Abstract

Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. M. Sander, “Fractal growth,” Sci. Am. 256, 94 (1987).
    [CrossRef]
  2. L. M. Sander, “Fractal growth processes,” Nature (London) 322, 789 (1986).
    [CrossRef]
  3. J. Nittman, G. Daccord, H. E. Stanley, “Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon,” Nature (London) 314, 141 (1985).
    [CrossRef]
  4. L. Niemeyer, L. Pietronero, H. J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett. 52, 1033 (1984).
    [CrossRef]
  5. P.-Z. Wong, “The statistical physics of sedimentary rock,” Phys. Today 41(12), 24 (1988).
    [CrossRef]
  6. M. F. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988).
  7. D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).
    [CrossRef]
  8. T. A. Witten, L. M. Sander, “Diffusion limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400 (1981).
    [CrossRef]
  9. M. F. Barnsley, S. Demko, “Iterated function systems and the global construction of fractals,” Proc. R. Soc. London Ser. A 399, 243 (1985).
    [CrossRef]
  10. H. E. Stanley, N. Ostrosky, On Growth and Form: Fractal and Non-Fractal Patterns in Physics (Nijhoff, Boston, Mass., 1986).
  11. A. J. Hurd, “Resource letter FR-1: fractals,” Am. J. Phys. 56, 969 (1988).
    [CrossRef]
  12. L. Kadanoff, “Where is the physics of fractals,” Phys. Today 39(2), 6 (1986).
    [CrossRef]
  13. P. H. Carter, R. Cawley, R. D. Mauldin, “Mathematics of dimension measurement for graphs of functions,” in Fractal Aspects of Materials, B. B. Mandelbrot, D. E. Passoja, eds. (Materials Research Society, Pittsburgh, Pa., 1985).
  14. B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
    [CrossRef] [PubMed]
  15. H. Haken, Information and Self-Organization: A Macroscopic Approach to Complex Systems, Vol. 40 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1988).
  16. R. S. MacKay, J. D. Meiss, Hamiltonian Dynamical Systems (Hilger, Philadelphia, 1987).
  17. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 of Springer Series in Applied Mathematical Sciences (Springer-Verlag, New York, 1983).
  18. S. M. Hammel, C. K. R. T. Jones, J. V. Moloney, “Global dynamical behavior of the optical field in a ring cavity,” J. Opt. Soc. Am. B 2, 552 (1985).
    [CrossRef]
  19. M. Hénon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys. 50, 69 (1976).
    [CrossRef]
  20. K. J. Falconer, The Geometry of Fractal Sets, Vol. 85 of Cambridge Tracts in Mathematics (Cambridge U. Press, Cambridge, 1985).
    [CrossRef]
  21. J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617 (1985).
    [CrossRef]
  22. A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).
  23. J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
    [CrossRef] [PubMed]
  24. R. Stoop, P. F. Meier, “Evaluation of Lyapunov exponents and scaling functions from time series,” J. Opt. Soc. Am. B 5, 1037 (1988).
    [CrossRef]
  25. P. Grassberger, I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983).
    [CrossRef]
  26. A. Cohen, I. Procaccia, “Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems,” Phys. Rev. A 31, 1872 (1985).
    [CrossRef] [PubMed]
  27. J. P. Eckmann, I. Procaccia, “Fluctuations of dynamical scaling indices in nonlinear systems,” Phys. Rev. A 34, 659 (1986).
    [CrossRef] [PubMed]
  28. P. Szépfalusy, T. Tél, “Dynamical fractal properties of one-dimensional maps,” Phys. Rev. A 35, 477 (1987).
    [CrossRef]
  29. N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
    [CrossRef]
  30. F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand, L.-S. Young, eds., Vol. 898 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), p. 366.
    [CrossRef]
  31. R. Mañé, “On the dimension of the compact invariant sets of certain non-linear maps,” in Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand, L.-S. Young, eds., Vol. 898 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), p. 320.
  32. A. M. Fraser, H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A 33, 1134 (1986).
    [CrossRef] [PubMed]
  33. W. Liebert, K. Pawelzik, H. G. Schuster, Institut für Theoretische Physik, Universität Frankfurt, Frankfurt, Federal Republic of Germany, “Optimal embeddings of chaotic attractors from topological considerations,” preprint (1989).
  34. D. S. Broomhead, G. P. King, “Extracting qualitative dynamics from experimental data,” Physica 20D, 217 (1986).
  35. S. Sato, M. Sano, Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems,” Prog. Theor. Phys. 77, 1 (1987).
    [CrossRef]
  36. A. I. Mees, P. E. Rapp, L. S. Jennings, “Singular-value decomposition and embedding dimension,” Phys. Rev. A 36, 340 (1987).
    [CrossRef] [PubMed]
  37. A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
    [CrossRef] [PubMed]
  38. A. M. Fraser, “Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria,” Physica 34D, 391 (1989).
  39. A. M. Fraser, “Information and entropy in strange attractors,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).
  40. J. D. Farmer, J. J. Sidorowich, “Exploiting chaos to predict the future and reduce noise,” in Evolution, Learning and Cognition, Y. C. Lee, ed. (World Scientific, Singapore, 1988), p. 227.
  41. F. Hausdorff, “Dimension und äusseres Mass,” Math. Annalen 79, 157 (1919).
    [CrossRef]
  42. J. D. Farmer, E. Ott, J. A. Yorke, “The dimension of chaotic attractors,” Physica 7D, 153 (1983).
  43. H. G. E. Hentschel, I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors,” Physica 8D, 435 (1983).
  44. P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett. A 97, 227 (1983).
    [CrossRef]
  45. G. Paladin, A. Vulpiani, “Anomalous scaling laws in multifractal objects,” Phys. Rep. 156, 147 (1987).
    [CrossRef]
  46. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
    [CrossRef] [PubMed]
  47. S. K. Sakar, “Multifractal description of singular measures in dynamical systems,” Phys. Rev. A 36, 4104 (1987).
    [CrossRef]
  48. M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
    [CrossRef] [PubMed]
  49. E. G. Gwinn, R. M. Westervelt, “Scaling structure of attractors at the transition from quasiperiodicity to chaos in electronic transport in Ge,” Phys. Rev. Lett. 59, 157 (1987).
    [CrossRef] [PubMed]
  50. J. A. Glazier, G. Gunaratne, A. Libchaber, “f(α) curves: experimental results,” Phys. Rev. A 37, 523 (1988).
    [CrossRef] [PubMed]
  51. M. J. Feigenbaum, M. H. Jensen, I. Procaccia, “Time ordering and the thermodynamics of strange sets: theory and experimental tests,” Phys. Rev. Lett. 57, 1503 (1986).
    [CrossRef] [PubMed]
  52. D. Katzen, I. Procaccia, “Phase transitions in the thermodynamic formalism of multifractals,” Phys. Rev. Lett. 58, 1169 (1987).
    [CrossRef] [PubMed]
  53. M. H. Jensen, L. P. Kadanoff, I. Procaccia, “Scaling structure and thermodynamics of strange sets,” Phys. Rev. A 36, 1409 (1987).
    [CrossRef] [PubMed]
  54. T. Bohr, M. H. Jensen, “Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets,” Phys. Rev. A 36, 4904 (1987).
    [CrossRef] [PubMed]
  55. A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970).
  56. H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
    [CrossRef]
  57. P. Grassberger, R. Badii, A. Politi, “Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors,” J. Stat. Phys. 51, 135 (1988).
    [CrossRef]
  58. P. Cvitanović, G. H. Gunarante, I. Procaccia, “Topological and metric properties of Hénon-type strange attractors,” preprint (University of Chicago, Chicago, Ill., 1988).
  59. G. Mayer-Kress, “Application of dimension algorithms to experimental chaos,” in Directions in Chaos, Hao Bailin, ed. (World Scientific, Singapore, 1987), p. 122.
    [CrossRef]
  60. P. Grassberger, I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346 (1983).
    [CrossRef]
  61. P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica 9D, 189 (1983).
  62. F. Takens, “Invariants related to dimension and entropy,” in Atas do 13° (Colóqkio Brasiliero do Matemática, Rio de Janeiro, 1983).
  63. P. Grassberger, “Finite sample corrections to entropy and dimension estimates,” Phys. Lett. A 128, 369 (1988).
    [CrossRef]
  64. J. Theiler, “Spurious dimension from correlation algorithms applied to limited time series data,” Phys. Rev. A 34, 2427 (1986).
    [CrossRef] [PubMed]
  65. K. Pawelzik, H. G. Schuster, “Generalized dimensions and entropies from a measured time series,” Phys. Rev. A 35, 481 (1987).
    [CrossRef] [PubMed]
  66. H. Atmanspacher, H. Scheingraber, W. Voges, “Global scaling properties of a chaotic attractor reconstructed from experimental data,” Phys. Rev. A 37, 1314 (1988).
    [CrossRef] [PubMed]
  67. Y. Termonia, Z. Alexandrowicz, “Fractal dimension of strange attractors from radius versus,” Phys. Rev. Lett. 51, 1265 (1983).
    [CrossRef]
  68. J. Guckenheimer, G. Buzyna, “Dimension measurements for geostrophic turbulence,” Phys. Rev. Lett. 51, 1483 (1983).
    [CrossRef]
  69. R. Badii, A. Politi, “Statistical description of chaotic attractors: the dimension function,” J. Stat. Phys. 40, 725 (1985).
    [CrossRef]
  70. P. Grassberger, “Generalizations of the Hausdorff dimension of fractal measures,” Phys. Lett. A 107, 101 (1985).
    [CrossRef]
  71. G. Broggi, “Evaluation of dimensions and entropies of chaotic systems,” J. Opt. Soc. Am. B 5, 1020 (1988).
    [CrossRef]
  72. W. van de Water, P. Schram, “Generalized dimensions from near-neighbor information,” Phys. Rev. A 37, 3118 (1988).
    [CrossRef] [PubMed]
  73. R. Badii, G. Broggi, “Measurement of the dimension spectrum f(α): fixed-mass approach,” Phys. Lett. A 131, 339 (1988).
    [CrossRef]
  74. K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
    [CrossRef]
  75. R. L. Somorjai, “Methods for estimating the intrinsic dimensionality of high-dimensional point sets,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 137.
    [CrossRef]
  76. J. L. Kaplan, J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen, H. O. Walther, eds., Vol. 730 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1979), p. 204.
    [CrossRef]
  77. P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
    [CrossRef]
  78. J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366 (1982).
  79. K. Ikeda, K. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955 (1986).
    [CrossRef]
  80. R. Badii, A. Politi, “Renyi dimensions from local expansion rates,” Phys. Rev. A 35, 1288 (1987).
    [CrossRef] [PubMed]
  81. D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
    [CrossRef] [PubMed]
  82. G. Gunarante, I. Procaccia, “Organization of chaos,” Phys. Rev. Lett. 59, 1377 (1987).
    [CrossRef]
  83. D. Auerbach, B. O’Shaughnessy, I. Procaccia, “Scaling structure of strange attractors,” Phys. Rev. A 37, 2234 (1988).
    [CrossRef] [PubMed]
  84. C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimension of chaotic attractors,” Phys. Rev. A 36, 3522 (1987).
    [CrossRef] [PubMed]
  85. C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimensions of multifractal chaotic attractors,” Phys. Rev. A 37, 1711 (1988).
    [CrossRef] [PubMed]
  86. K. Fukunaga, D. R. Olsen, “An algorithm for finding intrinsic dimensionality of data,” IEEE Trans. Comput. C-20, 176 (1971).
    [CrossRef]
  87. H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).
  88. D. S. Broomhead, R. Jones, G. P. King, “Topological dimension and local coordinates from time series data,” J. Phys. A 20, L563 (1987).
    [CrossRef]
  89. A. Čenys, K. Pyragas, “Estimation of the number of degrees of freedom from chaotic time series,” Phys. Lett. A 129, 227 (1988).
    [CrossRef]
  90. A. Passamante, T. Hediger, M. Gollub, “Fractal dimension and local intrinsic dimension,” Phys. Rev. A 39, 3640 (1989).
    [CrossRef] [PubMed]
  91. W. A. Brock, W. D. Dechert, J. A. Scheinkman, “A test for independence based on the correlation dimension,” preprint SSRI 8702 (University of Wisconsin, Madison, Wisc., 1987).
  92. A. Namajūnas, J. Pozžela, A. Tamaševičius, “An electronic technique for measuring phase space dimension from chaotic time series,” Phys. Lett. A 131, 85 (1988).
    [CrossRef]
  93. A. Destexhe, J. A. Sepulchre, A. Babloyantz, “A comparative study of the experimental quantification of deterministic chaos,” Phys. Lett. A 132, 101 (1988).
    [CrossRef]
  94. J. D. Farmer, J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845 (1987).
    [CrossRef] [PubMed]
  95. D. A. Russell, J. D. Hanson, E. Ott, “Dimension of strange attractors,” Phys. Rev. Lett. 45, 1175 (1980).
    [CrossRef]
  96. P. Grassberger, “On the fractal dimension of the Hénon attractor,” Phys. Lett. A 97, 224 (1983).
    [CrossRef]
  97. M. J. McGuinness, “A computation of the limit capacity of the Lorenz attractor,” Physica 16D, 265 (1985).
  98. W. E. Caswell, J. A. Yorke, “Invisible errors in dimension calculations: geometric and systematic effects,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 123.
    [CrossRef]
  99. A. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phys. Rev. Lett. 69, 1327 (1989).
    [CrossRef]
  100. F. Takens, “On the numerical determination of the dimension of an attractor,” in Dynamical Systems and Bifurcations, Groningen, 1984, B. L. J. Braaksma, H. W. Broer, F. Takens, eds., Vol. 1125 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1985).
    [CrossRef]
  101. J. Theiler, “Quantifying chaos: practical estimation of the correlation dimension,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1988).
  102. J. Theiler, “Statistical precision of dimension estimators,” Phys. Rev. A (to be published).
    [PubMed]
  103. R. Cawley, A. L. Licht, “Maximum likelihood method for evaluating correlation dimension,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds., Vol. 278 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), p. 90.
  104. S. Ellner, “Estimating attractor dimensions for limited data: a new method, with error estimates,” Phys. Lett. A 113, 128 (1988).
    [CrossRef]
  105. J. Theiler, “Lacunarity in a best estimator of fractal dimension,” Phys. Lett. A 133, 195 (1988).
    [CrossRef]
  106. J. Theiler, “Efficient algorithm for estimating the correlation dimension from a set of discrete points,” Phys. Rev. A 36, 4456 (1987).
    [CrossRef] [PubMed]
  107. S. Bingham, M. Kot, “Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity,” Phys. Lett. A 140, 327 (1989).
    [CrossRef]
  108. F. Hunt, F. Sullivan, “Efficient algorithms for computing fractal dimensions,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 74.
    [CrossRef]
  109. M. Franaszek, “Optimized algorithm for the calculation of correlation integrals,” Phys. Rev. A 39, 5540 (1989).
    [CrossRef]
  110. C.-K. and F. C. Moon, “An optical technique for measuring fractal dimensions of planar Poincaré maps,” Phys. Lett. A 114, 222 (1986).
    [CrossRef]
  111. P. Grassberger, “Do climatic attractors exist?” Nature (London) 323, 609 (1986).
    [CrossRef]
  112. A. R. Osborne, A. Provenzale, “Finite correlation dimension for stochastic systems with power-law spectra,” Physica 35D, 357 (1989).
  113. A. Brandstater, H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207 (1987).
    [CrossRef] [PubMed]
  114. N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
    [CrossRef]
  115. J. W. Havstad, C. L. Ehlers, “Attractor dimension of non-stationary dynamical systems from small data sets,” Phys. Rev. A 39, 845 (1989).
    [CrossRef] [PubMed]
  116. J. B. Ramsey, H.-J. Yuan, “Bias and error bars in dimension calculations and their evaluation in some simple models,” Phys. Lett. A 134, 287 (1989).
    [CrossRef]
  117. R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
    [CrossRef] [PubMed]
  118. F. Mitschke, M. Moller, W. Lange, “Measuring filtered chaotic signals,” Phys. Rev. A 37, 4518 (1988).
    [CrossRef] [PubMed]
  119. E. J. Kostelich, J. A. Yorke, “Noise reduction in dynamical systems,” Phys. Rev. A 38, 1649 (1988).
    [CrossRef] [PubMed]
  120. M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
    [CrossRef]
  121. L. A. Smith, “Intrinsic limits on dimension calculations,” Phys. Lett. A 133, 283 (1988).
    [CrossRef]
  122. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).
  123. B. B. Mandlebrot, “Corrélations et texture dans un nouveau modéle d’univers hiérarchisé, basé sur les ensembles trémas,” C. R. Acad. Sci. A 288, 81 (1979).
  124. Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
    [CrossRef]
  125. Y. Gefen, A. Aharony, B. B. Mandelbrot, “Phase transitions on fractals: III. Infinitely ramified lattices,” J. Phys. A 17, 1277 (1984).
    [CrossRef]
  126. R. Badii, A. Politi, “Intrinsic oscillations in measuring the fractal dimensions,” Phys. Lett. A 104, 303 (1984).
    [CrossRef]
  127. L. A. Smith, J.-D. Fournier, E. A. Spiegel, “Lacunarity and intermittency in fluid turbulence,” Phys. Lett. A 114, 465 (1986).
    [CrossRef]
  128. A. Arneodo, G. Grasseau, E. J. Kostelich, “Fractal dimensions and f(α) spectrum of the Hénon attractor,” Phys. Lett. A 124, 426 (1987).
    [CrossRef]
  129. D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
    [CrossRef]
  130. P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).
  131. J. Gleick, Chaos: Making a New Science (Viking, New York, 1987).
  132. H.-O. Peitgen, P. H. Richter, The Beauty of Fractals (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  133. H.-O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, New York, 1988).
  134. T. S. Parker, L. O. Chua, “Chaos: a tutorial for engineers,” Proc. IEEE 75, 982 (1987).
    [CrossRef]
  135. T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
    [CrossRef]
  136. N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
    [CrossRef]
  137. R. S. Shaw, “Strange attractors, chaotic behavior, and information flow,” Z. Naturforsch. 36a, 80 (1981).
  138. J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
    [CrossRef]
  139. E. Ott, “Strange attractors and chaotic motions of dynamical systems,” Rev. Mod. Phys. 53, 655 (1981).
    [CrossRef]
  140. A. V. Holden, ed., Chaos (Princeton U. Press, Princeton, N.J., 1986).
  141. H. G. Schuster, Deterministic Chaos: An Introduction (VCH, Weinheim, Federal Republic of Germany, 1988).
  142. H. Bai-Lin, Chaos (World Scientific, Singapore, 1984).
  143. P. Cvitanović, Universality in Chaos (Hilger, Bristol, UK, 1986).
  144. P. Grassberger, “Estimating the fractal dimensions and entropies of strange attractors,” in Chaos, A. V. Holden, ed. (Princeton U. Press, Princeton, N.J.1986), Chap. 14.
  145. G. Mayer-Kress, ed., Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  146. N. B. Abraham, J. P. Gollub, H. L. Swinney, “Testing nonlinear dynamics,” Physica 11D, 252 (1984).

