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

For citation:

Ponomarenko V. I., Prokhorov M. D. Reconstruction of time-delay systems equations over experimental time series. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 1, pp. 52-64. DOI: 10.18500/0869-6632-2002-10-1-52-64

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):
PDF icon and_2002-1-2_ponomarenko_prokhorov.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
537.86
DOI: 
10.18500/0869-6632-2002-10-1-52-64

Reconstruction of time-delay systems equations over experimental time series

Autors: 
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
Abstract: 

We propose а new method to reconstruct the time-delay differential equations over time series. The method is based on extreme location in the time series. On the base of information about time delay we determine nonlinear function and inertiality parameter. We verify our method by using it for the reconstruction of the time-delay differential equations from their chaotic solutions and for modelling experimental systems with delay-induced dynamics from their chaotic time series. 

Key words: 
-
Acknowledgments: 
This work was supported by the Russian Foundation for Basic Research (grant No. 02-02-17578), Young Scientists Grant No. 23 RAS, and CRDF grant REC-006.
Reference: 
  1. Mackey MC, Glass L. Oscillations and chaos in physiological control systems. Science. 1977;197(4300):287-289. DOI: 10.1126/science.267326.
  2. Gribkov D, Gribkova V. Learning dynamics from nonstationary time series: Analysis оf electroencephalograms. Phys. Rev. E. 2000;61(6):6538-6545. DOI: 10.1103/PhysRevE.61.6538.
  3. Kuznetsov SP. Complex dynamics of oscillators with delayed feedback (review). Radiophys. Quantum Electron. 1982;25(12):996-1009. DOI: 10.1007/BF01037379.
  4. Dmitriev AS, Kislov VY. Stochastic Oscillations in Radiophysics and Electronics. Moscow: Nauka; 1989. 280 p. (in Russian).
  5. 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. DOI: 10.1016/0030-4018(79)90090-7.
  6. Bezruchko BP, Smirnov DA. Constructing nonautonomous differential equations from experimental time series. Phys. Rev. Е. 2001;63(1):016207. DOI: 10.1103/PhysRevE.63.016207.
  7. Horbelt W, Timmer J, Biinner MJ, Meucci R, Ciofini M. Identifying physical properties оf а CO, laser by dynamical modeling оf measured time series. Phys. Rev. E. 2001;64(1):016222. DOI: 10.1103/physreve.64.016222.
  8. Anishchenko VS, Paviov AN, Janson NB. Global reconstruction in the presence of a priori information. Chaos, Solitons & Fractals. 1998;9(8):1267-1278. DOI: 10.1016/S0960-0779(98)00061-7.
  9. Farmer JD. Chaotic attractors of an infinite-dimensional dynamical system. Physica D. 1982;4(3):366-393. DOI: 10.1016/0167-2789(82)90042-2.
  10. Biinner MJ, Popp 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.
  11. Biinner MJ, Popp M, Meyer T, Kittel А, Parisi J. Tool to recover scalar time-delay systems from experimental time series. Phys. Rev. Е. 1996;54(4):R3082-R3085. DOI: 10.1103/PhysRevE.54.R3082.
  12. Fowler AC, Kember G. Delay recognition in chaotic time series. Phys. Lett. А. 1993;175(6):402-408. DOI: 10.1016/0375-9601(93)90991-8.
  13. Hegger R, Biinner MJ, Kantz H. Identifying and modeling delay feedback systems. Phys. Rev. Lett., 1998;81(3):558-561. DOI: 10.1103/PhysRevLett.81.558.
  14. Zhou C, Lai C-H. Extracting messages masked by chaotic signals оf time-delay systems. Phys. Rev. E. 1999;60(1):320-323. DOI: 10.1103/physreve.60.320.
  15. Tian Y-C, Gao F. Extraction of delay information from chaotic time series based оп information entropy. Physica D. 1997;108(1-2):113-118. DOI: 10.1016/S0167-2789(97)82008-8.
  16. Kaplan DT, Glass L. Coarse-grained embeddings of 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.
  17. Biinner MJ, Meyer T, Kittel A, Parisi J. Recovery of the time-evolution equation оf time-delay systems from time series. Phys. Rev. Е. 1997;56(5):5083-5089. DOI: 10.1103/PhysRevE.56.5083.
  18. Voss H, Kurths J. Reconstruction of non-linear time delay models from data by the use оf optimal transformations. Phys. Lett. А. 1997;234:336-344.
  19. 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.
  20. 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.
  21. Ohira Т‚ Sawatari R. Delay estimation from noisy time series. Phys. Rev. Е. 1997;55(3):R2077-R2080. DOI: 10.1103/PhysRevE.55.R2077.
  22. Bezruchko BP, Karavaev AS, Ponomarenko VI, Prokhorov MD. Reconstruction оf time-delay systems from chaotic time series. Phys. Rev. Е. 2001;64(5):056216. DOI: 10.1103/PhysRevE.64.056216.
  23. Karavaev AS, Ponomarenko VI, Prokhorov MD. Reconstruction of scalar time-delay system models. Tech. Phys. Lett. 2001;27(5):414-418. DOI: 10.1134/1.1376769.
  24. Mensour B, Longtin А. Synchronization оf delay-differential equations with application to private communication. Phys. Lett. А. 1998;244(1-3):59-70. DOI: 10.1016/S0375-9601(98)00271-0.
Received: 
21.02.2002
Accepted: 
20.03.2002
Available online: 
13.12.2023
Published: 
31.07.2002
  • 191 reads