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

For citation:

Bezruchko B. P., Seleznev E. P., Ponomarenko V. I., Prokhorov M. D., Smirnov D. A., Dikanev T. V., Sysoev I. V., Karavaev A. S. Special approaches to global reconstruction of equations from time series. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 3, pp. 137-158. DOI: 10.18500/0869-6632-2002-10-3-137-158

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: 

Special approaches to global reconstruction of equations from time series

Bezruchko Boris Petrovich, Saratov State University
Seleznev Evgeny Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
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
Smirnov Dmitrij Alekseevich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Dikanev Taras Viktorovich, Huawei Technologies Co in Russia
Sysoev Ilya Vyacheslavovich, Saratov State University
Karavaev Anatolij Sergeevich, Saratov State University

Some problems arising during global reconstruction from time series are illustrated by reconstruction of efalon equations and modeling оf real-world radiophysical systems. Efficiency оf specialized approaches oriented to modeling оf restricted classes of systems is demonstrated and new specific techniques are proposed.

Key words: 
The work was supported by the Russian Foundation for Basic Research (grants № 02-02-17578, 02-02-06502, 02-02-06503), Russian Academy оf Sciences (youth grant № 23), American Civilian Research аnd Development Foundation (award № REC006), аnd Federal Special Program «Integration» (Reg. № В0075).
  1. Box С, Jenkins С. Time Series Analysis: Forecasting and Control. Revised ed. San Francisco: Holden-Day; 1976. 712 p.
  2. Lorenz EN. Deterministic nonperiodic flow. J. оf the Atmospheric Sciences. 1963;20(2):130-141. DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
  3. Ruelle D, Takens Е. On the nature оf turbulence. Commun. Math. Phys. 1971;20(3):167-192. DOI: 10.1007/BF01646553.
  4. Crutchfield JP, McNamara BS. Equations of motion from а data series. Complex Systems. 1987;1:417-452.
  5. Farmer JD, Sidorowich JJ. Predicting chaotic time series. Phys. Rev. Lett. 1987;59(8):845-848. DOI: 10.1103/PhysRevLett.59.845.
  6. Casdagli M. Nonlinear prediction of chaotic time series. Physica D. 1989;35(3):335-356. DOI: 10.1016/0167-2789(89)90074-2.
  7. Cremers J, Hubler А. Construction of differential equations from experimental data. Z. Naturforschung A. 1987;42(8):797-802. DOI: 10.1515/zna-1987-0805.
  8. Biinner MJ, Popp M, Meyer T, Kittel А, Rau U, Parisi J. Recovery оf scalar time-delay systems from time series. Phys. Lett. A. 1996;211(6):345-349. DOI: 10.1016/0375-9601(96)00014-X.
  9. Voss H, Kurths J. Reconstruction оf non-linear time delay models from data by the use оf optimal transformations. Phys. Lett. A. 1997;234(5):336-344. DOI: 10.1016/S0375-9601(97)00598-7.
  10. Bir M, Hegger R, Kantz H. Fitting partial differential equations to spacetime dynamics. Phys. Rev. Е. 1999;59(1):337-343. DOI: 10.1103/PhysRevE.59.337.
  11. Breeden JL, Hubler A. Reconstructing equations of motion from experimental data with unobserved variables. Phys. Rev. A. 1990;42(10):5817-5826. DOI: 10.1103/PhysRevA.42.5817.