1989 (10)

A. M. Fraser, “Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria,” Physica 34D, 391 (1989).

A. Passamante, T. Hediger, M. Gollub, “Fractal dimension and local intrinsic dimension,” Phys. Rev. A 39, 3640 (1989).
[CrossRef] [PubMed]

A. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phys. Rev. Lett. 69, 1327 (1989).
[CrossRef]

S. Bingham, M. Kot, “Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity,” Phys. Lett. A 140, 327 (1989).
[CrossRef]

M. Franaszek, “Optimized algorithm for the calculation of correlation integrals,” Phys. Rev. A 39, 5540 (1989).
[CrossRef]

A. R. Osborne, A. Provenzale, “Finite correlation dimension for stochastic systems with power-law spectra,” Physica 35D, 357 (1989).

J. W. Havstad, C. L. Ehlers, “Attractor dimension of non-stationary dynamical systems from small data sets,” Phys. Rev. A 39, 845 (1989).
[CrossRef] [PubMed]

J. B. Ramsey, H.-J. Yuan, “Bias and error bars in dimension calculations and their evaluation in some simple models,” Phys. Lett. A 134, 287 (1989).
[CrossRef]

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

1988 (22)

A. J. Hurd, “Resource letter FR-1: fractals,” Am. J. Phys. 56, 969 (1988).
[CrossRef]

P.-Z. Wong, “The statistical physics of sedimentary rock,” Phys. Today 41(12), 24 (1988).
[CrossRef]

L. A. Smith, “Intrinsic limits on dimension calculations,” Phys. Lett. A 133, 283 (1988).
[CrossRef]

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

F. Mitschke, M. Moller, W. Lange, “Measuring filtered chaotic signals,” Phys. Rev. A 37, 4518 (1988).
[CrossRef] [PubMed]

E. J. Kostelich, J. A. Yorke, “Noise reduction in dynamical systems,” Phys. Rev. A 38, 1649 (1988).
[CrossRef] [PubMed]

G. Broggi, “Evaluation of dimensions and entropies of chaotic systems,” J. Opt. Soc. Am. B 5, 1020 (1988).
[CrossRef]

R. Stoop, P. F. Meier, “Evaluation of Lyapunov exponents and scaling functions from time series,” J. Opt. Soc. Am. B 5, 1037 (1988).
[CrossRef]

S. Ellner, “Estimating attractor dimensions for limited data: a new method, with error estimates,” Phys. Lett. A 113, 128 (1988).
[CrossRef]

J. Theiler, “Lacunarity in a best estimator of fractal dimension,” Phys. Lett. A 133, 195 (1988).
[CrossRef]

A. Namajūnas, J. Pozžela, A. Tamaševičius, “An electronic technique for measuring phase space dimension from chaotic time series,” Phys. Lett. A 131, 85 (1988).
[CrossRef]

A. Destexhe, J. A. Sepulchre, A. Babloyantz, “A comparative study of the experimental quantification of deterministic chaos,” Phys. Lett. A 132, 101 (1988).
[CrossRef]

W. van de Water, P. Schram, “Generalized dimensions from near-neighbor information,” Phys. Rev. A 37, 3118 (1988).
[CrossRef] [PubMed]

R. Badii, G. Broggi, “Measurement of the dimension spectrum f(α): fixed-mass approach,” Phys. Lett. A 131, 339 (1988).
[CrossRef]

D. Auerbach, B. O’Shaughnessy, I. Procaccia, “Scaling structure of strange attractors,” Phys. Rev. A 37, 2234 (1988).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimensions of multifractal chaotic attractors,” Phys. Rev. A 37, 1711 (1988).
[CrossRef] [PubMed]

A. Čenys, K. Pyragas, “Estimation of the number of degrees of freedom from chaotic time series,” Phys. Lett. A 129, 227 (1988).
[CrossRef]

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

J. A. Glazier, G. Gunaratne, A. Libchaber, “f(α) curves: experimental results,” Phys. Rev. A 37, 523 (1988).
[CrossRef] [PubMed]

P. Grassberger, R. Badii, A. Politi, “Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors,” J. Stat. Phys. 51, 135 (1988).
[CrossRef]

P. Grassberger, “Finite sample corrections to entropy and dimension estimates,” Phys. Lett. A 128, 369 (1988).
[CrossRef]

H. Atmanspacher, H. Scheingraber, W. Voges, “Global scaling properties of a chaotic attractor reconstructed from experimental data,” Phys. Rev. A 37, 1314 (1988).
[CrossRef] [PubMed]

1987 (22)

E. G. Gwinn, R. M. Westervelt, “Scaling structure of attractors at the transition from quasiperiodicity to chaos in electronic transport in Ge,” Phys. Rev. Lett. 59, 157 (1987).
[CrossRef] [PubMed]

K. Pawelzik, H. G. Schuster, “Generalized dimensions and entropies from a measured time series,” Phys. Rev. A 35, 481 (1987).
[CrossRef] [PubMed]

S. K. Sakar, “Multifractal description of singular measures in dynamical systems,” Phys. Rev. A 36, 4104 (1987).
[CrossRef]

D. Katzen, I. Procaccia, “Phase transitions in the thermodynamic formalism of multifractals,” Phys. Rev. Lett. 58, 1169 (1987).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, I. Procaccia, “Scaling structure and thermodynamics of strange sets,” Phys. Rev. A 36, 1409 (1987).
[CrossRef] [PubMed]

