Nonlinear Dynamics

, Volume 92, Issue 2, pp 683–697 | Cite as

Nonuniform reinjection probability density function in type V intermittency

  • Sergio Elaskar
  • Ezequiel del Río
  • L. Gutierrez Marcantoni
Original Paper
  • 43 Downloads

Abstract

In this paper, type V intermittency is studied using the M function methodology developed in the last years. This methodology is applied on two different maps to evaluate the reinjection probability density function (RPD), the probability density of laminar lengths and the characteristic relation. We have found that the RPD can be written as an exponential function, where the uniform reinjection is only a singular case. Also, the probability density of laminar lengths can be a nondifferentiable function when the local map has a nondifferentiable point inside the laminar interval. On the other hand, the characteristic relation is not unique, and it depends on the local map. Therefore, the behavior of the reinjection processes and the statistical properties for type V intermittency is wider than the previous studies have described. Finally, it is noted that the M function methodology is a suitable tool to analyze type V intermittency.

Keywords

Type V intermittency M function RPD Characteristic relation 

Notes

Acknowledgements

This research was supported by CONICET, Universidad Nacional de Córdoba, Universidad Politécnica de Madrid, and the Spanish Ministry of Science and Innovation (MICINN) under Project No. EPS2013-41078-R.

References

  1. 1.
    Schuster, H., Just, W.: Deterministic Chaos. Wiley VCH, Mörlenbach (2005)CrossRefMATHGoogle Scholar
  2. 2.
    Nayfeh, A., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)CrossRefMATHGoogle Scholar
  3. 3.
    Marek, M., Schreiber, I.: Chaotic Behaviour of Deterministic Dissipative Systems. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  4. 4.
    Elaskar, S., del Rio, E.: New Advances on Chaotic Intermittency and Its Applications. Springer, New York (2017)CrossRefMATHGoogle Scholar
  5. 5.
    Manneville, P., Pomeau, Y.: Intermittency and Lorenz model. Phys. Lett. A 75, 1–2 (1979)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Manneville, P.: Intermittency, self-similarity and 1/\(f\) spectrum in dissipative dynamical systems. J. Phys. 41, 1235–1243 (1980)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kaplan, H.: Return to type I intermittency. Phys. Rev. Lett. 68, 553–557 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Price, T., Mullin, P.: An experimental observation of a new type of intermittency. Physica D 48, 29–52 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Platt, N., Spiegel, E., Tresser, C.: On–off intermittency: a mechanism for bursting. Phys. Rev. Lett. 70, 279–282 (1993)CrossRefGoogle Scholar
  10. 10.
    Pikovsky, A., Osipov, G., Rosenblum, M., Zaks, M., Kurths, J.: Attractor–repeller collision and eyelet intermittency at the transition to phase synchronization. Phys. Rev. Lett. 79, 47–50 (1997)CrossRefGoogle Scholar
  11. 11.
    Lee, K., Kwak, Y., Lim, T.: Phase jumps near a phase synchronization transition in systems of two coupled chaotic oscillators. Phys. Rev. Lett. 81, 321–324 (1998)CrossRefGoogle Scholar
  12. 12.
    Hramov, A., Koronovskii, A., Kurovskaya, M., Boccaletti, S.: Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization. Phys. Rev. Lett. 97, 114101 (2006)CrossRefGoogle Scholar
  13. 13.
    Stavrinides, S., Anagnostopoulos, A.: Chapter 9: the route from synchronization to desynchronization of chaotic operating circuits and systems. In: Banerjee, S., Rondoni, L. (eds.) Applications of Chaos and Nonlinear Dynamics in Science and Engineering. Springer, Berlin (2013)Google Scholar
  14. 14.
    Dubois, M., Rubio, M., Berge, P.: Experimental evidence of intermittencies associated with a subharmonic bifurcation. Phys. Rev. Lett. 16, 1446–1449 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Malasoma, J., Werny, P., Boiron, M.: Multichannel type I intermittency in two models of Rayleigh–Benard convection. Phys. Rev. Lett. 51, 487–500 (2004)MATHGoogle Scholar
  16. 16.
    Stavrinides, S., Miliou, A., Laopoulos, T., Anagnostopoulos, A.: The intermittency route to chaos of an electronic digital oscillator. Int. J. Bifurc. Chaos 18, 1561–1566 (2008)CrossRefMATHGoogle Scholar
  17. 17.
    Sanmartin, J., Lopez-Rebollal, O., del Rio, E., Elaskar, S.: Hard transition to chaotic dynamics in Alfven wave-fronts. Phys. Plasmas 11, 2026–2035 (2004)CrossRefGoogle Scholar
  18. 18.
    Sanchez-Arriaga, G., Sanmartin, J., Elaskar, S.: Damping models in the truncated derivative nonlinear Schrödinger equation. Phys. Plasmas 14, 082108 (2007)CrossRefGoogle Scholar
  19. 19.
    Pizza, G., Frouzakis, C., Mantzaras, J.: Chaotic dynamics in premixed hydrogen/air channel flow combustion. Combust. Theor. Model. 16, 275–299 (2012)CrossRefMATHGoogle Scholar
  20. 20.