  12. Baake E, Baake M, Bock HJ, Briggs KM. Fitting ordinary differential equations to chaotic data. Phys. Rev. A. 1992;45(8):5524-5529. DOI: 10.1103/PhysRevA.45.5524.
  13. Brown R, Rulkov NF, Tracy ER. Modeling and synchronizing chaotic systems from time-series data. Phys. Rev. Е. 1994;49(5):3784-3800. DOI: 10.1103/PhysRevE.49.3784.
  14. Judd K, Mees AI. On selecting models for nonlinear time series. Physica D. 1995;82(4):426-444. DOI: 10.1016/0167-2789(95)00050-E.
  15. Small М, Judd K. Comparison оf new nonlinear modeling techniques with application to infant respiration. Physica D. 1998;117(1-4):283-298. DOI: 10.1016/S0167-2789(97)00311-4.
  16. Judd K, Mees АI. Embedding as а modeling problem. Physica D. 1998;120(3-4):273-286. DOI: 10.1016/S0167-2789(98)00089-X.
  17. Gouesbet С, Magquet J. Construction оf phenomenological models from numerical scalar time series. Physica D. 1992;58(1-4):202-215. DOI: 10.1016/0167-2789(92)90109-Z.
  18. Gouesbet G, Letellier С. Global vector-field approximation by using а multivariate polynomial L2 approximation оn nets. Phys.Rev. Е. 1994;49(6):4955-4972. DOI: 10.1103/PhysRevE.49.4955.
  19. Letellier C, Le Sceller L, Dutertre Р, Gouesbet G, Fei Z, Hudson JL. Topological characterization and global vector field reconstruction of an experimental electrochemical system. J. Phys. Chem. 1995;99(18):7016-7027. DOI: 10.1021/j100018a039.
  20. Letellier C, Le Sceller L, Gouesbet G, Lusseyran F, Kemoun А, Izrar В. Recovering deterministic behavior from experimental time series in mixing reactor. AIChE Journal. 1997;43(9):2194-2202. DOI: 10.1002/aic.690430906.
  21. Letellier C, Maquet J, Labro H, Le Sceller L, Gouesbet G, Argoul Е, Arneodo А. Analyzing chaotic behavior in а Belousov-Zhabotinskyi reaction by using а global vector field reconstruction. J. Phys. Chem. 1998;102:10265-10273.
  22. Timmer J. Modeling noisy time series: physiological tremor. Int. J. Bifurc. Chaos. 1998;8(7):1505-1516. DOI: 10.1142/S0218127498001157.
  23. Kadtke J. Classification of highly noisy signals using global dynamical models. Phys. Lett. A. 1995;203(4):196-202. DOI: 10.1016/0375-9601(95)00375-D.
  24. Kadtke J, Kremliovsky M. Estimating statistics for detecting determinism using global dynamical models. Phys.Lett. A. 1997;229(2):97-106. DOI: 10.1016/S0375-9601(97)00149-7.
  25. Gribkov DA, Gribkova VV, Kravtsov YA, Kuznetsov YI, Rzhanov AG. «Reconstruction оf dynamical system structure from time series». J. Commun. Technol. Electron. 1994;39(2):269-277.
  26. Gribkov DA, Gribkova VV, Kravtsov YA, Kuznetsov Yl, Rzhanov AG, Chepurnov AS. Constructing model оf system for stabilization оf resonance frequency and temperature of linear electron accelerator section from experimental data. Vestnik оf Moscow State University. Ser. 3. 1994;35(1):96-98.
  27. Anosov OL, Butkovskii ОY, Kravtsov YA. Nonlinear chaotic systems identification from observed time series. Math. Models Methods Appl. Sci. 1997;7(1):49-59. DOI: 10.1142/S0218202597000049.
  28. Anosov OL, Butkovskii OY, Kravtsov YA. Minimax procedure for identification of chaotic systems from an observable time sequence. J. Commun. Technol. Electron. 1997;42(3):313-319.
