefGoogle Scholar
  10. 10.
    Kye, W., Kim, Ch.: Characteristic relations of type-I intermittency in the presence of noise. Phys. Rev. E 62, 6304–6307 (2000)CrossRefGoogle Scholar
  11. 11.
    del Rio, E., Velarde, M., Rodríguez-Lozano, A.: Long time data series and difficulties with the characterization of chaotic attractors: a case with intermittency III. Chaos Solitons Fractals 4, 2169–2179 (1994)CrossRefMATHGoogle Scholar
  12. 12.
    Ono, Y., Fukushima, K., Yazaki, T.: Critical behavior for the onset of type-III intermittency observed in an electronic circuit. Phys. Rev. 52, 4520–4522 (1995)Google Scholar
  13. 13.
    del Rio, E., Elaskar, S.: New characteristic relations in type-II intermittency. Int. J. Bifurcation Chaos 20, 1185–1191 (2010)CrossRefMATHGoogle Scholar
  14. 14.
    Kwon, O., Kim, Ch., Lee, E., Lee, H.: Effects of reinjection on the scaling property of intermittency. Phys. Rev. E 53, 1253–1256 (1996)CrossRefGoogle Scholar
  15. 15.
    Elaskar, S., del Rio, E., Krause, G., Costa, A.: Effect of the lower boundary of reinjection and noise in type-II intermittency. Nonlinear Dyn. 79, 1411–1424 (2015)CrossRefGoogle Scholar
  16. 16.
    Elaskar, S., del Rio, E., Costa, A.: Reinjection probability density for type-III intermittency with noise and lower boundary of reinjection. J. Comput. Nonlinear Dyn. (2016, in press); ASME, doi:10.1115/1.4034732Google Scholar
  17. 17.
    Elaskar, S., del Rio, E., Donoso, J.: Reinjection probability density in type-III intermittency. Phys. A 390, 2759–2768 (2011)CrossRefGoogle Scholar
  18. 18.
    del Rio, E., Elaskar, S., Donoso, J.: Laminar length and characteristic relation in type-I intermittency. Commun. Numer. Simul. Nonlinear Sci. 19, 967–976 (2014)CrossRefGoogle Scholar
  19. 19.
    Krause, G., Elaskar, S., del Rio, E.: Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrodinger equation. Nonlinear Dyn. 77, 455–466 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    del Rio, E., Sanjuán, M., Elaskar, S.: Effect of noise on the reinjection probability density in intermittency. Commun. Numer. Simul. Nonlinear Sci. 17, 3587–3596 (2012)CrossRefMATHGoogle Scholar
  21. 21.
    del Rio, E., Elaskar, S., Makarov, V.: Theory of intermittency applied to classical pathological cases. Chaos 23, 033112 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rio, E., Elaskar, S.: On the theory of intermittency in 1D maps. Int. J. Bifurcation Chaos 26, 1650228–11 (2016)Google Scholar
  23. 23.
    Hirsch, J., Hubermann, B., Scalapino, D.: Theory of intermittency. Phys. Rev. A 25, 519–532 (1982)CrossRefGoogle Scholar
  24. 24.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2008)MATHGoogle Scholar
  25. 25.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1970)MATHGoogle Scholar
  26. 26.
    Wang, D., Mo, J., Zhao, X., Gu, H., Qu, S., Ren, W.: Intermittent chaotic neural firing characterized by non-smooth features. Chin. Phys. Lett. 27, 070503 (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sergio Elaskar
    • 1
  • Ezequiel del Río
    • 2
  1. 1.National University of CordobaCiudad de CórdobaArgentina
  2. 2.Polytechnic University of MadridMadridSpain

Personalised recommendations