T. Bohr, M. H. Jensen, “Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets,” Phys. Rev. A 36, 4904 (1987).
[CrossRef] [PubMed]

G. Paladin, A. Vulpiani, “Anomalous scaling laws in multifractal objects,” Phys. Rep. 156, 147 (1987).
[CrossRef]

P. Szépfalusy, T. Tél, “Dynamical fractal properties of one-dimensional maps,” Phys. Rev. A 35, 477 (1987).
[CrossRef]

S. Sato, M. Sano, Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems,” Prog. Theor. Phys. 77, 1 (1987).
[CrossRef]

A. I. Mees, P. E. Rapp, L. S. Jennings, “Singular-value decomposition and embedding dimension,” Phys. Rev. A 36, 340 (1987).
[CrossRef] [PubMed]

D. S. Broomhead, R. Jones, G. P. King, “Topological dimension and local coordinates from time series data,” J. Phys. A 20, L563 (1987).
[CrossRef]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimension of chaotic attractors,” Phys. Rev. A 36, 3522 (1987).
[CrossRef] [PubMed]

R. Badii, A. Politi, “Renyi dimensions from local expansion rates,” Phys. Rev. A 35, 1288 (1987).
[CrossRef] [PubMed]

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

G. Gunarante, I. Procaccia, “Organization of chaos,” Phys. Rev. Lett. 59, 1377 (1987).
[CrossRef]

J. D. Farmer, J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845 (1987).
[CrossRef] [PubMed]

J. Theiler, “Efficient algorithm for estimating the correlation dimension from a set of discrete points,” Phys. Rev. A 36, 4456 (1987).
[CrossRef] [PubMed]

A. Brandstater, H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207 (1987).
[CrossRef] [PubMed]

A. Arneodo, G. Grasseau, E. J. Kostelich, “Fractal dimensions and f(α) spectrum of the Hénon attractor,” Phys. Lett. A 124, 426 (1987).
[CrossRef]

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

T. S. Parker, L. O. Chua, “Chaos: a tutorial for engineers,” Proc. IEEE 75, 982 (1987).
[CrossRef]

L. M. Sander, “Fractal growth,” Sci. Am. 256, 94 (1987).
[CrossRef]

1986 (15)

L. M. Sander, “Fractal growth processes,” Nature (London) 322, 789 (1986).
[CrossRef]

L. Kadanoff, “Where is the physics of fractals,” Phys. Today 39(2), 6 (1986).
[CrossRef]

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
[CrossRef] [PubMed]

J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
[CrossRef]

L. A. Smith, J.-D. Fournier, E. A. Spiegel, “Lacunarity and intermittency in fluid turbulence,” Phys. Lett. A 114, 465 (1986).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

C.-K. and F. C. Moon, “An optical technique for measuring fractal dimensions of planar Poincaré maps,” Phys. Lett. A 114, 222 (1986).
[CrossRef]

P. Grassberger, “Do climatic attractors exist?” Nature (London) 323, 609 (1986).
[CrossRef]

K. Ikeda, K. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955 (1986).
[CrossRef]

A. M. Fraser, H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A 33, 1134 (1986).
[CrossRef] [PubMed]

D. S. Broomhead, G. P. King, “Extracting qualitative dynamics from experimental data,” Physica 20D, 217 (1986).

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

J. P. Eckmann, I. Procaccia, “Fluctuations of dynamical scaling indices in nonlinear systems,” Phys. Rev. A 34, 659 (1986).
[CrossRef] [PubMed]

M. J. Feigenbaum, M. H. Jensen, I. Procaccia, “Time ordering and the thermodynamics of strange sets: theory and experimental tests,” Phys. Rev. Lett. 57, 1503 (1986).
[CrossRef] [PubMed]

J. Theiler, “Spurious dimension from correlation algorithms applied to limited time series data,” Phys. Rev. A 34, 2427 (1986).
[CrossRef] [PubMed]

1985 (10)

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

A. Cohen, I. Procaccia, “Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems,” Phys. Rev. A 31, 1872 (1985).
[CrossRef] [PubMed]

R. Badii, A. Politi, “Statistical description of chaotic attractors: the dimension function,” J. Stat. Phys. 40, 725 (1985).
[CrossRef]

P. Grassberger, “Generalizations of the Hausdorff dimension of fractal measures,” Phys. Lett. A 107, 101 (1985).
[CrossRef]

M. J. McGuinness, “A computation of the limit capacity of the Lorenz attractor,” Physica 16D, 265 (1985).

S. M. Hammel, C. K. R. T. Jones, J. V. Moloney, “Global dynamical behavior of the optical field in a ring cavity,” J. Opt. Soc. Am. B 2, 552 (1985).
[CrossRef]

J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).

J. Nittman, G. Daccord, H. E. Stanley, “Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon,” Nature (London) 314, 141 (1985).
[CrossRef]

M. F. Barnsley, S. Demko, “Iterated function systems and the global construction of fractals,” Proc. R. Soc. London Ser. A 399, 243 (1985).
[CrossRef]

1984 (5)

N. B. Abraham, J. P. Gollub, H. L. Swinney, “Testing nonlinear dynamics,” Physica 11D, 252 (1984).

L. Niemeyer, L. Pietronero, H. J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett. 52, 1033 (1984).
[CrossRef]

P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).

Y. Gefen, A. Aharony, B. B. Mandelbrot, “Phase transitions on fractals: III. Infinitely ramified lattices,” J. Phys. A 17, 1277 (1984).
[CrossRef]

R. Badii, A. Politi, “Intrinsic oscillations in measuring the fractal dimensions,” Phys. Lett. A 104, 303 (1984).
[CrossRef]

1983 (11)

Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
[CrossRef]

P. Grassberger, “On the fractal dimension of the Hénon attractor,” Phys. Lett. A 97, 224 (1983).
[CrossRef]

P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983).
[CrossRef]

J. D. Farmer, E. Ott, J. A. Yorke, “The dimension of chaotic attractors,” Physica 7D, 153 (1983).

H. G. E. Hentschel, I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors,” Physica 8D, 435 (1983).

P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett. A 97, 227 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica 9D, 189 (1983).

Y. Termonia, Z. Alexandrowicz, “Fractal dimension of strange attractors from radius versus,” Phys. Rev. Lett. 51, 1265 (1983).
[CrossRef]

J. Guckenheimer, G. Buzyna, “Dimension measurements for geostrophic turbulence,” Phys. Rev. Lett. 51, 1483 (1983).
[CrossRef]

1982 (2)

H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
[CrossRef]

J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366 (1982).

1981 (4)

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

E. Ott, “Strange attractors and chaotic motions of dynamical systems,” Rev. Mod. Phys. 53, 655 (1981).
[CrossRef]

R. S. Shaw, “Strange attractors, chaotic behavior, and information flow,” Z. Naturforsch. 36a, 80 (1981).

T. A. Witten, L. M. Sander, “Diffusion limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400 (1981).
[CrossRef]

1980 (2)

D. A. Russell, J. D. Hanson, E. Ott, “Dimension of strange attractors,” Phys. Rev. Lett. 45, 1175 (1980).
[CrossRef]

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

1979 (2)

K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
[CrossRef]

B. B. Mandlebrot, “Corrélations et texture dans un nouveau modéle d’univers hiérarchisé, basé sur les ensembles trémas,” C. R. Acad. Sci. A 288, 81 (1979).

1976 (1)

M. Hénon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys. 50, 69 (1976).
[CrossRef]

1971 (1)

K. Fukunaga, D. R. Olsen, “An algorithm for finding intrinsic dimensionality of data,” IEEE Trans. Comput. C-20, 176 (1971).
[CrossRef]

1919 (1)

F. Hausdorff, “Dimension und äusseres Mass,” Math. Annalen 79, 157 (1919).
[CrossRef]

Abraham, N. B.

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

N. B. Abraham, J. P. Gollub, H. L. Swinney, “Testing nonlinear dynamics,” Physica 11D, 252 (1984).

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Aharony, A.

Y. Gefen, A. Aharony, B. B. Mandelbrot, “Phase transitions on fractals: III. Infinitely ramified lattices,” J. Phys. A 17, 1277 (1984).
[CrossRef]

Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
[CrossRef]

Albano, A. M.

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Alexandrowicz, Z.

Y. Termonia, Z. Alexandrowicz, “Fractal dimension of strange attractors from radius versus,” Phys. Rev. Lett. 51, 1265 (1983).
[CrossRef]

Arneodo, A.

A. Arneodo, G. Grasseau, E. J. Kostelich, “Fractal dimensions and f(α) spectrum of the Hénon attractor,” Phys. Lett. A 124, 426 (1987).
[CrossRef]

Atmanspacher, H.

H. Atmanspacher, H. Scheingraber, W. Voges, “Global scaling properties of a chaotic attractor reconstructed from experimental data,” Phys. Rev. A 37, 1314 (1988).
[CrossRef] [PubMed]

Atten, P.

P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).

Auerbach, D.

D. Auerbach, B. O’Shaughnessy, I. Procaccia, “Scaling structure of strange attractors,” Phys. Rev. A 37, 2234 (1988).
[CrossRef] [PubMed]

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

Babloyantz, A.

A. Destexhe, J. A. Sepulchre, A. Babloyantz, “A comparative study of the experimental quantification of deterministic chaos,” Phys. Lett. A 132, 101 (1988).
[CrossRef]

Badii, R.

R. Badii, G. Broggi, “Measurement of the dimension spectrum f(α): fixed-mass approach,” Phys. Lett. A 131, 339 (1988).
[CrossRef]

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

P. Grassberger, R. Badii, A. Politi, “Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors,” J. Stat. Phys. 51, 135 (1988).
[CrossRef]

R. Badii, A. Politi, “Renyi dimensions from local expansion rates,” Phys. Rev. A 35, 1288 (1987).
[CrossRef] [PubMed]

R. Badii, A. Politi, “Statistical description of chaotic attractors: the dimension function,” J. Stat. Phys. 40, 725 (1985).
[CrossRef]

R. Badii, A. Politi, “Intrinsic oscillations in measuring the fractal dimensions,” Phys. Lett. A 104, 303 (1984).
[CrossRef]

Bailey, T. A.

K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
[CrossRef]

Bai-Lin, H.

H. Bai-Lin, Chaos (World Scientific, Singapore, 1984).

Barnsley, M. F.

M. F. Barnsley, S. Demko, “Iterated function systems and the global construction of fractals,” Proc. R. Soc. London Ser. A 399, 243 (1985).
[CrossRef]

M. F. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988).

Bessis, D.

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

Bingham, S.

S. Bingham, M. Kot, “Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity,” Phys. Lett. A 140, 327 (1989).
[CrossRef]

Bohr, T.

T. Bohr, M. H. Jensen, “Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets,” Phys. Rev. A 36, 4904 (1987).
[CrossRef] [PubMed]

Brandstater, A.

A. Brandstater, H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207 (1987).
[CrossRef] [PubMed]

Brock, W. A.

W. A. Brock, W. D. Dechert, J. A. Scheinkman, “A test for independence based on the correlation dimension,” preprint SSRI 8702 (University of Wisconsin, Madison, Wisc., 1987).

Broggi, G.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

G. Broggi, “Evaluation of dimensions and entropies of chaotic systems,” J. Opt. Soc. Am. B 5, 1020 (1988).
[CrossRef]

R. Badii, G. Broggi, “Measurement of the dimension spectrum f(α): fixed-mass approach,” Phys. Lett. A 131, 339 (1988).
[CrossRef]

Broomhead, D. S.

D. S. Broomhead, R. Jones, G. P. King, “Topological dimension and local coordinates from time series data,” J. Phys. A 20, L563 (1987).
[CrossRef]

D. S. Broomhead, G. P. King, “Extracting qualitative dynamics from experimental data,” Physica 20D, 217 (1986).

Buzyna, G.

J. Guckenheimer, G. Buzyna, “Dimension measurements for geostrophic turbulence,” Phys. Rev. Lett. 51, 1483 (1983).
[CrossRef]

Caputo, J. G.

P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).

Carter, P. H.

P. H. Carter, R. Cawley, R. D. Mauldin, “Mathematics of dimension measurement for graphs of functions,” in Fractal Aspects of Materials, B. B. Mandelbrot, D. E. Passoja, eds. (Materials Research Society, Pittsburgh, Pa., 1985).