  29. Pavlov AN, Janson NB. Application of technique for mathematical model reconstruction to electrocardiogram. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(1):93-108.
  30. Pavlov AN, Janson NB, Anishchenko VS. Application оf statistical methods for solution ю problem оf global reconstruction. Tech. Phys. Lett. 1997;23(8):7-13.
  31. Anishchenko VS, Pavlov AN. Global reconstruction in application to multichannel communication. Phys. Rev. Е. 1998;57(2):2455-2457. DOI: 10.1103/PhysRevE.57.2455.
  32. Anishchenko VS, Pavlov АМ, Janson NB. Global reconstruction in the presence of а priori information. Chaos, Solitons & Fractals. 1998;9(8):1267-1278. DOI: 10.1016/S0960-0779(98)00061-7.
  33. Hegger R, Kantz H, Schmuser F, Diestelhorst M, Kapsch К-Р, Beige H. Dynamical properties of a ferroelectric capacitors observed through nonlinear time series analysis. Chaos. 1998;8(3):727-754. DOI: 10.1063/1.166356.
  34. Bezruchko BP, Seleznev YP, Smirnov DA. Reconstruction оf equations of nonautonomous nonlinear oscillator from а time series: models, experiment. Izvestiya VUZ. Applied Nonlinear Dynamics. 1999;7(1):49-67.
  35. Biinner MJ, Meyer T, Kittel А, Parisi J. Recovery оf the time-evolution equation of time-delay systems from time series. Phys. Rev. Е. 1997;56(5):5083-5089. DOI: 10.1103/PhysRevE.56.5083.
  36. Hegger R, Bunner MJ, Kantz H, Giaquinta А. Identifying and modelling delay feedback systems. Phys. Rev. Lett. 1998;81(3):558 -561. DOI: 10.1103/PhysRevLett.81.558.
  37. Pavlov AN, Janson NB, Anishchenko VS. Reconstruction of dynamical systems. J. Commun. Technol. Electron. 1999;44(9):1075-1092.
  38. Anosov OL, Butkovskii OY, Kravtsov YА. Reconstruction оf dynamical systems from chaotic time series (brief review). Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(1):29-51.
  39. Biinner MJ, Ciofini M, Giaquinta А, Hegger R, Kantz H, Meucci R, Politi A. Reconstruction of systems with delayed feedback: (I) Theory. Eur. Phys. J. D. 2000;10(2):165-176. DOI: 10.1007/s100530050538. Reconstruction оf systems with delayed feedback: (II) Applications. Eur. Phys. J. D. 2000;10(2):177-185. DOI: 10.1007/s100530050539.
  40. Kantz H, Schreiber T. Nonlinear Time Series Analysis. Cambridge: Cambridge University Press; 1997. 369 p.
  41. Voss HU, Schwache А, Kurths J, Mitschke Е. Equations оf motion from chaotic data: A driven optical fiber ring resonator. Phys. Lett. A. 1999;256(1):47-54. DOI: 10.1016/S0375-9601(99)00219-4. Timmer J, Rust H, Horbelt W, Voss HU. Parametric, nonparametric and parametric modelling of а chaotic circuit time series. Phys. Lett. A. 2000;274(3-4):123-130. DOI: 10.1016/S0375-9601(00)00548-X.
  42. Horbelt W, Timmer J, Bunner MJ, Meucci В, Ciofini M. Identifying physical properties of а CO,-laser by dynamical modeling of measured time series. Phys. Rev. E. 2001;64(1):016222. DOI: 10.1103/PhysRevE.64.016222.
  43. Horbelt W, Timmer J, Voss H. Parameter estimation in nonlinear delayed feedback systems from noisy data. Phys. Lett. A. 2002;299(5-6):513-521. DOI: 10.1016/S0375-9601(02)00748-X.
  44. Bezruchko В, Smirnov D. Constructing nonautonomous differential equations from а time series. Phys. Rev. Е. 2001;63(1):016207. DOI: 10.1103/PhysRevE.63.016207.
