Abstract
An approach to the long-term prognosis of qualitative behavior of a dynamic system (DS) is proposed, which is based on the nonlinear-dynamical analysis of a weakly nonstationary chaotic time series (TS). A method for constructing prognostic models using the observed evolution of a single dynamic variable is described, which employs the proposed approach for prediction of bifurcations of low-dimensional DSs. The method is applied to analyze the TS generated by the Roessler system and the system of equations modeling photochemical processes in the mesosphere. The analysis is performed for a TS calculated in the case of a slow variation in the control parameter of the system. The duration of the “observed” TS is limited such that the system demonstrates only one, chaotic, type of behavior without any bifurcations during the observed TS. The proposed algorithm allows us to predict correctly the bifurcation sequences for both systems at times much longer than the duration of the observed TS, to point out the expected instants of specific bifurcation transitions and accuracy of determining these instants, as well as to calculate the probabilities to observe the predicted regimes of the system's behavior at the time of interest.
Similar content being viewed by others
REFERENCES
H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag, New York (1997).
F. Takens, in: D.A. Rand and L.-S. Young (eds.), Dynamical Systems and Turbulence, Warwick, 1980. Lecture Notes in Mathematics, Vol. 898, Springer, Berlin (1981), p. 366.
J. D. Farmer and J. J. Sidorowich, Phys. Rev. Lett., 59 845 (1987).
R. Manuka and R. Savit, Physica D, 99, 134 (1996).
T. Schreiber, Phys. Rev. Lett., 78, 843 (1997).
A. Witt, J. Kurths, and A. Pikovsky, Phys. Rev. E, 58, 1800 (1998).
N. B. Yanson, et.al., Pis'ma Zh. Tekh. Fiz., 25, 74 (1999).
J. Stark, et.al., Nonlinear Analysis, Theory, Methods & Applications, 30, 5303 (1997).
V. S. Anishenko, T. E. Vadivasova, and V. V. Astakhov, Nonlinear Dynamics of Chaotic and Stochastic Systems [in Russian], Saratov Univ. Press, Saratov (1999).
O. E. Róssler, Phys. Lett. A, 57, 397 (1976).
B. Fichtelmann and G. Sonnemann, Ann. Geophys., 10, 719 (1992).
G. Sonnemann and B. Fichtelmann, J. Geophys. Res. D, 102, 1193 (1997).
A. M. Fraser and H. L. Swinney, Phys. Rev. A, 33, 1134 (1986).
M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Phys. Rev. A, 45, 3403 (1992).
V. S. Anishenko and A. N. Pavlov, Phys. Rev. E, 57, 2455 (1998).
G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York (1961).
A. M. Feigin, I. B. Konovalov, and Y. I. Molkov, J. Geophys. Res. D, 103, 25447 (1998).
I. B. Konovalov and A. M. Feigin, Nonlinear Processes in Geophysics, 7, 87 (2000).
P. Yang, G. P. Brasseur, and J. C. Gille, Physica D, 76, 331 (1994).
I.-F. Li, P. Biswas, and S. Islam, Atmos. Environ., 28, 1707 (1994).
J. D. Neelin and M. Latif, Physics Today, No. 12, 32 (1998).
B. Wang, A. Barcilon, and Z. Fang, J. Atmos. Sci., 56, 5 (1999).
H. D. I. Abarbanel, R. Huerta, M. I. Rabinovich, et.al., Neural Comput., 8, No. 8, 1567 (1996).
G. W. Frank, T. Lookman, M. A. H. Nerenberg, et.al., Physica D, 46, 427 (1990).
H. N. Srivastava, S. N. Bhattacharya, and K. C. Sinha Ray, Geophys. Res. Lett., 23, 3519 (1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Feigin, A.M., Molkov, Y.I., Mukhin, D.N. et al. Prognosis of Qualitative Behavior of a Dynamic System by the Observed Chaotic Time Series. Radiophysics and Quantum Electronics 44, 348–367 (2001). https://doi.org/10.1023/A:1017988912081
Issue Date:
DOI: https://doi.org/10.1023/A:1017988912081