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On determining the dimension of chaotic flows. (English) Zbl 1194.37053

Summary: We describe a method for determining the approximate fractal dimension of an attractor. Our technique fits linear subspaces of appropriate dimension to sets of points on the attractor. The deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques. We show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time-varying quantity is available for analysis. These methods are applied to several dissipative dynamical systems.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Lorentz, E. N., J. Atmos. Sci., 20, 130 (1963)
[2] Ruelle, D.; Takens, F., Comm. Math. Phys., 50, 69 (1976)
[3] Huberman, B. A.; Crutchfield, J. P., Phys. Rev. Lett., 43, 1743 (1979)
[4] Huberman, B. A.; Crutchfield, J. P.; Packard, N. H., Appl. Phys. Lett., 37, 750 (1980)
[5] Gollub, J.; Swinney, H., Phys. Rev. Lett., 35, 927 (1975)
[6] Walden, R. W.; Donnelly, R. J., Phys. Rev. Lett., 42, 301 (1979)
[7] Rössler, O. E., Phys. Lett., 57A, 196 (1976)
[8] Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S., Phys. Rev. Lett., 45, 712 (1980)
[9] Roux, J. C.; Rossi, A.; Bachelart, S.; Vidal, C., Phys. Lett., 77A, 391 (1980)
[10] Shimada, I.; Nagashima, T., Prog. Theor. Phys., 61, 1605 (1979)
[11] Bennetin, G.; Galgani, L.; Strelcyn, J. M., Phys. Rev., A14, 2338 (1976)
[12] Mandelbrot, B. B., Fractals: Form, Chance, and Dimension (1977), W.H. Freeman: W.H. Freeman San Francisco · Zbl 0376.28020
[13] Eq. (2) defines a set’s capacity; Eq. (2) defines a set’s capacity
[14] Mori, H., Prog. Theor. Phys., 63, 1044 (1980)
[15] Farmer, J. D.; Packard, N. H., Chaotic Attractors in Infinite-Dimensional Systems I: Differential Delay Equations, (Physica D (1980), UCSC), preprint, submitted to
[16] Kaplan, J.; Yorke, J., Chaotic Behavior of Multidimensional Difference Equations, Springer Lecture Notes in Mathematics, 730, 204 (1979)
[17] Crutchfield, J. P., Senior Thesis (1979), University of California at Santa Cruz
[18] Farmer, J. D.; Crutchfield, J. P.; Froehling, H.; Packard, N. H.; Shaw, R. S., Annals, N.Y. Acad. Sci., 357, 453 (1980)
[19] Williams, R. F., Berkeley Turbulence Seminar, 1976-1977, (Benard, P.; Ratiu, T., Springer Lecture Notes in Mathematics, 615 (1977))
[20] Shaw, R., Z. Naturforsch., 36a, 80 (1981)
[21] Abraham, R., (Jantsch, E.; Waddington, C. H., Evolution and Consciousness (1976), Addison Wesley: Addison Wesley Reading, Mass)
[22] D. Ruelle, private communication.; D. Ruelle, private communication.
[23] Poincaré, H., The Foundations of Science (1913), Science: Science Garrison, N.Y · JFM 44.0086.16
[24] Shepard, R. N., Psychometrika, 27, 219 (1962)
[25] Fukunaga, K.; Olsen, D. R., IEEE Trans. Comp., C-20, 176 (1971)
[26] Trunk, G. V., IEEE Trans. Comp., C-25, 165 (1976)
[27] White, L. J.; Ksienski, A. A., Pattern Recognition, 6, 35 (1974)
[28] Takens, F., Detecting Strange Attractors in Turbulence (1980), preprint · Zbl 0513.58032
[29] Hénon, M., Comm. Math. Phys., 50, 69 (1976)
[30] Simó, C., On the Hénon-Pomeau Attractor (1979), preprint
[31] Rössler, O. E., Phys. Lett., 71A, 155 (1979)
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