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

For citation:

Ponomarenko V. I., Prokhorov M. D., Karavaev A. S., Seleznev E. P., Dikanev T. V. Recovery of dynamical models of time-delay systems from time series. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 3, pp. 56-66. DOI: 10.18500/0869-6632-2003-11-3-56-66

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):
Article type: 

Recovery of dynamical models of time-delay systems from time series

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
Karavaev Anatolij Sergeevich, Saratov State University
Seleznev Evgeny Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Dikanev Taras Viktorovich, Huawei Technologies Co in Russia

We develop the method for the estimation оf the parameters оf time-delay systems from time series. The method is based on the statistical analysis of time intervals between extrema in the time series and the projection of the infinite-dimensional phase space оf а time-delay system to suitably chosen low-dimensional subspaces. We verify our method by using it for the reconstruction оf different time-delay differential equations from their chaotic solutions.

Key words: 
The authors thank B.P. Bezruchko for stimulating discussions. This work was supported by the Russian Foundation о Fundamental Research, Grant № 03-02-17593, U.S. Civilian Research Development Foundation for the Independent States of the Former Soviet Union, Award № REC-006, State contract M with Minpromnauki RF, аnd FNP «Dynasty» аnd ICFFM, Grant № 245 662.
  1. Kuang Y. Delay Differential Equations With Applications in Population Dynamics. Boston: Academic Press; 1993. 398 p.
  2. Ikeda K. Multiple-valued stationary state and its instability оf the transmitted light by а ring cavity system. Opt. Commun. 1979;30(2):257–261.
  3. Lang R, Kobayashi K. External optical feedback effects оп semiconducior injection lasers. IEEE J. Quantum Electron. 1980;16(3):347–355. DOI: 10.1109/JQE.1980.1070479.
  4. Mackey MC, Glass L. Oscillations and chaos in physiological control systems. Science. 1977;197(4300):287–289. DOI: 10.1126/science.267326.
  5. Fowler AC, Kember С. Delay recognition in chaotic time series. Phys. Lett. А. 1993;175(6):402–408. DOI: 10.1016/0375-9601(93)90991-8.
  6. Hegger R, Bunner MJ, Kaniz H, Giaquinta А. Identifying and modeling delay feedback systems. Phys. Rev. Lett. 1998;81(3):558–561. DOI: 10.1103/PhysRevLett.81.558.
  7. Zhou C, Lai C-H. Extracting messages masked by chaotic signals оf time-delay systems. Phys. Rev. Е. 1999;60(1):320–323. DOI: 10.1103/PhysRevE.60.320.
  8. Udaltsov VS, Goedgebuer J-P, Larger L, Cuenot J-B, Levy Р, Rhodes WT. Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations. Phys. Rev. А. 2003;308(1):54–60. DOI: 10.1016/S0375-9601(02)01776-0.
  9. Tian Y-C, Gao F. Extraction of delay information from chaotic time series based on information entropy. Physica D. 1997;108(1–2):113–118. DOI: 10.1016/S0167-2789(97)82008-8.
  10. Kaplan DT, Glass L. Coarse-grained embeddings оf time series: random walks, gaussian random process, and deterministic chaos. Physica D. 1993;64(4):431–454. DOI: 10.1016/0167-2789(93)90054-5.
  11. Bunner MJ, Рорр M, Meyer T, Kittel А, Rau U, Parisi J. Recovery оf scalar time-delay systems from time series. Phys. Lett. А. 1996;211(6):345–349. DOI: 10.1016/0375-9601(96)00014-X.
  12. Bunner MJ, Popp M, Meyer T, Kittel A, Parisi J. Tool to recover scalar time-delay systems from experimental time series. Phys. Rev. E. 1996;54(4):R3082–R3085. DOI: 10.1103/PhysRevE.54.R3082.
  13. Bunner MJ, Meyer T, Kittel А, Parisi J. Recovery оf the time-evolution equation оf time-delay systems from time series. Phys. Rev. E. 1997;56(5):5083–5089. DOI: 10.1103/PhysRevE.56.5083.
  14. Bunner MJ, Ciofini M, Giaquinta А, Hegger R, Kantz H, Meucci R, Politi А. Reconstruction оf systems with delayed feedback: (I) Theory. Eur. Phys. J. D. 2000;10(2):165–176. DOI: 10.1007/s100530050538.
  15. Voss H, Kurths J. Reconstruction of nonlinear time delay models from data by the use оf optimal transformations. Phys. Lett. А. 1997;234:336–344.
  16. Ellner SP, Kendall BE, Wood SN, McCauley E, Briggs CJ. Inferring mechanism from time-series data: delay differential equations. Physica D. 1997;110(3–4):182–194. DOI: 10.1016/S0167-2789(97)00123-1.
  17. Eurich CW, Milton JG. Noise-induced transitions in human postural sway. Phys. Rev. E. 1996;54(6):6681–6684. DOI: 10.1103/PhysRevE.54.6681.
  18. Ohira T, Sawatari R. Delay estimation from noisy time series. Phys. Rev. Е. 1997;55(3):R2077–R2080. DOI: 10.1103/PhysRevE.55.R2077.
  19. Bezruchko BP, Karavaev AS, Ponomarenko V, Prokhorov MD. Reconstruction оf time-delay systems from chaotic time series. Phys. Rev. Е. 2001;64(5):056216. DOI: 10.1103/PhysRevE.64.056216.
  20. Ponomarenko VI, Prokhorov MD. Extracting information masked by the chaotic signal оf а time-delay system. Phys. Rev. Е. 2002;66(2):026215. DOI: 10.1103/physreve.66.026215.
  21. Voss H, Kurths J. Reconstruction of nonlinear time-delayed feedback models from optical data. Chaos, Solitons and Fractals. 1999;10(4–5):805–809. DOI: 10.1016/S0960-0779(98)00030-7.
Available online: