×

zbMATH — the first resource for mathematics

Local Lyapunov exponents computed from observed data. (English) Zbl 0802.58041
The Lyapunov exponents of a dynamical system quantify the growth or decay of infinitesimal perturbations to orbits of that system, the long-time growth rates refer to invariants of the dynamical system (standard exponents, named global). The paper develops the authors’ earlier introduced notion [ibid. 1, No. 2, 175-199 (1991; Zbl 0797.58053)] of local Lyapunov exponents, which determine how a perturbation to a system orbit will grow in a finite time. The local exponents are shown to be derivable from observations of a scalar data set (e.g. a reconstruction of the dynamics from an experimental signal enters the scene). In several examples like Hénon map, Lorenz system, Ikeda map, it is demonstrated that the method allows one to accurately reproduce results determined when the dynamics is known beforehand.

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Eckmann, J.-P., and D. Ruelle, ?Ergodic Theory of Chaos and Strange Attractors,?Rev. Mod. Phys. 57, 617 (1985). · Zbl 0989.37516
[2] Abarbanel, H. D. I., R. Brown, and M. B. Kennel, ?Variation of Lyapunov Exponents on a Strange Attractor,?Journal of Nonlinear Science,1, 175-199 (1991). · Zbl 0797.58053
[3] Brown, R., P. Bryant, and H. D. I. Abarbanel, ?Computing the Lyapunov Spectrum of a Dynamical System from Observed Time Series,?Phys. Rev. A 43, 2787 (1991). An earlier paper with many of the main results is inPhys. Rev. Lett 65, 1523 (1990).
[4] Maãné, R., ?On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps,? inDynamical Systems and Turbulence, Warwick 1980, eds. D. Rand and L. S. Young,Lecture Notes in Mathematics 898, 230 Berlin: Springer, (1981).
[5] Takens, E., ?Detecting Strange Attractors in Turbulence,? inDynamical Systems and Turbulence, Warwick 1980, eds. D. Rand and L. S. Young,Lecture Notes in Mathematics 898, 366 Berlin: Springer, (1981).
[6] Parlitz, U.,Int. J. of Bif. Chaos,2, 155-165 (1992). · Zbl 0878.34047
[7] Abarbanel, H. D. I., and M. M. Sushchik, ?True Local Lyapunov Exponents and Models of Chaotic Systems Based on Observations,? to be submitted toJournal of Nonlinear Science, May 1992.
[8] Fraser, A. M., and H. L. Swinney,Phys. Rev.,33A, 134 (1986); Fraser, A. M.,IEEE Trans, on Info. Theory 35, 245 (1989); Fraser, A. M.,Physica 34D, 391 (1989).
[9] Abarbanel, H. D. L., R. Brown, and M. B. Kennel, ?Lyapunov Exponents in Chaotic Systems: Their Importance and Their Evaluation Using Observed Data,?Int. Journal Mod. Phys. B5, 1347-1375 (1991). · Zbl 0737.58051
[10] Kennel, Matthew, B., R. Brown, and H. D. I. Abarbanel, ?Determining Minimum Embedding Dimension Using a Geometrical Construction.,?Phys. Rev. A,45, 3403-3411 (1992).
[11] Oseledec, V. I., ?A Multiplicative Ergodic Theorem. Lyapunov Characteristic Numbers for Dynamical Systems,?Trudy Mosk. Mat. Obsc. 19, 197 (1968);Moscow Math. Soc. 19, 197 (1968).
[12] Eckmann, J. -P., S. O. Kamphorst, D. Ruelle, and S. Ciliberto, ?Lyapunov Exponents from Time Series,?Phys. Rev. A34, 4971 (1986).
[13] [13]Sano, M., and Y. Sawada, ?Measurement of Lyapunov Spectrum from a Chaotic Time Series,?Phys. Rev. Lett. 55, 1082 (1985). This paper also introduces the local-phase space method for finding DF(y(n)), but uses only local linear maps for this purpose. The method is unable to determine all exponents, just the largest exponent. The comparison between this method and the one used here is discussed in [9].
[14] Ford, J. R., and W. R. Borland, ?Common Los Alamos Mathematical Software Compendium,? Document Number CIC # 148 (April 1988), Los Alamos National Laboratory, Los Alamos, New Mexico.
[15] Stoer, J., and R. Burlisch,Introduction to Numerical Analysis, Springer-Verlag, New York. (1980).
[16] [16]Landa, P. S., and M. Rozenblum, ?Time Series Analysis for System Identification and Diagnostics,?Physica D 48, 232-254 (1991). · Zbl 0717.92002
[17] Lorenz, E. N., ?Deterministic Nonperiodic Flow,?J. Atmos. Sci. 20, 130 (1963). · Zbl 1417.37129
[18] Henon, M., ?A Two-Dimensional Mapping with a Strange Attractor,?Commun. Math. Phys. 50, 69 (1976). · Zbl 0576.58018
[19] Ikeda, K., ?Multiple-Valued Stationary State and Its Instability of the Transmitted Light by a Ring Cavity System,?Opt. Commun. 30, 257 (1979).
[20] [20]Ellner, S., A. R. Gallant, D. F. McGaffrey, and D. Nychka, ?Convergence Rates and Data Requirements for Jacobian-based Estimates of Lyapunov Exponents from Data,?Phys. Lett. A,153, 357-363 (1991).
[21] McGaffrey, D. F., S. Ellner, A. R. Gallant, and D. W. Nychka, ?Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression,?Journal of the American Statistical Association, to appear. · Zbl 0782.62045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.