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A practical method for calculating largest Lyapunov exponents from small data sets. (English) Zbl 0779.58030
This paper addresses the problem of calculating the largest Lyapunov exponents of a dynamical system from small sets of experimental data. The authors have devised an algorithm which they claim to be fast, easily implemented, and robust to changes in embedding dimension, size of the data set, reconstruction delay, and noise level. The current most popular algorithm for quantifying chaotic behavior is the Grassberger-Procaccia method, but this algorithm is sensitive to small variations in parameters, and generally requires long, noise-free time series. The authors of this paper note that calculation of the largest Lyapunov exponents is the most direct method for quantifying chaotic behavior manifested in time series produced by dynamical systems. However, the largest Lyapunov exponent has proven difficult to estimate for small data sets, and existing algorithms to compute it are computationally intensive and difficult to implement. The new algorithm which the authors present is direct and efficient, and is capable of estimating the correlation dimension as well as the largest Lyapunov exponent.

MSC:
37D45Strange attractors, chaotic dynamics
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References:
[1] Abarbanel, H. D. I.; Brown, R.; Kadtke, J. B.: Prediction in chaotic nonlinear systems: methods for time series with broadband Fourier spectra. Phys. rev. A 41, 1782 (1990)
[2] Abraham, N. B.; Albano, A. M.; Das, B.; De Guzman, G.; Yong, S.; Gioggia, R. S.; Puccioni, G. P.; Tredicce, J. R.: Calculating the dimension of attractors from small data sets. Phys. lett. A 114, 217 (1986)
[3] Albano, A. M.; Muench, J.; Schwartz, C.; Mees, A. I.; Rapp, P. E.: Singular-value decomposition and the grassberger-procaccia algorithm. Phys. rev. A 38, 3017 (1988)
[4] Albano, A. M.; Passamante, A.; Farrell, M. E.: Using higher-order correlations to define an embedding window. Physica D 54, 85 (1991) · Zbl 0748.93005
[5] Benettin, G.; Froeschle, C.; Scheidecker, J. P.: Kolmogorov entropy of a dynamical system with increasing number of degrees of freedom. Phys. rev. A 19, 2454 (1979)
[6] Bennettin, G.; Galgani, L.; Strelcyn, J. -M.: Kolmogorov entropy and numerical experiments. Phys. rev. A 14, 2338 (1976)
[7] Briggs, K.: An improved method for estimating Lyapunov exponents of chaotic time series. Phys. lett. A 151, 27 (1990)
[8] Broomhead, D. S.; King, G. P.: Extracting qualitative dynamics from experimental data. Physica D 20, 217 (1986) · Zbl 0603.58040
[9] Brown, R.; Bryant, P.; Abarbanel, H. D. I.: Computing the Lyapunov spectrum of a dynamical system from observed time series. Phys. rev. A 43, 2787 (1991)
[10] Casdagli, M.: Nonlinear prediction of chaotic time series. Physica D 35, 335 (1989) · Zbl 0671.62099
[11] Chen, P.: Empirical and theoretical evidence of economic chaos. Sys. dyn. Rev. 4, 81 (1988)
[12] Deppisch, J.; Bauer, H. -U.; Geisel, T.: Hierarchical training of neural networks and prediction of chaotic time series. Phys. lett. A 158, 57 (1991)
[13] Eckmann, J. -P.; Kamphorst, S. O.; Ruelle, D.; Ciliberto, S.: Lyapunov exponents from time series. Phys. rev. A 34, 4971 (1986)
[14] Eckmann, J. -P.; Ruelle, D.: Fundamental limtations for estimating dimensions and Lyapunov exponents in dynamical systems. Physica D 56, 185 (1992) · Zbl 0759.58030
[15] Eckmann, J. -P.; Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. mod. Phys. 57, 617 (1985) · Zbl 0989.37516
[16] Ellner, S.; Gallant, A. R.; Mccaffrey, D.; Nychka, D.: Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data. Phys. lett. A 153, 357 (1991)
[17] Farmer, J. D.; Sidorowich, J. J.: Predicting chaotic time series. Phys. rev. Lett. 59, 845 (1987)
[18] Frank, G. W.; Lookman, T.; Nerenberg, M. A. H.; Essex, C.; Lemieux, J.; Blume, W.: Chaotic time series analysis of epileptic seizures. Physica D 46, 427 (1990) · Zbl 0713.92010
[19] Fraser, A. M.; Swinney, H. L.: Independent coordinates for strange attractors from mutual information. Phys. rev. A 33, 1134 (1986) · Zbl 1184.37027
[20] Grassberger, P.; Procaccia, I.: Characterization of strange attractors. Phys. rev. Lett. 50, 346 (1983) · Zbl 0593.58024
[21] Grassberger, P.; Procaccia, I.: Estimation of the Kolmogorov entropy from a chaotic signal. Phys. rev. A 28, 2591 (1983) · Zbl 0593.58024
[22] Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. math. Phys. 50, 69 (1976) · Zbl 0576.58018
[23] Liebert, W.; Schuster, H. G.: Proper choice of the time delay for the analysis of chaotic time series. Phys. lett. A 142, 107 (1989)
[24] Lorenz, E. N.: Deterministic nonperiodic flow. J. atmos. Sci. 20, 130 (1963)
[25] Mackey, M. C.; Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287 (1977)
[26] Oseledec, V. I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow math. Soc. 19, 197 (1968)
[27] Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S.: Geometry from a time series. Phys. rev. Lett. 45, 712 (1980)
[28] Ramsey, J. B.; Yuan, H. -J.: The statistical properties of dimension calculations using small data sets. Nonlinearity 3, 155 (1990) · Zbl 0736.58031
[29] Rauf, F.; Ahmed, H. M.: Calculation of Lyapunov exponents through nonlinear adaptive filters. Proc. IEEE int. Symp. on circuits and systems (1991)
[30] Rössler, O. E.: An equation for hyperchaos. Phys. lett. A 71, 155 (1979) · Zbl 0996.37502
[31] Rössler, O. E.: An equation for continuous chaos. Phys. lett. A 57, 397 (1976)
[32] Sano, M.; Sawada, Y.: Measurement of the Lyapunov spectrum from a chaotic time series. Phys. rev. Lett. 55, 1082 (1985)
[33] Sato, S.; Sano, M.; Sawada, Y.: Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems. Prog. theor. Phys. 77, 1 (1987)
[34] Shimada, I.; Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. theor. Phys. 61, 1605 (1979) · Zbl 1171.34327
[35] Stoop, R.; Parisi, J.: Calculation of Lyapunov exponents avoiding spurious elements. Physica D 50, 89 (1991) · Zbl 0732.58026
[36] Sugihara, G.; May, R. M.: Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344, 734 (1990)
[37] Takens, F.: Detecting strange attractors in turbulence. Lecture notes in mathematics 898, 366 (1981) · Zbl 0513.58032
[38] Wales, D. J.: Calculating the rate loss of information from chaotic time series by forecasting. Nature 350, 485 (1991)
[39] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A.: Determining Lyapunov exponents from a time series. Physica D 16, 285 (1985) · Zbl 0585.58037
[40] Wright, J.: Method for calculating a Lyapunov exponent. Phys. rev. A 29, 2924 (1984)
[41] Zeng, X.; Eykholt, R.; Pielke, R. A.: Estimating the Lyapunov-exponent spectrum from short time series of low precision. Phys. rev. Lett. 66, 3229 (1991)