Caswell, W. E.

W. E. Caswell, J. A. Yorke, “Invisible errors in dimension calculations: geometric and systematic effects,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 123.
[CrossRef]

Cawley, R.

P. H. Carter, R. Cawley, R. D. Mauldin, “Mathematics of dimension measurement for graphs of functions,” in Fractal Aspects of Materials, B. B. Mandelbrot, D. E. Passoja, eds. (Materials Research Society, Pittsburgh, Pa., 1985).

R. Cawley, A. L. Licht, “Maximum likelihood method for evaluating correlation dimension,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds., Vol. 278 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), p. 90.

Cenys, A.

A. Čenys, K. Pyragas, “Estimation of the number of degrees of freedom from chaotic time series,” Phys. Lett. A 129, 227 (1988).
[CrossRef]

Chhabra, A.

A. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phys. Rev. Lett. 69, 1327 (1989).
[CrossRef]

Chua, L. O.

T. S. Parker, L. O. Chua, “Chaos: a tutorial for engineers,” Proc. IEEE 75, 982 (1987).
[CrossRef]

T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
[CrossRef]

Ciliberto, S.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
[CrossRef] [PubMed]

Cohen, A.

A. Cohen, I. Procaccia, “Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems,” Phys. Rev. A 31, 1872 (1985).
[CrossRef] [PubMed]

Crutchfield, J. P.

J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
[CrossRef]

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Cvitanovic, P.

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

P. Cvitanović, Universality in Chaos (Hilger, Bristol, UK, 1986).

P. Cvitanović, G. H. Gunarante, I. Procaccia, “Topological and metric properties of Hénon-type strange attractors,” preprint (University of Chicago, Chicago, Ill., 1988).

Daccord, G.

J. Nittman, G. Daccord, H. E. Stanley, “Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon,” Nature (London) 314, 141 (1985).
[CrossRef]

Das, B.

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Dechert, W. D.

W. A. Brock, W. D. Dechert, J. A. Scheinkman, “A test for independence based on the correlation dimension,” preprint SSRI 8702 (University of Wisconsin, Madison, Wisc., 1987).

DeGuzman, G.

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

Demko, S.

M. F. Barnsley, S. Demko, “Iterated function systems and the global construction of fractals,” Proc. R. Soc. London Ser. A 399, 243 (1985).
[CrossRef]

Derighetti, B.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Destexhe, A.

A. Destexhe, J. A. Sepulchre, A. Babloyantz, “A comparative study of the experimental quantification of deterministic chaos,” Phys. Lett. A 132, 101 (1988).
[CrossRef]

Dubes, R. C.

K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
[CrossRef]

Dubuc, B.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

Eckmann, J. P.

J. P. Eckmann, I. Procaccia, “Fluctuations of dynamical scaling indices in nonlinear systems,” Phys. Rev. A 34, 659 (1986).
[CrossRef] [PubMed]

Eckmann, J.-P.

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
[CrossRef] [PubMed]

J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

Ehlers, C. L.

J. W. Havstad, C. L. Ehlers, “Attractor dimension of non-stationary dynamical systems from small data sets,” Phys. Rev. A 39, 845 (1989).
[CrossRef] [PubMed]

Ellner, S.

S. Ellner, “Estimating attractor dimensions for limited data: a new method, with error estimates,” Phys. Lett. A 113, 128 (1988).
[CrossRef]

Falconer, K. J.

K. J. Falconer, The Geometry of Fractal Sets, Vol. 85 of Cambridge Tracts in Mathematics (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

Farmer, D.

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

Farmer, J. D.

J. D. Farmer, J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845 (1987).
[CrossRef] [PubMed]

J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
[CrossRef]

J. D. Farmer, E. Ott, J. A. Yorke, “The dimension of chaotic attractors,” Physica 7D, 153 (1983).

J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366 (1982).

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

J. D. Farmer, J. J. Sidorowich, “Exploiting chaos to predict the future and reduce noise,” in Evolution, Learning and Cognition, Y. C. Lee, ed. (World Scientific, Singapore, 1988), p. 227.

Feigenbaum, M. J.

M. J. Feigenbaum, M. H. Jensen, I. Procaccia, “Time ordering and the thermodynamics of strange sets: theory and experimental tests,” Phys. Rev. Lett. 57, 1503 (1986).
[CrossRef] [PubMed]

Fournier, J.-D.

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

L. A. Smith, J.-D. Fournier, E. A. Spiegel, “Lacunarity and intermittency in fluid turbulence,” Phys. Lett. A 114, 465 (1986).
[CrossRef]

Franaszek, M.

M. Franaszek, “Optimized algorithm for the calculation of correlation integrals,” Phys. Rev. A 39, 5540 (1989).
[CrossRef]

Fraser, A. M.

A. M. Fraser, “Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria,” Physica 34D, 391 (1989).

A. M. Fraser, H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A 33, 1134 (1986).
[CrossRef] [PubMed]

A. M. Fraser, “Information and entropy in strange attractors,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

Fredrickson, P.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
[CrossRef]

Froehling, H.

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

Fukunaga, K.

K. Fukunaga, D. R. Olsen, “An algorithm for finding intrinsic dimensionality of data,” IEEE Trans. Comput. C-20, 176 (1971).
[CrossRef]

Gagne, Y.

P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).

Gefen, Y.

Y. Gefen, A. Aharony, B. B. Mandelbrot, “Phase transitions on fractals: III. Infinitely ramified lattices,” J. Phys. A 17, 1277 (1984).
[CrossRef]

Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
[CrossRef]

Gioggia, R. S.

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Glazier, J. A.

J. A. Glazier, G. Gunaratne, A. Libchaber, “f(α) curves: experimental results,” Phys. Rev. A 37, 523 (1988).
[CrossRef] [PubMed]

Gleick, J.

J. Gleick, Chaos: Making a New Science (Viking, New York, 1987).

Gollub, J. P.

N. B. Abraham, J. P. Gollub, H. L. Swinney, “Testing nonlinear dynamics,” Physica 11D, 252 (1984).

Gollub, M.

A. Passamante, T. Hediger, M. Gollub, “Fractal dimension and local intrinsic dimension,” Phys. Rev. A 39, 3640 (1989).
[CrossRef] [PubMed]

Grassberger, P.

P. Grassberger, “Finite sample corrections to entropy and dimension estimates,” Phys. Lett. A 128, 369 (1988).
[CrossRef]

P. Grassberger, R. Badii, A. Politi, “Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors,” J. Stat. Phys. 51, 135 (1988).
[CrossRef]

P. Grassberger, “Do climatic attractors exist?” Nature (London) 323, 609 (1986).
[CrossRef]

P. Grassberger, “Generalizations of the Hausdorff dimension of fractal measures,” Phys. Lett. A 107, 101 (1985).
[CrossRef]

P. Grassberger, I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983).
[CrossRef]

P. Grassberger, “On the fractal dimension of the Hénon attractor,” Phys. Lett. A 97, 224 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett. A 97, 227 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica 9D, 189 (1983).

P. Grassberger, “Estimating the fractal dimensions and entropies of strange attractors,” in Chaos, A. V. Holden, ed. (Princeton U. Press, Princeton, N.J.1986), Chap. 14.

Grasseau, G.

A. Arneodo, G. Grasseau, E. J. Kostelich, “Fractal dimensions and f(α) spectrum of the Hénon attractor,” Phys. Lett. A 124, 426 (1987).
[CrossRef]

Grebogi, C.

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimensions of multifractal chaotic attractors,” Phys. Rev. A 37, 1711 (1988).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimension of chaotic attractors,” Phys. Rev. A 36, 3522 (1987).
[CrossRef] [PubMed]

Greenside, H. S.

H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
[CrossRef]

Guckenheimer, J.

J. Guckenheimer, G. Buzyna, “Dimension measurements for geostrophic turbulence,” Phys. Rev. Lett. 51, 1483 (1983).
[CrossRef]

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 of Springer Series in Applied Mathematical Sciences (Springer-Verlag, New York, 1983).

Gunarante, G.

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

G. Gunarante, I. Procaccia, “Organization of chaos,” Phys. Rev. Lett. 59, 1377 (1987).
[CrossRef]

Gunarante, G. H.

P. Cvitanović, G. H. Gunarante, I. Procaccia, “Topological and metric properties of Hénon-type strange attractors,” preprint (University of Chicago, Chicago, Ill., 1988).

Gunaratne, G.

J. A. Glazier, G. Gunaratne, A. Libchaber, “f(α) curves: experimental results,” Phys. Rev. A 37, 523 (1988).
[CrossRef] [PubMed]

Gwinn, E. G.

E. G. Gwinn, R. M. Westervelt, “Scaling structure of attractors at the transition from quasiperiodicity to chaos in electronic transport in Ge,” Phys. Rev. Lett. 59, 157 (1987).
[CrossRef] [PubMed]

Haken, H.

H. Haken, Information and Self-Organization: A Macroscopic Approach to Complex Systems, Vol. 40 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1988).

Halsey, T. C.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

Hammel, S. M.

Hanson, J. D.

D. A. Russell, J. D. Hanson, E. Ott, “Dimension of strange attractors,” Phys. Rev. Lett. 45, 1175 (1980).
[CrossRef]

Hausdorff, F.

F. Hausdorff, “Dimension und äusseres Mass,” Math. Annalen 79, 157 (1919).
[CrossRef]

Havstad, J. W.

J. W. Havstad, C. L. Ehlers, “Attractor dimension of non-stationary dynamical systems from small data sets,” Phys. Rev. A 39, 845 (1989).
[CrossRef] [PubMed]

Hediger, T.

A. Passamante, T. Hediger, M. Gollub, “Fractal dimension and local intrinsic dimension,” Phys. Rev. A 39, 3640 (1989).
[CrossRef] [PubMed]

Hénon, M.

M. Hénon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys. 50, 69 (1976).
[CrossRef]

Hentschel, H. G. E.

H. G. E. Hentschel, I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors,” Physica 8D, 435 (1983).

Holmes, P.

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 of Springer Series in Applied Mathematical Sciences (Springer-Verlag, New York, 1983).

Hübner, U.

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

Hunt, F.

F. Hunt, F. Sullivan, “Efficient algorithms for computing fractal dimensions,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 74.
[CrossRef]

Hurd, A. J.

A. J. Hurd, “Resource letter FR-1: fractals,” Am. J. Phys. 56, 969 (1988).
[CrossRef]

Ikeda, K.

K. Ikeda, K. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955 (1986).
[CrossRef]

Jain, A. K.

K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
[CrossRef]

Jennings, L. S.

A. I. Mees, P. E. Rapp, L. S. Jennings, “Singular-value decomposition and embedding dimension,” Phys. Rev. A 36, 340 (1987).
[CrossRef] [PubMed]

Jensen, M. H.

M. H. Jensen, L. P. Kadanoff, I. Procaccia, “Scaling structure and thermodynamics of strange sets,” Phys. Rev. A 36, 1409 (1987).
[CrossRef] [PubMed]

T. Bohr, M. H. Jensen, “Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets,” Phys. Rev. A 36, 4904 (1987).
[CrossRef] [PubMed]

M. J. Feigenbaum, M. H. Jensen, I. Procaccia, “Time ordering and the thermodynamics of strange sets: theory and experimental tests,” Phys. Rev. Lett. 57, 1503 (1986).
[CrossRef] [PubMed]

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

Jensen, R. V.

A. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phys. Rev. Lett. 69, 1327 (1989).
[CrossRef]

Jones, C. K. R. T.

Jones, R.

D. S. Broomhead, R. Jones, G. P. King, “Topological dimension and local coordinates from time series data,” J. Phys. A 20, L563 (1987).
[CrossRef]

Kadanoff, L.

L. Kadanoff, “Where is the physics of fractals,” Phys. Today 39(2), 6 (1986).
[CrossRef]

Kadanoff, L. P.

M. H. Jensen, L. P. Kadanoff, I. Procaccia, “Scaling structure and thermodynamics of strange sets,” Phys. Rev. A 36, 1409 (1987).
[CrossRef] [PubMed]

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

Kamphorst, S. O.

