×

zbMATH — the first resource for mathematics

The dimension of chaotic attractors. (English) Zbl 0561.58032
Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica D 7, 153-180 (1983).
[For the entire collection see Zbl 0536.00007.]
The authors review most of the notions of dimension for chaotic attractors that are in current use, describe rigorous and numerical results on the relations between different types of dimension, and formulate the conjecture that for typical chaotic attractors all dimensions defined by purely metric properties take on a common value (the ”fractal dimension”), while on the other hand frequency dependent dimensions take on a (generally different) common value. They support this conjecture by numerical computations on a class of generalized baker’s transformations. The frequency dependent dimensions typically coincide with the Lyapunov-dimension. The feasability of computing the different dimension numbers is discussed. The reader might wish to compare these results to those obtained by P. Grassberger and I. Procaccia [Physica D 9, 189–208 (1983; Zbl 0593.58024)] where a further frequency dependent dimension number is defined.
Reviewer: G.Keller

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A05 Dynamical aspects of measure-preserving transformations
28D05 Measure-preserving transformations
37D99 Dynamical systems with hyperbolic behavior
References:
[1] Shaw, R.: Strange attractors, chaotic behavior, and information flow. Z. naturforsch. 36a, 80 (1981) · Zbl 0599.58033
[2] Ott, E.: Strange attractors and chaotic motions of dynamical systems. Rev. mod. Phys. 53, 655 (1981) · Zbl 1114.37303
[3] Helleman, R.: Self-generated chaotic behavior in nonlinear mechanics. Fundamental problems in stat. Mech. 5, 165-233 (1980)
[4] Yorke, J. A.; Yorke, E. D.: Chaotic bahavior and fluid dynamics. Topics in applied physics 45, 77-95 (1981)
[5] Mandelbrot, B.: Fractals: form, chance, and dimension. (1977) · Zbl 0376.28020
[6] Mandelbrot, B.: The fractal geometry of nature. (1982) · Zbl 0504.28001
[7] Sinai, Ja.: Gibbs measure in ergodic theory. Russ. math. Surveys 4, 21-64 (1972) · Zbl 0246.28008
[8] P. Frederickson, J. Kaplan, E. Yorke, and J. Yorke, ”The Lyapunov Dimension of Strange Attractors”, J. Diff. Eqns. in press. · Zbl 0515.34040
[9] Farmer, J. D.: Dimension, fractal measures, and chaotic dynamics. Evolution of order and chaos, 228 (1982)
[10] Farmer, J. D.: Information dimension and the probabilistic structure of chaos. UCSC doctoral disseration (1981) and Z. Naturforsch. 37a, 1304-1325 (1982)
[11] Alexander, J.; Yorke, J.: The fat Baker’s transformation. (1982)
[12] L.S. Young, ”Dimension, Entropy, and Lyapunov Exponents”, to appear in Ergodic Theory and Dynamical Systems. · Zbl 0523.58024
[13] Hurwicz, W.; Wallman, H.: Dimension theory. (1948) · Zbl 0036.12501
[14] Kolmogroov, A. N.: A new invariant for transitive dynamical systems. Dokl. akad. Nauk SSSR 119, 861-864 (1958) · Zbl 0083.10602
[15] Hausdorff: Dimension und \(Au{\beta}\)eres \(Ma{\beta}\). Math. annalen. 79, 157 (1918) · JFM 46.0292.01
[16] Bowen, R.; Ruelle, D.: The ergodic theory of axiom-A flows. Inv. math. 29, 181-202 (1975) · Zbl 0311.58010
[17] Renyi, A.: See alsoacta Mathematica (Hungary). Acta Mathematica (Hungary) 10, 193 (1959)
[18] Shannon, C.: A mathematical theory of communication. Bell tech. Jour. 27, 623-656 (1948) · Zbl 1154.94303
[19] Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Comm. math. Phys. 81, 229-238 (1981) · Zbl 0486.58021
[20] F. Takens, ”Invariants Related to Dimension and Entropy” , to appear in Atas do 13 Colognio Brasiliero de Mathematica.
[21] Janssen, T.; Tjon, J.: Bifurcations of lattice structure. (1982)
[22] Kaplan, J.; Yorke, J.: H.o.peitgenh.o.waltherfunctional differential equations and the approximation of fixed points, Proceedings. Lecture notes in math. 730, 228 (1978)
[23] Mori, H.: Prog. theor. Phys.. 63, 3 (1980)
[24] Oseledec, V. I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow math. Soc. 19, 197 (1968)
[25] C. Grebogi, E. Ott, and J. Yorke, ”Chaotic Attractors in Crisis”, in this volume. · Zbl 0561.58029
[26] Douady, A.; Oesterle, J.: Dimension de Hausdorff des attracteurs. Comptes rendus des seances de l’academie des sciences 24, 1135-1138 (1980) · Zbl 0443.58016
[27] Kolmogorov, A. N.: Dolk. akad. Nauk SSSR. 124, 754 (1959)
[28] Sinai, Ya.G.: Dolk. akad. Nauk SSSR. 124, 768 (1959)
[29] Crutchfield, J.; Packard, N.: Symbolic dynamics of one-dimensional maps: entropies, finite precision, and noise. Int’l. J. Theo. phys. 21, 433 (1982) · Zbl 0508.58029
[30] S. Pelikan, private communication.
[31] Billingsley, P.: Ergodic theory and information. (1965) · Zbl 0141.16702
[32] Good, I. J.: The fractional dimensional theory of continued fractions. Proc. camb. Phil. soc. 37, 199-228 (1941) · Zbl 0061.09408
[33] Eggleston, H. G.: The fractional dimension of a set defined by decimal properties. Quart. J. Math. Oxford ser. 20, 31-36 (1949) · Zbl 0031.20801
[34] Besicovitch, A.: On the sum of digits of real numbers represented in the dyadic system. Math. annalen. 110, 321 (1934) · Zbl 0009.39503
[35] Kaplan, J. L.; Mallet-Paret, J.; Yorke, J. A.: The Lyapunov dimension of a nowhere differentiable attracting torus. (1982) · Zbl 0558.58018
[36] Arnold, V. I.; Avez: Ergodic theory in classical mechanics. (1968)
[37] Russel, D.; Hansen, J.; Ott, E.: Dimensionality and Lyapunov numbers of strange attractors. Phys. rev. Lett. 45, 1175 (1980)
[38] Farmer, J. D.: Chaotic attractors of an infinite dimensional dynamical system. Physica 4D, 366-393 (1982) · Zbl 1194.37052
[39] Greenside, H.; 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)
[40] Froehling, H.; Crutchfield, J.; Farmer, J. D.; Packard, N.; Shaw, R.: On determining the dimension of chaotic flows. Physica 3D, 605 (1981) · Zbl 1194.37053
[41] R. Kautz, private communication.
[42] Chorin, A. J.: The evolution of a turbulent vortex. Comm. math. Phys. 83, 517-535 (1982) · Zbl 0494.76024
[43] Bennettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.: Also seemeccanica. Meccanica 15, 9 (1980)
[44] Shimada, I.; Nagashima, T.: Prog. theor. Phys.. 61, 228 (1979)
[45] Mandelbrot, B. B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. fluid mechanics 62, 331-358 (1974) · Zbl 0289.76031
[46] Mandelbrot, B. B.: Fractals and turbulence: attractors and dispersion. Lecture notes in mathematics 615, 83-93 (1976, 1977)
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.