zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Determining Lyapunov exponents from a time series. (English) Zbl 0585.58037
Authors’ summary: ”We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskij reaction and Couette-Taylor flow.”
Reviewer: K.Furutani

37-99Dynamic systems and ergodic theory (MSC2000)
Full Text: DOI
[1] Abraham, N. B.; Gollub, J. P.; Swinney, H. L.: Testing nonlinear dynamics. Physica 11D, 252 (1984) · Zbl 0582.58021
[2] Roux, J. -C.; Simoyi, R. H.; Swinney, H. L.: Observation of a strange attractor. Physica 8D, 257 (1983) · Zbl 0538.58024
[3] Brandstater, A.; Swift, J.; Swinney, H. L.; Wolf, A.; Farmer, J. D.; Jen, E.; Crutchfield, J. P.: Low-dimensional chaos in a hydrodynamic system. Phys. rev. Lett. 51, 1442 (1983)
[4] Malraison, B.; Atten, P.; Berge, P.; Dubois, M.: Turbulence-dimension of strange attractors: an experimental determination for the chaotic regime of two convective systems. J. physique letters 44, L-897 (1983)
[5] Guckenheimer, J.; Buzyna, G.: Dimension measurements for geostrophic turbulence. Phys. rev. Lett. 51, 1438 (1983)
[6] Gollub, J. P.; Romer, E. J.; Socolar, J. E.: Trajectory divergence for coupled relaxation oscillators: measurements and models. J. stat. Phys. 23, 321 (1980)
[7] Oseledec, V. I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow math. Soc. 19, 197 (1968)
[8] Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J. -M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Meccanica 15, 9 (1980) · Zbl 0488.70015
[9] Shimada, I.; Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. theor. Phys. 61, 1605 (1979) · Zbl 1171.34327
[10] Shaw, R.: Strange attractors, chaotic behavior, and information flow. Z. naturforsch. 36A, 80 (1981) · Zbl 0599.58033
[11] Hudson, J. L.; Mankin, J. C.: Chaos in the Belousov-zhabotinskii reaction. J. chem. Phys. 74, 6171 (1981)
[12] Nagashima, H.: Experiment on chaotic response of forced Belousov-zhabotinskii reaction. J. phys. Soc. Japan 51, 21 (1982)
[13] Wolf, A.; Swift, J.: Progress in computing Lyapunov exponents from experimental data. Statistical physics and chaos in fusion plasmas (1984)
[14] Wright, J.: Method for calculating a Lyapunov exponent. Phys. rev. 29, 2923 (1984)
[15] Blacher, S.; Perdang, J.: Power of chaos. Physica 3D, 512 (1981) · Zbl 1194.37046
[16] Crutchfield, J. P.; Packard, N. H.: Symbolic dynamics of noisy chaos. Physica 7D, 201 (1983) · Zbl 0508.58029
[17] Grassberger, P.; Procaccia, I.: Estimation of the Kolmogorov entropy from a chaotic signal. Phys. rev. 28, 2591 (1983) · Zbl 0593.58024
[18] Shaw, R.: The dripping faucet. (1984)
[19] Farmer, J. D.; Ott, E.; Yorke, J. A.: The dimension of chaotic attractors. Physica 7D, 153 (1983) · Zbl 0561.58032
[20] S. Ciliberto and J.P. Gollub, ”Chaotic Mode Competition in Parametrically Forced Surface Waves”--preprint.
[21] Grassberger, P.; Procaccia, I.: Characterization of strange attractors. Phys. rev. Lett. 50, 346 (1983) · Zbl 0593.58024
[22] Haken, H.: At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point. Phys. lett. 94A, 71 (1983)
[23] Lorenz, E. N.: Deterministic nonperiodic flow. J. atmos. Sci. 20, 130 (1983)
[24] Rossler, O. E.: An equation for hyperchaos. Phys. lett. 71A, 155 (1979)
[25] Hénon, M.: A two-dimensional mapping with a strange attractor. Comm. math. Phys. 50, 69 (1976) · Zbl 0576.58018
[26] Rossler, O. E.: An equation for continuous chaos. Phys. lett. 57A, 397 (1976)
[27] Mackey, M. C.; Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287 (1977)
[28] Frederickson, P.; Kaplan, J.; Yorke, E.; Yorke, J.: The Lyapunov dimension of strange attractors. J. diff. Eqs. 49, 185 (1983) · Zbl 0515.34040
[29] Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Comm. math. Phys. 81, 229 (1981) · Zbl 0486.58021
[30] Russell, D. A.; Hanson, J. D.; Ott, E.: Dimension of strange attractors. Phys. rev. Lett. 45, 1175 (1980)
[31] D. Ruelle, private communication.
[32] After R. Shaw, unpublished.
[33] Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S.: Geometry from a time series. Phys. rev. Lett. 45, 712 (1980)
[34] Takens, F.: Detecting strange attractors in turbulence. Lecture notes in mathematics 898, 366 (1981) · Zbl 0513.58032
[35] Greenside, H. S.; Wolf, A.; Swift, J.; Pignataro, T.: The impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors. Phys. rev. 25, 3453 (1982)
[36] Roux, J. -C.; Rossi, A.: Quasiperiodicity in chemical dynamics. Nonequilibrium dynamics in chemical systems, 141 (1984)
[37] Crutchfield, J. P.; Farmer, J. D.; Huberman, B. A.: Fluctuations and simple chaotic dynamics. Phys. rep. 92, 45 (1982)
[38] Ruelle, D.: Applications conservant une mesure absolument continué par rapport à dx sur [0,1]. Comm. math. Phys. 55, 47 (1977) · Zbl 0362.28013
[39] Knuth, D. E.: The art of computer programming, vol. 3 -- sorting and searching. (1975) · Zbl 0335.68018