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
[CrossRef] [PubMed]

Kaplan, J. L.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
[CrossRef]

J. L. Kaplan, J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen, H. O. Walther, eds., Vol. 730 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1979), p. 204.
[CrossRef]

Katzen, D.

D. Katzen, I. Procaccia, “Phase transitions in the thermodynamic formalism of multifractals,” Phys. Rev. Lett. 58, 1169 (1987).
[CrossRef] [PubMed]

King, G. P.

D. S. Broomhead, R. Jones, G. P. King, “Topological dimension and local coordinates from time series data,” J. Phys. A 20, L563 (1987).
[CrossRef]

D. S. Broomhead, G. P. King, “Extracting qualitative dynamics from experimental data,” Physica 20D, 217 (1986).

Kostelich, E. J.

E. J. Kostelich, J. A. Yorke, “Noise reduction in dynamical systems,” Phys. Rev. A 38, 1649 (1988).
[CrossRef] [PubMed]

A. Arneodo, G. Grasseau, E. J. Kostelich, “Fractal dimensions and f(α) spectrum of the Hénon attractor,” Phys. Lett. A 124, 426 (1987).
[CrossRef]

Kot, M.

S. Bingham, M. Kot, “Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity,” Phys. Lett. A 140, 327 (1989).
[CrossRef]

Lange, W.

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

F. Mitschke, M. Moller, W. Lange, “Measuring filtered chaotic signals,” Phys. Rev. A 37, 4518 (1988).
[CrossRef] [PubMed]

Libchaber, A.

J. A. Glazier, G. Gunaratne, A. Libchaber, “f(α) curves: experimental results,” Phys. Rev. A 37, 523 (1988).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

Licht, A. L.

R. Cawley, A. L. Licht, “Maximum likelihood method for evaluating correlation dimension,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds., Vol. 278 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), p. 90.

Liebert, W.

W. Liebert, K. Pawelzik, H. G. Schuster, Institut für Theoretische Physik, Universität Frankfurt, Frankfurt, Federal Republic of Germany, “Optimal embeddings of chaotic attractors from topological considerations,” preprint (1989).

MacKay, R. S.

R. S. MacKay, J. D. Meiss, Hamiltonian Dynamical Systems (Hilger, Philadelphia, 1987).

Malraison, B.

P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).

Mandelbrot, B. B.

Y. Gefen, A. Aharony, B. B. Mandelbrot, “Phase transitions on fractals: III. Infinitely ramified lattices,” J. Phys. A 17, 1277 (1984).
[CrossRef]

Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
[CrossRef]

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

Mandlebrot, B. B.

B. B. Mandlebrot, “Corrélations et texture dans un nouveau modéle d’univers hiérarchisé, basé sur les ensembles trémas,” C. R. Acad. Sci. A 288, 81 (1979).

Mañé, R.

R. Mañé, “On the dimension of the compact invariant sets of certain non-linear maps,” in Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand, L.-S. Young, eds., Vol. 898 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), p. 320.

Matsumoto, K.

K. Ikeda, K. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955 (1986).
[CrossRef]

Mauldin, R. D.

P. H. Carter, R. Cawley, R. D. Mauldin, “Mathematics of dimension measurement for graphs of functions,” in Fractal Aspects of Materials, B. B. Mandelbrot, D. E. Passoja, eds. (Materials Research Society, Pittsburgh, Pa., 1985).

Mayer-Kress, G.

G. Mayer-Kress, “Application of dimension algorithms to experimental chaos,” in Directions in Chaos, Hao Bailin, ed. (World Scientific, Singapore, 1987), p. 122.
[CrossRef]

McGuinness, M. J.

M. J. McGuinness, “A computation of the limit capacity of the Lorenz attractor,” Physica 16D, 265 (1985).

Mees, A. I.

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

A. I. Mees, P. E. Rapp, L. S. Jennings, “Singular-value decomposition and embedding dimension,” Phys. Rev. A 36, 340 (1987).
[CrossRef] [PubMed]

Meier, P. F.

Meir, Y.

Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
[CrossRef]

Meiss, J. D.

R. S. MacKay, J. D. Meiss, Hamiltonian Dynamical Systems (Hilger, Philadelphia, 1987).

Mello, T.

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Mitschke, F.

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

F. Mitschke, M. Moller, W. Lange, “Measuring filtered chaotic signals,” Phys. Rev. A 37, 4518 (1988).
[CrossRef] [PubMed]

Moller, M.

F. Mitschke, M. Moller, W. Lange, “Measuring filtered chaotic signals,” Phys. Rev. A 37, 4518 (1988).
[CrossRef] [PubMed]

Möller, M.

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

Moloney, J. V.

Moon, C.-K. and F. C.

C.-K. and F. C. Moon, “An optical technique for measuring fractal dimensions of planar Poincaré maps,” Phys. Lett. A 114, 222 (1986).
[CrossRef]

Muench, J.

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

Namajunas, A.

A. Namajūnas, J. Pozžela, A. Tamaševičius, “An electronic technique for measuring phase space dimension from chaotic time series,” Phys. Lett. A 131, 85 (1988).
[CrossRef]

Niemeyer, L.

L. Niemeyer, L. Pietronero, H. J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett. 52, 1033 (1984).
[CrossRef]

Nittman, J.

J. Nittman, G. Daccord, H. E. Stanley, “Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon,” Nature (London) 314, 141 (1985).
[CrossRef]

O’Shaughnessy, B.

D. Auerbach, B. O’Shaughnessy, I. Procaccia, “Scaling structure of strange attractors,” Phys. Rev. A 37, 2234 (1988).
[CrossRef] [PubMed]

Olsen, D. R.

K. Fukunaga, D. R. Olsen, “An algorithm for finding intrinsic dimensionality of data,” IEEE Trans. Comput. C-20, 176 (1971).
[CrossRef]

Osborne, A. R.

A. R. Osborne, A. Provenzale, “Finite correlation dimension for stochastic systems with power-law spectra,” Physica 35D, 357 (1989).

Ostrosky, N.

H. E. Stanley, N. Ostrosky, On Growth and Form: Fractal and Non-Fractal Patterns in Physics (Nijhoff, Boston, Mass., 1986).

Ott, E.

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimensions of multifractal chaotic attractors,” Phys. Rev. A 37, 1711 (1988).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimension of chaotic attractors,” Phys. Rev. A 36, 3522 (1987).
[CrossRef] [PubMed]

J. D. Farmer, E. Ott, J. A. Yorke, “The dimension of chaotic attractors,” Physica 7D, 153 (1983).

E. Ott, “Strange attractors and chaotic motions of dynamical systems,” Rev. Mod. Phys. 53, 655 (1981).
[CrossRef]

D. A. Russell, J. D. Hanson, E. Ott, “Dimension of strange attractors,” Phys. Rev. Lett. 45, 1175 (1980).
[CrossRef]

Packard, N. H.

J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
[CrossRef]

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Paladin, G.

G. Paladin, A. Vulpiani, “Anomalous scaling laws in multifractal objects,” Phys. Rep. 156, 147 (1987).
[CrossRef]

Parker, T. S.

T. S. Parker, L. O. Chua, “Chaos: a tutorial for engineers,” Proc. IEEE 75, 982 (1987).
[CrossRef]

T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
[CrossRef]

Passamante, A.

A. Passamante, T. Hediger, M. Gollub, “Fractal dimension and local intrinsic dimension,” Phys. Rev. A 39, 3640 (1989).
[CrossRef] [PubMed]

Pawelzik, K.

K. Pawelzik, H. G. Schuster, “Generalized dimensions and entropies from a measured time series,” Phys. Rev. A 35, 481 (1987).
[CrossRef] [PubMed]

W. Liebert, K. Pawelzik, H. G. Schuster, Institut für Theoretische Physik, Universität Frankfurt, Frankfurt, Federal Republic of Germany, “Optimal embeddings of chaotic attractors from topological considerations,” preprint (1989).

Peitgen, H.-O.

H.-O. Peitgen, P. H. Richter, The Beauty of Fractals (Springer-Verlag, Berlin, 1986).
[CrossRef]

Pettis, K. W.

K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
[CrossRef]

Pietronero, L.

L. Niemeyer, L. Pietronero, H. J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett. 52, 1033 (1984).
[CrossRef]

Pignataro, T.

H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
[CrossRef]

Politi, A.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

P. Grassberger, R. Badii, A. Politi, “Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors,” J. Stat. Phys. 51, 135 (1988).
[CrossRef]

R. Badii, A. Politi, “Renyi dimensions from local expansion rates,” Phys. Rev. A 35, 1288 (1987).
[CrossRef] [PubMed]

R. Badii, A. Politi, “Statistical description of chaotic attractors: the dimension function,” J. Stat. Phys. 40, 725 (1985).
[CrossRef]

R. Badii, A. Politi, “Intrinsic oscillations in measuring the fractal dimensions,” Phys. Lett. A 104, 303 (1984).
[CrossRef]

Pozžela, J.

A. Namajūnas, J. Pozžela, A. Tamaševičius, “An electronic technique for measuring phase space dimension from chaotic time series,” Phys. Lett. A 131, 85 (1988).
[CrossRef]

Procaccia, I.

D. Auerbach, B. O’Shaughnessy, I. Procaccia, “Scaling structure of strange attractors,” Phys. Rev. A 37, 2234 (1988).
[CrossRef] [PubMed]

D. Katzen, I. Procaccia, “Phase transitions in the thermodynamic formalism of multifractals,” Phys. Rev. Lett. 58, 1169 (1987).
[CrossRef] [PubMed]

G. Gunarante, I. Procaccia, “Organization of chaos,” Phys. Rev. Lett. 59, 1377 (1987).
[CrossRef]

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, I. Procaccia, “Scaling structure and thermodynamics of strange sets,” Phys. Rev. A 36, 1409 (1987).
[CrossRef] [PubMed]

M. J. Feigenbaum, M. H. Jensen, I. Procaccia, “Time ordering and the thermodynamics of strange sets: theory and experimental tests,” Phys. Rev. Lett. 57, 1503 (1986).
[CrossRef] [PubMed]

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

J. P. Eckmann, I. Procaccia, “Fluctuations of dynamical scaling indices in nonlinear systems,” Phys. Rev. A 34, 659 (1986).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

A. Cohen, I. Procaccia, “Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems,” Phys. Rev. A 31, 1872 (1985).
[CrossRef] [PubMed]

H. G. E. Hentschel, I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors,” Physica 8D, 435 (1983).

P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica 9D, 189 (1983).

P. Grassberger, I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983).
[CrossRef]

P. Cvitanović, G. H. Gunarante, I. Procaccia, “Topological and metric properties of Hénon-type strange attractors,” preprint (University of Chicago, Chicago, Ill., 1988).

Provenzale, A.

A. R. Osborne, A. Provenzale, “Finite correlation dimension for stochastic systems with power-law spectra,” Physica 35D, 357 (1989).

Puccioni, G. P.

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

Pyragas, K.

A. Čenys, K. Pyragas, “Estimation of the number of degrees of freedom from chaotic time series,” Phys. Lett. A 129, 227 (1988).
[CrossRef]

Quiniou, J. F.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

Ramsey, J. B.

J. B. Ramsey, H.-J. Yuan, “Bias and error bars in dimension calculations and their evaluation in some simple models,” Phys. Lett. A 134, 287 (1989).
[CrossRef]

Rapp, P. E.

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

A. I. Mees, P. E. Rapp, L. S. Jennings, “Singular-value decomposition and embedding dimension,” Phys. Rev. A 36, 340 (1987).
[CrossRef] [PubMed]

Ravani, M.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Renyi, A.

A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970).

Richter, P. H.

H.-O. Peitgen, P. H. Richter, The Beauty of Fractals (Springer-Verlag, Berlin, 1986).
[CrossRef]

Roques-Carmes, C.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

Rubio, M. A.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Ruelle, D.

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
[CrossRef] [PubMed]

J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

Russell, D. A.

D. A. Russell, J. D. Hanson, E. Ott, “Dimension of strange attractors,” Phys. Rev. Lett. 45, 1175 (1980).
[CrossRef]

Sakar, S. K.

S. K. Sakar, “Multifractal description of singular measures in dynamical systems,” Phys. Rev. A 36, 4104 (1987).
[CrossRef]

Sander, L. M.