  45. Bezruchko B, Dikanev T, Smirnov D. Role of transient processes for reconstruction оf model equations from time series. Phys. Rev. Е. 2001;64(3):036210. DOI: 10.1103/PhysRevE.64.036210.
  46. Smirnov Р, Bezruchko В, Seleznev Y. Choice оf dynamical variables for global reconstruction of model equations from time series. Phys. Rev. Е. 2002;65(2):026205. DOI: 10.1103/physreve.65.026205.
  47. Bezruchko B, Karavaev А, Ponomarenko V, Prokhorov M. Reconstruction оf time-delay systems from chaotic time series. Phys. Rev. Е. 2001;64(5):056216. DOI: 10.1103/PhysRevE.64.056216.
  48. Ponomarenko V, Prokhorov M. Extracting information masked by chaotic signal of time-delay system. Phys. Rev. Е. 2002;66(2):026215. DOI: 10.1103/physreve.66.026215.
  49. Le Sceller L, Letellier С, Gouesbet С. Structure selection for global vector field reconstruction by using the identification оf fixed points. Phys. Rev. Е. 1999;60(2):1600-1606. DOI: 10.1103/PhysRevE.60.1600.
  50. Aguirre LA, Freitas US, Letellier C, Maquet J. Structure-selection techniques applied to continuous-time nonlinear models. Physica D. 2001;158(1-4):1-18. DOI: 10.1016/S0167-2789(01)00313-X.
  51. Menard O, Letellier C, Maquet J, Le Sceller L, Gouesbet С. Analysis of a nonsynchronized sinusoidally driven dynamical system. Int. J. Bifurc. Chaos. 2000;10(7):1759-1772. DOI: 10.1142/S0218127400001080.
  52. Judd K, Small M. Towards long-term prediction. Physica D. 2000;136(1-2):31-44. DOI: 10.1016/S0167-2789(99)00152-9.
  53. Small M, Judd K, Mees A. Modeling continuous processes from data. Phys. Rev. E. 2002;65(4):046704. DOI: 10.1103/PhysRevE.65.046704.
  54. Takens Е. Detecting strange attractors in turbulence. In: Rang D, Young LS, editors. Dynamical Systems and Turbulence. Warwick, 1980. Lecture Notes in Mathematics, 1981. Vol. 898. Berlin: Springer; 1981. P. 366-381. DOI: 10.1007/BFb0091924.
  55. Sauer T, Yorke JA, Casdagli M. Embedology. J. Stat. Phys. 1991;65(3-4):579-616. DOI: 10.1007/BF01053745.
  56. Janson NB, Pavlov AN, Anishchenko VS. One method for restoring inhomogeneous attractors. Int. J. Bifurc. Chaos. 1998;8(4):825-833. DOI: 10.1142/S0218127498000620.
  57. McSharry PE, Smith LA. Better nonlinear models from noisy data: Attractors with maximum likelihood. Phys. Rev. Lett. 1999;83(21):4285-4288. DOI: 10.1103/PhysRevLett.83.4285.
  58. Letellier C, Macquet J, Le Sceller L, Gouesbet G, Aguirre LA. On the non-equivalence of observables in phase space reconstructions from recorded time series. J. Phys. A: Math. Gen. 1998;31(39):7913-7927. DOI: 10.1088/0305-4470/31/39/008.
  59. Kaplan DT. Exceptional events as evidence for determinism. Physica D. 1994;73(1-2):738-748. DOI: 10.1016/0167-2789(94)90224-0.
  60. Hasler M. Electric circuits with chaotic behavior. Proc. IEEE. 1978;75(2):40-55.
  61. Bezruchko BP, Seleznev YP. Complex dynamics оf driven oscillator with piecewise-linear characteristics. Tech. Phys. Lett. 1994;20(19):75-79.
  62. Levenberg K. A method for the solution оf certain problems in least squares. Quarterly оf Applied Mathematics. 1944;2:164-168. DOI: 10.1090/qam/10666.
  63. Bezruchko BP, Sysoev IV, Smirnov DA. Reconstruction оf model equations for driven systems under regular driving. Proceeding оf the 6th International School Chaos’2001. Saratov; 2001. Р. 17-18 (in Russian).
Available online: