ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Sysoev I. V., Ponomarenko V. I., Prokhorov M. D. Reconstruction of unidirectionally coupled time-delayed systems of first order from time series of the driven system. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 1, pp. 84-93. DOI: 10.18500/0869-6632-2017-25-1-84-93

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 112)
Article type: 

Reconstruction of unidirectionally coupled time-delayed systems of first order from time series of the driven system

Sysoev Ilya Vyacheslavovich, Saratov State University
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Prokhorov Mihail Dmitrievich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

Time-delayed systems, including coupled ones, became popular models of different physical and biological objects. Often One or few variables of such models cannot be directly measured, these variables are called hidden variables. However, reconstruction of models from experimental signals in presence of hidden variables can be very suitable for model verification and indirect measurement. Current study considers reconstruction of parameters of both systems and hidden variable of the driving system from time series of the driven system in ensemble of two unidirectionally coupled first order time-delayed systems. Initial condition approach was used; this method considers initial conditions for hidden variables as additional unknown parameters. The method was adapted for time-delayed systems: vector of initial conditions was used instead of a single initial condition. Time series of the driven system, parameters of nonlinear function of both systems and the coupling coefficient were shown to be reconstructable from a time series of driven system in periodical regime, if starting guesses for the hidden variable were set using a priori information about the model. The space of starting guesses for parameters was studied; the probability of successful reconstruction was shown to be approximately 1/4 for starting guesses for parameters distant from their true values in 50 % of their absolute values. The fundamental possibility of reconstruction of system with one time delay in presence of hidden variables from scalar time series was shown.  

  1. Swameye I., Muller T.G., Timmer J., et al. Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by data based modeling. Proc. Natl. Acad. Sci. USA. 2003. Vol. 100. P. 1028–1033.
  2. Volnova A.B., Lenkov D.N. Absence epilepsy: Mechanisms of hypersynchronization of neuronal networks. Medical Academic Journal 2012. Vol. 12, Iss. 1. P. 7–19.
  3. Jensen K.S., Mosekilde E., Holstein-Rathlou N.-H. Self-sustained oscillations and chaotic behaviour in kidney pressure regulation. Mondes Develop. Vol. 54/55. P. 91–109.
  4. Glass L., Mackey M.C. Pathological physiological conditions resulting from instabilities in physiological control systems. Ann. NY. Acad. Sci. 1979. Vol. 316. P. 214–235.
  5. Milton J., Jung P. Epilepsy as a Dynamical Disease. New York: Springer-Verlag Berlin Heidelberg, 2003.
  6. Jimmi H. Talla Mbe, Alain F. Talla, Geraud R. Goune Chengui, Aurelien Coillet, Laurent Larger, Paul Woafo, Yanne K. Chembo. Mixed-mode oscillations in slow- fast delayed optoelectronic systems. Phys. Rev. E. 2015. Vol. 91, 012902.
  7. Khorev V.S., Prokhorov M.D., Ponomarenko V.I. Determination of the delay time and feedback strength of a semiconductor laser with optical feedback from time series of radiation intensity. Technical Physics Letters. 2016. Vol. 42, Iss. 3. P. 68–75.
  8. Gouesbet G., Letellier C. Global vector-field approximation by using a multivariate polynomial approximation on nets. Phys. Rev. E. 1994. Vol. 49. P. 4955–4972.
  9. Packard N., Crutchfield J., Farmer J., Shaw R. Geometry from a time series. Phys. Rev. Lett. 1980. Vol. 45. P. 712–716.
  10. Baake E., Baake M., Bock H.G., Briggs K.M. Fitting ordinary differential equations to chaotic data. Phys. Rev. A. 1992. Vol. 45, Iss. 8. P. 5524–5529.
  11. Bezruchko B.P., Smirnov D.A., Sysoev I.V. Reconstruction with hidden variables: Modified Bock’s approach. Izv. VUZ. Applied Nonlinear Dynamics. 2004. Vol. 12, Iss. 6. P. 93–104.
  12. Prokhorov M.D., Ponomarenko V.I. Estimation of coupling between time-delay systems from time series. Phys. Rev. E. 2005. Vol. 72. 016210.
  13. Rontani D., Locquet A., Sciamanna M., Citrin D.S., Ortin S. Time-delay identification in a chaotic semiconductor laser with optical feedback: A dynamical point of view. IEEE Journal of Quantum Electronics. 2009. Vol. 45, Iss. 7. P. 879–891.
  14. Ponomarenko V.I., Prokhorov M.D. Recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes. Phys. Rev. E. 2008. Vol. 78. 066207.
  15. Zhiglyavskiy A.A., Zhilinskas A.G. Methods of Finding the Global Extremum. Moscow: Nauka, FizMatLit, 1991. 248 p.
  16. Prokhorov M.D., Ponomarenko V.I., Karavaev A.S., Bezruchko B.P. Reconstruction of time-delayed feedback systems from time series. Physica D. 2005. Vol. 203, Iss. 3–4. P. 209–223.
  17. Prokhorov M.D., Ponomarenko V.I. Reconstruction of time-delay systems using small impulsive disturbances. Phys. Rev. E. 2009. Vol. 80, Iss. 6. 066206.
  18. Ponomarenko V.I., Prokhorov M.D. Recovery of equations of coupled time-delay systems from time series. Technical Physics Letters. 2005. Vol. 31, Iss. 1. P. 64-67.
Short text (in English):
(downloads: 112)