L. M. Sander, “Fractal growth,” Sci. Am. 256, 94 (1987).
[CrossRef]

L. M. Sander, “Fractal growth processes,” Nature (London) 322, 789 (1986).
[CrossRef]

T. A. Witten, L. M. Sander, “Diffusion limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400 (1981).
[CrossRef]

Sano, M.

S. Sato, M. Sano, Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems,” Prog. Theor. Phys. 77, 1 (1987).
[CrossRef]

Sato, S.

S. Sato, M. Sano, Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems,” Prog. Theor. Phys. 77, 1 (1987).
[CrossRef]

Sawada, Y.

S. Sato, M. Sano, Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems,” Prog. Theor. Phys. 77, 1 (1987).
[CrossRef]

Scheingraber, H.

H. Atmanspacher, H. Scheingraber, W. Voges, “Global scaling properties of a chaotic attractor reconstructed from experimental data,” Phys. Rev. A 37, 1314 (1988).
[CrossRef] [PubMed]

Scheinkman, J. A.

W. A. Brock, W. D. Dechert, J. A. Scheinkman, “A test for independence based on the correlation dimension,” preprint SSRI 8702 (University of Wisconsin, Madison, Wisc., 1987).

Schram, P.

W. van de Water, P. Schram, “Generalized dimensions from near-neighbor information,” Phys. Rev. A 37, 3118 (1988).
[CrossRef] [PubMed]

Schuster, H. G.

K. Pawelzik, H. G. Schuster, “Generalized dimensions and entropies from a measured time series,” Phys. Rev. A 35, 481 (1987).
[CrossRef] [PubMed]

W. Liebert, K. Pawelzik, H. G. Schuster, Institut für Theoretische Physik, Universität Frankfurt, Frankfurt, Federal Republic of Germany, “Optimal embeddings of chaotic attractors from topological considerations,” preprint (1989).

H. G. Schuster, Deterministic Chaos: An Introduction (VCH, Weinheim, Federal Republic of Germany, 1988).

Schwartz, C.

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

Sepulchre, J. A.

A. Destexhe, J. A. Sepulchre, A. Babloyantz, “A comparative study of the experimental quantification of deterministic chaos,” Phys. Lett. A 132, 101 (1988).
[CrossRef]

Servizi, G.

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

Shaw, R.

J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
[CrossRef]

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

Shaw, R. S.

R. S. Shaw, “Strange attractors, chaotic behavior, and information flow,” Z. Naturforsch. 36a, 80 (1981).

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Shraiman, B. I.

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

Sidorowich, J. J.

J. D. Farmer, J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845 (1987).
[CrossRef] [PubMed]

J. D. Farmer, J. J. Sidorowich, “Exploiting chaos to predict the future and reduce noise,” in Evolution, Learning and Cognition, Y. C. Lee, ed. (World Scientific, Singapore, 1988), p. 227.

Smith, L. A.

L. A. Smith, “Intrinsic limits on dimension calculations,” Phys. Lett. A 133, 283 (1988).
[CrossRef]

L. A. Smith, J.-D. Fournier, E. A. Spiegel, “Lacunarity and intermittency in fluid turbulence,” Phys. Lett. A 114, 465 (1986).
[CrossRef]

Somorjai, R. L.

R. L. Somorjai, “Methods for estimating the intrinsic dimensionality of high-dimensional point sets,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 137.
[CrossRef]

Spiegel, E. A.

L. A. Smith, J.-D. Fournier, E. A. Spiegel, “Lacunarity and intermittency in fluid turbulence,” Phys. Lett. A 114, 465 (1986).
[CrossRef]

Stanley, H. E.

J. Nittman, G. Daccord, H. E. Stanley, “Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon,” Nature (London) 314, 141 (1985).
[CrossRef]

H. E. Stanley, N. Ostrosky, On Growth and Form: Fractal and Non-Fractal Patterns in Physics (Nijhoff, Boston, Mass., 1986).

Stauffer, D.

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).
[CrossRef]

Stavans, J.

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

Stoop, R.

Sullivan, F.

F. Hunt, F. Sullivan, “Efficient algorithms for computing fractal dimensions,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 74.
[CrossRef]

Swift, J.

H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
[CrossRef]

Swift, J. B.

A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).

Swinney, H. L.

A. Brandstater, H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207 (1987).
[CrossRef] [PubMed]

A. M. Fraser, H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A 33, 1134 (1986).
[CrossRef] [PubMed]

A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).

N. B. Abraham, J. P. Gollub, H. L. Swinney, “Testing nonlinear dynamics,” Physica 11D, 252 (1984).

Szépfalusy, P.

P. Szépfalusy, T. Tél, “Dynamical fractal properties of one-dimensional maps,” Phys. Rev. A 35, 477 (1987).
[CrossRef]

Takens, F.

F. Takens, “On the numerical determination of the dimension of an attractor,” in Dynamical Systems and Bifurcations, Groningen, 1984, B. L. J. Braaksma, H. W. Broer, F. Takens, eds., Vol. 1125 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1985).
[CrossRef]

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand, L.-S. Young, eds., Vol. 898 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), p. 366.
[CrossRef]

F. Takens, “Invariants related to dimension and entropy,” in Atas do 13° (Colóqkio Brasiliero do Matemática, Rio de Janeiro, 1983).

Tamaševicius, A.

A. Namajūnas, J. Pozžela, A. Tamaševičius, “An electronic technique for measuring phase space dimension from chaotic time series,” Phys. Lett. A 131, 85 (1988).
[CrossRef]

Tarroja, M. F. H.

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Tél, T.

P. Szépfalusy, T. Tél, “Dynamical fractal properties of one-dimensional maps,” Phys. Rev. A 35, 477 (1987).
[CrossRef]

Termonia, Y.

Y. Termonia, Z. Alexandrowicz, “Fractal dimension of strange attractors from radius versus,” Phys. Rev. Lett. 51, 1265 (1983).
[CrossRef]

Theiler, J.

J. Theiler, “Lacunarity in a best estimator of fractal dimension,” Phys. Lett. A 133, 195 (1988).
[CrossRef]

J. Theiler, “Efficient algorithm for estimating the correlation dimension from a set of discrete points,” Phys. Rev. A 36, 4456 (1987).
[CrossRef] [PubMed]

J. Theiler, “Spurious dimension from correlation algorithms applied to limited time series data,” Phys. Rev. A 34, 2427 (1986).
[CrossRef] [PubMed]

J. Theiler, “Statistical precision of dimension estimators,” Phys. Rev. A (to be published).
[PubMed]

J. Theiler, “Quantifying chaos: practical estimation of the correlation dimension,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1988).

Tredicce, J. R.

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

Tricot, C.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

Tufillaro, N.

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

Turchetti, G.

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

Vaienti, S.

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

van de Water, W.

W. van de Water, P. Schram, “Generalized dimensions from near-neighbor information,” Phys. Rev. A 37, 3118 (1988).
[CrossRef] [PubMed]

Vastano, J. A.

A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).

Voges, W.

H. Atmanspacher, H. Scheingraber, W. Voges, “Global scaling properties of a chaotic attractor reconstructed from experimental data,” Phys. Rev. A 37, 1314 (1988).
[CrossRef] [PubMed]

Vulpiani, A.

G. Paladin, A. Vulpiani, “Anomalous scaling laws in multifractal objects,” Phys. Rep. 156, 147 (1987).
[CrossRef]

Westervelt, R. M.

E. G. Gwinn, R. M. Westervelt, “Scaling structure of attractors at the transition from quasiperiodicity to chaos in electronic transport in Ge,” Phys. Rev. Lett. 59, 157 (1987).
[CrossRef] [PubMed]

Wiesmann, H. J.

L. Niemeyer, L. Pietronero, H. J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett. 52, 1033 (1984).
[CrossRef]

Witten, T. A.

T. A. Witten, L. M. Sander, “Diffusion limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400 (1981).
[CrossRef]

Wolf, A.

A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).

H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
[CrossRef]

Wong, P.-Z.

P.-Z. Wong, “The statistical physics of sedimentary rock,” Phys. Today 41(12), 24 (1988).
[CrossRef]

Yorke, E. D.

P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
[CrossRef]

Yorke, J. A.

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimensions of multifractal chaotic attractors,” Phys. Rev. A 37, 1711 (1988).
[CrossRef] [PubMed]

E. J. Kostelich, J. A. Yorke, “Noise reduction in dynamical systems,” Phys. Rev. A 38, 1649 (1988).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimension of chaotic attractors,” Phys. Rev. A 36, 3522 (1987).
[CrossRef] [PubMed]

P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
[CrossRef]

J. D. Farmer, E. Ott, J. A. Yorke, “The dimension of chaotic attractors,” Physica 7D, 153 (1983).

W. E. Caswell, J. A. Yorke, “Invisible errors in dimension calculations: geometric and systematic effects,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 123.
[CrossRef]

J. L. Kaplan, J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen, H. O. Walther, eds., Vol. 730 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1979), p. 204.
[CrossRef]

Young, S.

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

Yuan, H.-J.

J. B. Ramsey, H.-J. Yuan, “Bias and error bars in dimension calculations and their evaluation in some simple models,” Phys. Lett. A 134, 287 (1989).
[CrossRef]

Zucker, S. W.

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

Am. J. Phys. (1)

A. J. Hurd, “Resource letter FR-1: fractals,” Am. J. Phys. 56, 969 (1988).
[CrossRef]

C. R. Acad. Sci. A (1)

B. B. Mandlebrot, “Corrélations et texture dans un nouveau modéle d’univers hiérarchisé, basé sur les ensembles trémas,” C. R. Acad. Sci. A 288, 81 (1979).

Commun. Math. Phys. (1)

M. Hénon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys. 50, 69 (1976).
[CrossRef]

IEEE Trans. Comput. (1)

K. Fukunaga, D. R. Olsen, “An algorithm for finding intrinsic dimensionality of data,” IEEE Trans. Comput. C-20, 176 (1971).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

K. W. Pettis, T. A. Bailey, A. K. Jain, R. C. Dubes, “An intrinsic dimensionality estimator from near neighbor information,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25 (1979).
[CrossRef]

J. Diff, Eq. (1)

P. Fredrickson, J. L. Kaplan, E. D. Yorke, J. A. Yorke, “The Liapunov dimension of strange attractors,” J. Diff, Eq. 49, 185 (1983).
[CrossRef]

J. Mec. Theor. Appl. (1)

P. Atten, J. G. Caputo, B. Malraison, Y. Gagne, “Détermination de dimension d’attracteurs pour différents écoulements,” J. Mec. Theor. Appl. 133 (Suppl.) (1984).

