Izvestiya of Saratov University.

Physics

ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)


For citation:

Ponomarenko V. I., Prohorov M. D., Navrotskaya E. V. Method of TimeDelay Systems Recovery from Time Series with Known Type of Model Equation. Izvestiya of Sarat. Univ. Physics. , 2011, vol. 11, iss. 2, pp. 72-78. DOI: 10.18500/1817-3020-2011-11-2-72-78

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 52)
Language: 
Russian
Heading: 
UDC: 
537.86

Method of TimeDelay Systems Recovery from Time Series with Known Type of Model Equation

Autors: 
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences
Prohorov Mikhail Dmitrievich, Saratov Branch of Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences
Navrotskaya Elena Vladimirovna, Saratov State University
Abstract: 

We propose the method for the reconstruction of firstorder timedelay systems from their time series. The method is based on taking into account the type of the system equation at the regression model construction. The method allows one to recover the delay time, the parameter characterizing the inertial properties of the system and the nonlinear function. It can be applied to the recovery of timedelay systems performing chaotic and periodic oscillations.

Reference: 
  1. Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston : Academic Press, 1993. 398 p.
  2. Glass L., Mackey M. C. From Clocks to Chaos : The Rhythms of Life. Princeton : Princeton University Press, 1988. 248 p.
  3. Кузнецов С. П. Сложная динамика генераторов с запаздывающей обратной связью (обзор) // Изв. вузов. Радиофизика. 1982. Т. 25. С. 1410–1428.
  4. Ikeda K. Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system // Opt. Commun. 1979. Vol. 30. P. 257–261.
  5. Lang R., Kobayashi K. External optical feedback effects on semiconductor injection lasers // IEEE J. Quantum Electron. 1980. Vol. 16. P. 347–355.
  6. Mackey M. C., Glass L. Oscillations and chaos in physiological control systems // Science. 1977. Vol. 197. P. 287–289.
  7. Bocharov G. A., Rihan F. A. Numerical modelling in biosciences using delay differential equations / / J. Comp. Appl. Math. 2000. Vol. 125. P. 183–199.
  8. Farmer J. D. Chaotic attractors of an infi nite-dimensional dynamical system // Physica D. 1982. Vol. 4. P. 366–393.
  9. Fowler A. C., Kember G. Delay recognition in chaotic time series // Phys. Lett. A. 1993. Vol. 175. P. 402–408.
  10. Hegger R., Bünner M. J., Kantz H., Giaquinta A. Identifying and modeling delay feedback systems // Phys. Rev. Lett. 1998. Vol. 81. P. 558–561.
  11. Udaltsov V. S., Goedgebuer J. -P., Larger L., Cuenot J.-B., Levy P., Rhodes W. T. Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations // Phys. Lett. A. 2003. Vol. 308. P. 54–60.
  12. Bünner M. J., Ciofi ni M., Giaquinta A., Hegger R., Kantz H., Meucci R., Politi A. Reconstruction of systems with delayed feedback : (I) Theory // Eur. Phys. J. D. 2000. Vol. 10. P. 165–176.
  13. Tian Y. -C., Gao F. Extraction of delay information from chaotic time series based on information entropy // Physica D. 1997. Vol. 108. P. 113–118.
  14. Kaplan D. T., Glass L. Coarse-grained embeddings of time series : Random walks, gaussian random process, and deterministic chaos // Physica D. 1993. Vol. 64. P. 431–454.
  15. Bünner M. J., Meyer Th., Kittel A., Parisi J. Recovery of the time-evolution equation of time-delay systems from time series // Phys. Rev. E. 1997. Vol. 56. P. 5083–5089.
  16. Voss H., Kurths J. Reconstruction of non-linear time delay models from data by the use of optimal transformations // Phys. Lett. A. 1997. Vol. 234. P. 336–344.
  17. Ellner S. P., Kendall B. E., Wood S. N., McCauley E., Briggs C. J. Inferring mechanism from time-series data : Delay differential equations // Physica D. 1997. Vol. 110. P. 182–194.
  18. Voss H. U., Schwache A., Kurths J., Mitschke F. Equations of motion from chaotic data : A driven optical fi ber ring resonator // Phys. Lett. A. 1999. Vol. 256. P. 47–54.
  19. Horbelt W., Timmer J., Voss H. U. Parameter estimation in nonlinear delayed feedback systems from noisy data // Phys. Lett. A. 2002. Vol. 299. P. 513–521.
  20. Dai C., Chen W., Li L., Zhu Y., Yang Y. Seeker optimization algorithm for parameter estimation of time-delay chaotic systems // Phys. Rev. E. 2011. Vol. 83. 036203.
  21. Zunino L., Soriano M. C., Fischer I., Rosso O. A., Mirasso C. R. Permutation-information-theory approach to unveil delay dynamics from time-series analysis // Phys. Rev. E. 2010. Vol. 82. 046212.
  22. Ma H., Xu B., Lin W., Feng J. Adaptive identifi cation of time delays in nonlinear dynamical models // Phys. Rev. E. 2010. Vol. 82. 066210.
  23. Ortí n S., Gutié rrez J. M., Pesquera L., Vasquez H. Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction // Physica A. 2005. Vol. 351. P. 133–141.
  24. Bezruchko B. P., Karavaev A. S., Ponomarenko V. I., Prokhorov M. D. Reconstruction of time-delay systems from chaotic time series // Phys. Rev. E. 2001. Vol. 64. 056216.
  25. Пономаренко В. И., Прохоров М. Д., Караваев А. С., Безручко Б. П. Определение параметров систем с запаздывающей обратной связью по хаотическим временным реализациям // ЖЭТФ. 2005. Т. 127, вып. 3. С. 515–527.
  26. Siefert M. Practical criterion for delay estimation using random perturbations // Phys. Rev. E. 2007. Vol. 76. 026215.
  27. Yu D., Frasca M., Liu F. Control-based method to identify underlying delays of a nonlinear dynamical system // Phys. Rev. E. 2008. Vol. 78. 046209.
  28. Пономаренко В. И., Прохоров М. Д., Селезнев Е. П. Оценка характеристик автоколебательных систем с запаздыванием в периодическом режиме // Изв. вузов. Прикладная нелинейная динамика. 2007. Т. 15, №. 6. С. 86–92.
  29. Ponomarenko V. I., Prokhorov M. D. Recovery of systems with a linear fi lter and nonlinear delay feedback in periodic regimes // Phys. Rev. E. 2008. Vol. 78. 066207.
  30. Prokhorov M. D., Ponomarenko V. I. Reconstruction of time-delay systems using small impulsive disturbances // Phys. Rev. E. 2009. Vol. 80. 066206.