    Nishiura, Y., Ueyama, D., Yanagita, T.: Chaotic pulses for discrete reaction diffusion systems. SIAM J. Appl. Dyn. Syst. 4, 723–754 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    de Anna, P., Le Borgne, T., Dentz, M., Tartakovsky, A., Bolster, D., Davy, P.: Flow intermittency, dispersion and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110, 184502 (2013)CrossRefGoogle Scholar
  22. 22.
    Stan, C., Cristescu, C., Dimitriu, D.: Analysis of the intermittency behavior in a low-temperature discharge plasma by recurrence plot quantification. Phys. Plasmas 17, 042115 (2010)CrossRefGoogle Scholar
  23. 23.
    Chian, A.: Complex System Approach to Economic Dynamics. Lecture Notes in Economics and Mathematical Systems, pp. 39–50. Springer, Berlin (2007)Google Scholar
  24. 24.
    Zebrowski, J., Baranowski, R.: Type I intermittency in nonstationary systems: models and human heart-rate variability. Physica A 336, 74–86 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Paradisi, P., Allegrini, P., Gemignani, A., Laurino, M., Menicucci, D., Piarulli, A.: Scaling and intermittency of brains events as a manifestation of consciousness. AIP Conf. Proc. 1510, 151–161 (2012)Google Scholar
  26. 26.
    del Rio, E., Elaskar, S.: New characteristic relation in type II intermittency. Int. J. Bifurc. Chaos 20, 1185–1191 (2010)CrossRefMATHGoogle Scholar
  27. 27.
    Elaskar, S., del Rio, E., Donoso, J.: Reinjection probability density in type III intermittency. Physica A 390, 2759–2768 (2011)CrossRefGoogle Scholar
  28. 28.
    del Rio, E., Sanjuan, M., Elaskar, S.: Effect of noise on the reinjection probability density in intermittency. Commun. Nonlinear Sci. Numer. Simul. 17, 3587–3596 (2012)CrossRefMATHGoogle Scholar
  29. 29.
    Elaskar, S., del Rio, E.: Intermittency reinjection probability function with and without noise effects. In: Latest Trends in Circuits, Automatics Control and Signal Processing, pp. 145–154. ISBN: 978-1-61804-131-9, Barcelona (2012)Google Scholar
  30. 30.
    del Rio, E., Elaskar, S., Makarov, S.: Theory of intermittency applied to classical pathological cases. Chaos 23, 033112 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    del Rio, E., Elaskar, S., Donoso, J.: Laminar length and characteristic relation in type I intermittency. Commun. Nonlinear Sci. Numer. Simul. 19, 967–976 (2014)CrossRefGoogle Scholar
  32. 32.
    Krause, G., Elaskar, S., del Rio, E.: Type I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation. Nonlinear Dyn. 77, 455–466 (2014)CrossRefGoogle Scholar
  33. 33.
    Krause, G., Elaskar, S., del Rio, E.: Noise effect on statistical properties of type I intermittency. Physica A 402, 318–329 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Elaskar, S., del Río, 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
  35. 35.
    del Río, E., Elaskar, S.: On the intermittency theory. Int. J. Bifurc. Chaos 26, 1650228 (2016)CrossRefMATHGoogle Scholar
  36. 36.
    del Rio, E., Elaskar S.: The intermittency route to chaos. In: Skiadas, C.H., Skiadas, C. (eds.) Handbook of Applications of Chaos Theory, pp. 3–20. CRC Press Book. ISBN 9781466590434. Paris (2016)Google Scholar
  37. 37.
    Elaskar, S., del Río, E., Costa, A.: Reinjection probability density for type III intermittency with noise and lower boundary of reinjection. J. Comp. Nonlinear Dyn. 12, 031020-11 (2017)Google Scholar
  38. 38.
    Bauer, M., Habip, S., He, D., Martiessen, W.: New type of intermittency in discontinuous maps. Phys. Rev. Lett. 68, 1625–1628 (1992)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    He, D., Bauer, M., Habip, S., Kruger, U., Martiessen, W., Christiansen, B., Wang, B.: Type V intermittency. Phys. Lett. A 171, 61–65 (1992)CrossRefGoogle Scholar
  40. 40.
    Fan, J., Ji, F., Guan, S., Wang, B., He, D.: The distribution of laminar lengths in type V intermittency. Phys. Lett. A 182, 232–237 (1993)CrossRefGoogle Scholar
  41. 41.
    Wu, S., He, D.: Characteristics of period-doubling bifurcation cascades in quasidiscontinuous systems. Commun. Theor. Phys. 35, 275–282 (2001)CrossRefMATHGoogle Scholar
  42. 42.
    Wang, D., Mo, J., Zhao, X., Gu, H., Qu, S., Ren, W.: Intermittent chaotic neural firing characterized by non-smooth like features. Chin. Phys. Lett. 27, 070503 (2010)CrossRefGoogle Scholar
  43. 43.
    Gu, H., Xiao, W.: Difference between intermittent chaotic bursting and spiking of neural firing patterns. Int. J. Bifurc. Chaos 24, 1450082 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Bai-lin, H.: Elementary Simbolic Dynamics and Chaos in Dissipative Systems. World Scientific, Singapore (1989)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Aeronáutica, FCEFyN, Instituto de Estudios Avanzados en Ingeniería y TecnologíaCONICET and Universidad Nacional de CórdobaCórdobaArgentina
  2. 2.Departamento de Física Aplicada, ETSIAEUniversidad Politécnica de MadridMadridSpain

Personalised recommendations