J. Opt. Soc. Am. B (3)

J. Phys. A (2)

Y. Gefen, A. Aharony, B. B. Mandelbrot, “Phase transitions on fractals: III. Infinitely ramified lattices,” J. Phys. A 17, 1277 (1984).
[CrossRef]

D. S. Broomhead, R. Jones, G. P. King, “Topological dimension and local coordinates from time series data,” J. Phys. A 20, L563 (1987).
[CrossRef]

J. Stat. Phys. (3)

K. Ikeda, K. Matsumoto, “Study of a high-dimensional chaotic attractor,” J. Stat. Phys. 44, 955 (1986).
[CrossRef]

P. Grassberger, R. Badii, A. Politi, “Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors,” J. Stat. Phys. 51, 135 (1988).
[CrossRef]

R. Badii, A. Politi, “Statistical description of chaotic attractors: the dimension function,” J. Stat. Phys. 40, 725 (1985).
[CrossRef]

Math. Annalen (1)

F. Hausdorff, “Dimension und äusseres Mass,” Math. Annalen 79, 157 (1919).
[CrossRef]

Nature (London) (3)

L. M. Sander, “Fractal growth processes,” Nature (London) 322, 789 (1986).
[CrossRef]

J. Nittman, G. Daccord, H. E. Stanley, “Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon,” Nature (London) 314, 141 (1985).
[CrossRef]

P. Grassberger, “Do climatic attractors exist?” Nature (London) 323, 609 (1986).
[CrossRef]

Phys. Lett. A (19)

C.-K. and F. C. Moon, “An optical technique for measuring fractal dimensions of planar Poincaré maps,” Phys. Lett. A 114, 222 (1986).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, G. DeGuzman, S. Young, R. S. Gioggia, G. P. Puccioni, J. R. Tredicce, “Calculating the dimension of attractors from small data sets,” Phys. Lett. A 114, 217 (1986).
[CrossRef]

J. B. Ramsey, H.-J. Yuan, “Bias and error bars in dimension calculations and their evaluation in some simple models,” Phys. Lett. A 134, 287 (1989).
[CrossRef]

R. Badii, A. Politi, “Intrinsic oscillations in measuring the fractal dimensions,” Phys. Lett. A 104, 303 (1984).
[CrossRef]

L. A. Smith, J.-D. Fournier, E. A. Spiegel, “Lacunarity and intermittency in fluid turbulence,” Phys. Lett. A 114, 465 (1986).
[CrossRef]

A. Arneodo, G. Grasseau, E. J. Kostelich, “Fractal dimensions and f(α) spectrum of the Hénon attractor,” Phys. Lett. A 124, 426 (1987).
[CrossRef]

M. Möller, W. Lange, F. Mitschke, N. B. Abraham, U. Hübner, “Errors from digitizing and noise in estimating attractor dimensions,” Phys. Lett. A 138, 176 (1989).
[CrossRef]

L. A. Smith, “Intrinsic limits on dimension calculations,” Phys. Lett. A 133, 283 (1988).
[CrossRef]

R. Badii, G. Broggi, “Measurement of the dimension spectrum f(α): fixed-mass approach,” Phys. Lett. A 131, 339 (1988).
[CrossRef]

A. Čenys, K. Pyragas, “Estimation of the number of degrees of freedom from chaotic time series,” Phys. Lett. A 129, 227 (1988).
[CrossRef]

P. Grassberger, “On the fractal dimension of the Hénon attractor,” Phys. Lett. A 97, 224 (1983).
[CrossRef]

A. Namajūnas, J. Pozžela, A. Tamaševičius, “An electronic technique for measuring phase space dimension from chaotic time series,” Phys. Lett. A 131, 85 (1988).
[CrossRef]

A. Destexhe, J. A. Sepulchre, A. Babloyantz, “A comparative study of the experimental quantification of deterministic chaos,” Phys. Lett. A 132, 101 (1988).
[CrossRef]

S. Ellner, “Estimating attractor dimensions for limited data: a new method, with error estimates,” Phys. Lett. A 113, 128 (1988).
[CrossRef]

J. Theiler, “Lacunarity in a best estimator of fractal dimension,” Phys. Lett. A 133, 195 (1988).
[CrossRef]

S. Bingham, M. Kot, “Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity,” Phys. Lett. A 140, 327 (1989).
[CrossRef]

P. Grassberger, “Generalizations of the Hausdorff dimension of fractal measures,” Phys. Lett. A 107, 101 (1985).
[CrossRef]

P. Grassberger, “Finite sample corrections to entropy and dimension estimates,” Phys. Lett. A 128, 369 (1988).
[CrossRef]

P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett. A 97, 227 (1983).
[CrossRef]

Phys. Rep. (1)

G. Paladin, A. Vulpiani, “Anomalous scaling laws in multifractal objects,” Phys. Rep. 156, 147 (1987).
[CrossRef]

Phys. Rev. A (31)

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A 33, 1141 (1986).
[CrossRef] [PubMed]

S. K. Sakar, “Multifractal description of singular measures in dynamical systems,” Phys. Rev. A 36, 4104 (1987).
[CrossRef]

A. I. Mees, P. E. Rapp, L. S. Jennings, “Singular-value decomposition and embedding dimension,” Phys. Rev. A 36, 340 (1987).
[CrossRef] [PubMed]

A. M. Albano, J. Muench, C. Schwartz, A. I. Mees, P. E. Rapp, “Singular-value decomposition and the Grassberger–Procaccia algorithm,” Phys. Rev. A 38, 3017 (1988).
[CrossRef] [PubMed]

P. Grassberger, I. Procaccia, “Estimation of the Kolmogorov entropy from a chaotic signal,” Phys. Rev. A 28, 2591 (1983).
[CrossRef]

A. Cohen, I. Procaccia, “Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems,” Phys. Rev. A 31, 1872 (1985).
[CrossRef] [PubMed]

J. P. Eckmann, I. Procaccia, “Fluctuations of dynamical scaling indices in nonlinear systems,” Phys. Rev. A 34, 659 (1986).
[CrossRef] [PubMed]

P. Szépfalusy, T. Tél, “Dynamical fractal properties of one-dimensional maps,” Phys. Rev. A 35, 477 (1987).
[CrossRef]

A. M. Fraser, H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A 33, 1134 (1986).
[CrossRef] [PubMed]

J. Theiler, “Spurious dimension from correlation algorithms applied to limited time series data,” Phys. Rev. A 34, 2427 (1986).
[CrossRef] [PubMed]

K. Pawelzik, H. G. Schuster, “Generalized dimensions and entropies from a measured time series,” Phys. Rev. A 35, 481 (1987).
[CrossRef] [PubMed]

H. Atmanspacher, H. Scheingraber, W. Voges, “Global scaling properties of a chaotic attractor reconstructed from experimental data,” Phys. Rev. A 37, 1314 (1988).
[CrossRef] [PubMed]

J. A. Glazier, G. Gunaratne, A. Libchaber, “f(α) curves: experimental results,” Phys. Rev. A 37, 523 (1988).
[CrossRef] [PubMed]

M. H. Jensen, L. P. Kadanoff, I. Procaccia, “Scaling structure and thermodynamics of strange sets,” Phys. Rev. A 36, 1409 (1987).
[CrossRef] [PubMed]

T. Bohr, M. H. Jensen, “Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets,” Phys. Rev. A 36, 4904 (1987).
[CrossRef] [PubMed]

H. S. Greenside, A. Wolf, J. Swift, T. Pignataro, “Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors,” Phys. Rev. A 25, 3453 (1982).
[CrossRef]

B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, S. W. Zucker, “Evaluating the fractal dimension of profiles,” Phys. Rev. A 39, 1500 (1989).
[CrossRef] [PubMed]

J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, S. Ciliberto, “Liapunov exponents from a time series,” Phys. Rev. A 34, 4971 (1986).
[CrossRef] [PubMed]

J. Theiler, “Efficient algorithm for estimating the correlation dimension from a set of discrete points,” Phys. Rev. A 36, 4456 (1987).
[CrossRef] [PubMed]

A. Passamante, T. Hediger, M. Gollub, “Fractal dimension and local intrinsic dimension,” Phys. Rev. A 39, 3640 (1989).
[CrossRef] [PubMed]

D. Auerbach, B. O’Shaughnessy, I. Procaccia, “Scaling structure of strange attractors,” Phys. Rev. A 37, 2234 (1988).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimension of chaotic attractors,” Phys. Rev. A 36, 3522 (1987).
[CrossRef] [PubMed]

C. Grebogi, E. Ott, J. A. Yorke, “Unstable periodic orbits and the dimensions of multifractal chaotic attractors,” Phys. Rev. A 37, 1711 (1988).
[CrossRef] [PubMed]

W. van de Water, P. Schram, “Generalized dimensions from near-neighbor information,” Phys. Rev. A 37, 3118 (1988).
[CrossRef] [PubMed]

R. Badii, A. Politi, “Renyi dimensions from local expansion rates,” Phys. Rev. A 35, 1288 (1987).
[CrossRef] [PubMed]

F. Mitschke, M. Moller, W. Lange, “Measuring filtered chaotic signals,” Phys. Rev. A 37, 4518 (1988).
[CrossRef] [PubMed]

E. J. Kostelich, J. A. Yorke, “Noise reduction in dynamical systems,” Phys. Rev. A 38, 1649 (1988).
[CrossRef] [PubMed]

D. Bessis, J.-D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimension of strange sets,” Phys. Rev. A 36, 20 (1987).
[CrossRef]

J. W. Havstad, C. L. Ehlers, “Attractor dimension of non-stationary dynamical systems from small data sets,” Phys. Rev. A 39, 845 (1989).
[CrossRef] [PubMed]

M. Franaszek, “Optimized algorithm for the calculation of correlation integrals,” Phys. Rev. A 39, 5540 (1989).
[CrossRef]

A. Brandstater, H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett. (17)

Y. Gefen, Y. Meir, A. Aharony, B. B. Mandelbrot, “Geometric implementation of hypercubic lattices with noninteger dimension,” Phys. Rev. Lett. 50, 145 (1983).
[CrossRef]

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M. A. Rubio, “Dimension increase in filtered chaotic signals,” Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunarante, I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387 (1987).
[CrossRef] [PubMed]

G. Gunarante, I. Procaccia, “Organization of chaos,” Phys. Rev. Lett. 59, 1377 (1987).
[CrossRef]

A. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phys. Rev. Lett. 69, 1327 (1989).
[CrossRef]

J. D. Farmer, J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845 (1987).
[CrossRef] [PubMed]

D. A. Russell, J. D. Hanson, E. Ott, “Dimension of strange attractors,” Phys. Rev. Lett. 45, 1175 (1980).
[CrossRef]

L. Niemeyer, L. Pietronero, H. J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett. 52, 1033 (1984).
[CrossRef]

T. A. Witten, L. M. Sander, “Diffusion limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400 (1981).
[CrossRef]

M. J. Feigenbaum, M. H. Jensen, I. Procaccia, “Time ordering and the thermodynamics of strange sets: theory and experimental tests,” Phys. Rev. Lett. 57, 1503 (1986).
[CrossRef] [PubMed]

D. Katzen, I. Procaccia, “Phase transitions in the thermodynamic formalism of multifractals,” Phys. Rev. Lett. 58, 1169 (1987).
[CrossRef] [PubMed]

Y. Termonia, Z. Alexandrowicz, “Fractal dimension of strange attractors from radius versus,” Phys. Rev. Lett. 51, 1265 (1983).
[CrossRef]

J. Guckenheimer, G. Buzyna, “Dimension measurements for geostrophic turbulence,” Phys. Rev. Lett. 51, 1483 (1983).
[CrossRef]

P. Grassberger, I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, J. Stavans, “Global universality at the onset of chaos: results of a forced Rayleigh–Bernard experiment,” Phys. Rev. Lett. 55, 2798 (1985).
[CrossRef] [PubMed]

E. G. Gwinn, R. M. Westervelt, “Scaling structure of attractors at the transition from quasiperiodicity to chaos in electronic transport in Ge,” Phys. Rev. Lett. 59, 157 (1987).
[CrossRef] [PubMed]

Phys. Today (2)

L. Kadanoff, “Where is the physics of fractals,” Phys. Today 39(2), 6 (1986).
[CrossRef]

P.-Z. Wong, “The statistical physics of sedimentary rock,” Phys. Today 41(12), 24 (1988).
[CrossRef]

Physica (11)

A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica 16D, 285 (1985).

A. M. Fraser, “Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria,” Physica 34D, 391 (1989).

D. S. Broomhead, G. P. King, “Extracting qualitative dynamics from experimental data,” Physica 20D, 217 (1986).

P. Grassberger, I. Procaccia, “Measuring the strangeness of strange attractors,” Physica 9D, 189 (1983).

J. D. Farmer, E. Ott, J. A. Yorke, “The dimension of chaotic attractors,” Physica 7D, 153 (1983).

H. G. E. Hentschel, I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors,” Physica 8D, 435 (1983).

M. J. McGuinness, “A computation of the limit capacity of the Lorenz attractor,” Physica 16D, 265 (1985).

H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard, R. Shaw, “On determining the dimension of chaotic flows,” Physica 3D, 605 (1981).

J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366 (1982).

A. R. Osborne, A. Provenzale, “Finite correlation dimension for stochastic systems with power-law spectra,” Physica 35D, 357 (1989).

N. B. Abraham, J. P. Gollub, H. L. Swinney, “Testing nonlinear dynamics,” Physica 11D, 252 (1984).

Proc. IEEE (1)

T. S. Parker, L. O. Chua, “Chaos: a tutorial for engineers,” Proc. IEEE 75, 982 (1987).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

M. F. Barnsley, S. Demko, “Iterated function systems and the global construction of fractals,” Proc. R. Soc. London Ser. A 399, 243 (1985).
[CrossRef]

Prog. Theor. Phys. (1)

S. Sato, M. Sano, Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems,” Prog. Theor. Phys. 77, 1 (1987).
[CrossRef]

Rev. Mod. Phys. (2)

J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

E. Ott, “Strange attractors and chaotic motions of dynamical systems,” Rev. Mod. Phys. 53, 655 (1981).
[CrossRef]

Sci. Am. (2)

J. P. Crutchfield, J. D. Farmer, N. H. Packard, R. Shaw, “Chaos,” Sci. Am. 255, 46 (1986).
[CrossRef]

L. M. Sander, “Fractal growth,” Sci. Am. 256, 94 (1987).
[CrossRef]

Z. Naturforsch. (1)

R. S. Shaw, “Strange attractors, chaotic behavior, and information flow,” Z. Naturforsch. 36a, 80 (1981).

Other (38)

A. V. Holden, ed., Chaos (Princeton U. Press, Princeton, N.J., 1986).

H. G. Schuster, Deterministic Chaos: An Introduction (VCH, Weinheim, Federal Republic of Germany, 1988).

H. Bai-Lin, Chaos (World Scientific, Singapore, 1984).

P. Cvitanović, Universality in Chaos (Hilger, Bristol, UK, 1986).

P. Grassberger, “Estimating the fractal dimensions and entropies of strange attractors,” in Chaos, A. V. Holden, ed. (Princeton U. Press, Princeton, N.J.1986), Chap. 14.

G. Mayer-Kress, ed., Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986).
[CrossRef]

T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, T. Mello, M. F. H. Tarroja, N. Tufillaro, R. S. Gioggia, “Definitions of chaos and measuring its characteristics,” in Optical Chaos, J. Chros-towski, N. B. Abraham, eds. Proc. Soc. Photo-Opt. In-strum. Eng.667, 2 (1986).
[CrossRef]

J. Gleick, Chaos: Making a New Science (Viking, New York, 1987).

H.-O. Peitgen, P. H. Richter, The Beauty of Fractals (Springer-Verlag, Berlin, 1986).
[CrossRef]

H.-O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, New York, 1988).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

W. A. Brock, W. D. Dechert, J. A. Scheinkman, “A test for independence based on the correlation dimension,” preprint SSRI 8702 (University of Wisconsin, Madison, Wisc., 1987).

R. L. Somorjai, “Methods for estimating the intrinsic dimensionality of high-dimensional point sets,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 137.
[CrossRef]

J. L. Kaplan, J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen, H. O. Walther, eds., Vol. 730 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1979), p. 204.
[CrossRef]

W. E. Caswell, J. A. Yorke, “Invisible errors in dimension calculations: geometric and systematic effects,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 123.
[CrossRef]

F. Takens, “On the numerical determination of the dimension of an attractor,” in Dynamical Systems and Bifurcations, Groningen, 1984, B. L. J. Braaksma, H. W. Broer, F. Takens, eds., Vol. 1125 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1985).
[CrossRef]

J. Theiler, “Quantifying chaos: practical estimation of the correlation dimension,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1988).

J. Theiler, “Statistical precision of dimension estimators,” Phys. Rev. A (to be published).
[PubMed]

R. Cawley, A. L. Licht, “Maximum likelihood method for evaluating correlation dimension,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds., Vol. 278 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), p. 90.

F. Hunt, F. Sullivan, “Efficient algorithms for computing fractal dimensions,” in Dimensions and Entropies in Chaotic Systems—Quantification of Complex Behavior, G. Mayer-Kress, ed., Vol. 32 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1986), p. 74.
[CrossRef]

F. Takens, “Invariants related to dimension and entropy,” in Atas do 13° (Colóqkio Brasiliero do Matemática, Rio de Janeiro, 1983).

A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970).

P. Cvitanović, G. H. Gunarante, I. Procaccia, “Topological and metric properties of Hénon-type strange attractors,” preprint (University of Chicago, Chicago, Ill., 1988).

G. Mayer-Kress, “Application of dimension algorithms to experimental chaos,” in Directions in Chaos, Hao Bailin, ed. (World Scientific, Singapore, 1987), p. 122.
[CrossRef]

W. Liebert, K. Pawelzik, H. G. Schuster, Institut für Theoretische Physik, Universität Frankfurt, Frankfurt, Federal Republic of Germany, “Optimal embeddings of chaotic attractors from topological considerations,” preprint (1989).

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand, L.-S. Young, eds., Vol. 898 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), p. 366.
[CrossRef]

R. Mañé, “On the dimension of the compact invariant sets of certain non-linear maps,” in Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand, L.-S. Young, eds., Vol. 898 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), p. 320.

A. M. Fraser, “Information and entropy in strange attractors,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

J. D. Farmer, J. J. Sidorowich, “Exploiting chaos to predict the future and reduce noise,” in Evolution, Learning and Cognition, Y. C. Lee, ed. (World Scientific, Singapore, 1988), p. 227.

K. J. Falconer, The Geometry of Fractal Sets, Vol. 85 of Cambridge Tracts in Mathematics (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

H. Haken, Information and Self-Organization: A Macroscopic Approach to Complex Systems, Vol. 40 of Springer Series in Synergetics (Springer-Verlag, Berlin, 1988).

R. S. MacKay, J. D. Meiss, Hamiltonian Dynamical Systems (Hilger, Philadelphia, 1987).

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 of Springer Series in Applied Mathematical Sciences (Springer-Verlag, New York, 1983).

H. E. Stanley, N. Ostrosky, On Growth and Form: Fractal and Non-Fractal Patterns in Physics (Nijhoff, Boston, Mass., 1986).

P. H. Carter, R. Cawley, R. D. Mauldin, “Mathematics of dimension measurement for graphs of functions,” in Fractal Aspects of Materials, B. B. Mandelbrot, D. E. Passoja, eds. (Materials Research Society, Pittsburgh, Pa., 1985).

M. F. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988).

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Self-similar fractals. (a) The Sierpinski gasket. Here, the inner triangles are small copies of the full figure. (b) Prince William Sound. The sound has been darkened to highlight the fractal appearance of the coastline.

Fig. 2
Fig. 2

Two strange attractors. (a) The Ikeda map. The complex map, zn+1 = a + Rzn exp{i[ϕp/(1 + |zn|2)]}, which derives from a model of the plane-wave interactivity field in an optical ring laser,18 is iterated many times, and the points [Re(zn), Im(zn)] are plotted for n ≥ 1000. Here, a = 1.0, R = 0.9, ϕ = 0.4, p = 6. The fractal dimension of this attractor is approximately 1.7. (b) The Hénon map.19 This map, xn+1 = 1.0 – axn2 + yn; yn+1 = bxn, with a = 1.4 and b = 0.3, gives an attractor with fractal dimension of approximately 1.3. Note how much thinner the attractor of lower dimension appears.

Fig. 3
Fig. 3

Generalized dimension. (a) Dq as a function of q for a typical multifractal. Also shown are both f and α as a function of q for the same multifractal. (b) f as a function of α. The curve is always convex upward, and the peak of the curve occurs at q = 0. At this point f is equal to the fractal dimension D0. Also, the f(α) curve is tangent to the curve f = α, and the point of tangency occurs at q = 1. In general, the left-hand branch corresponds to q > 0 and the right-hand branch to q < 0.

Fig. 4
Fig. 4

Sources of error in the correlation integral. (a) Ideally, the correlation integral C(N, r) scales as rm for m < ν, and as rν for m > ν over a range from C(N, r) = 2/N2 to saturation at C(N, r) = 1. Here the dimension is somewhere between 2 and 3. This idealization, however, is only approximated by correlation integrals computed from actual samples of time series data. (b) An actual correlation integral for a two-dimensional chaotic attractor is shown here, with embedding dimensions m = 1 through m = 6. The finite sample size leads to poor statistics at small r, and the finite size of the attractor (the edge effect) limits the scaling at large r. Nonetheless, the slopes are more or less constant over a range of C(N, r) of the order of N2. (c) The effect of noise. With σ the amplitude of the noise, one sees that, for rσ, a slope that approaches the embedding dimension m is observed. For rσ, the effect of the noise is unimportant. (d) The effect of discretization is to introduce a stair step into the correlation integral (solid curve). The steps are all of equal width, but the log–log plot magnifies those at small r. The effect is minimized if one plots log C(N, r) versus log r for r = (k + ½), where k is an integer and is the level of discretization (dashed–dotted curve). (e) Lacunarity leads to an intrinsic oscillation in the correlation integral that inhibits accurate determinations of slope. The example here is the correlation integral of the middle-thirds Cantor set. (f) Autocorrelation in the time-series data can lead to an anomalous shoulder in the correlation integral. This effect is most highly pronounced for high-dimensional attractors. In this case the input time series was autocorrelated Gaussian noise, and the correlation integral was computed for various (large) embedding dimensions of m = 4 to m = 32. Equation (76) corrects for this effect. (g) If the time-series data arise from a nonchaotic attractor, then the scaling of C(N, r) as rν begins to break down for C(N, r) < 1/N. The dotted–dashed curve here has a slope of 2, corresponding to the two-dimensional quasi-periodic input data.

Equations (76)

Equations on this page are rendered with MathJax. Learn more.

lim t μ L [ ϕ t ( ) ] = 0.
μ ( ) = μ [ ϕ t ( ) ] ,
μ ( ) = lim T 1 T 0 T I [ ϕ t ( X 0 ) ] d t ,
ϕ t ( X 0 + ) = ϕ t ( X 0 ) + J ( t ) + O ( | | 2 ) ,
J ( t ) = ϕ t ( X 0 ) X 0 = X ( t ) X ( 0 ) .
J i j ( t ) = X i ( t ) X j ( 0 ) ,
λ n = lim t 1 t log | n th eigenvalue of J ( t ) | .
π : R M R
X ̂ ( t ) = [ x ( t ) , x ( t τ ) , , x ( t ( m 1 ) τ ) ] T ,
π ( m ) : R M R m
X ̂ ( t ) = { x ( t ) , e h τ x ( t τ ) , , exp [ h ( m 1 ) τ ] x [ t ( m 1 ) τ ] } T .
bulk size dimension .
dimension = lim size 0 log bulk log size ,
δ ( ) sup { ¯ X Y ¯ : X , Y } ,
¯ X Y ¯ ( i = 1 m | X i Y i | s ) 1 / s ,
B X ( r ) = μ [ X ( r ) ] .
D P ( X ) = lim r 0 log B X ( r ) log r .
D P = A D p ( X ) d μ ( X ) .
Γ ( A , D , r ) = inf C ( r , A ) i δ i D ,
Γ ( A , D ) = lim sup r 0 Γ ( A , D , r ) = { for D < D H 0 for D > D H ,
Γ ( A , D , r ) = i δ i D = i r D = n ( r ) r D ,
n ( r ) r D H
D H = lim r 0 log [ 1 / n ( r ) ] log r .
D q = 1 q 1 lim r 0 log i P i q log r .
D = lim r 0 log ( max P i i ) log r ,
D = lim r 0 log ( min P i i ) log r .
Γ q ( A , D , r ) = i μ i q δ i ( 1 q ) D ,
D q = 1 1 q inf { ( 1 q ) D : lim r 0 Γ q ( A , D , r ) = 0 } .
α = τ q f = α q τ
q = f α τ = α q f .
f ( α ) = min q { q α τ ( q ) } ,
τ ( q ) = min α { q α f ( α ) } .
i P i q = n ( α , r ) r q α d α
r f ( α ) r q α d α r θ ,
S α = { X A : D p ( X ) = α }
S ( r ) = i P i log 2 P i ,
D I = lim r 0 S ( r ) log 2 r
= lim r 0 i P i log 2 P i log 2 r .
S q ( r ) = 1 q 1 log i P i q ,
D q = lim r 0 S q ( r ) log r = 1 q 1 lim r 0 log i P i q log r ,
C ( r ) = B X ( r ) .
ν = lim r 0 log C ( r ) log r .
B X j ( r ) # { X i : i j and ¯ X i X j ¯ r } N 1 = 1 N 1 i = 1 i j N Θ ( r ¯ X i X j ¯ ) ,
C ( N , r ) = 1 N j = 1 N B X j ( r ) = 1 N ( N 1 ) i j Θ ( r ¯ X i X j ¯ ) .
C ( N , r ) = # of distances less than r # of distances altogether .
G q ( r ) = [ B X ( r ) q 1 ] 1 / ( q 1 ) .
D q = lim r 0 log G q ( r ) log r .
G 1 ( r ) = lim q 1 G q ( r ) = exp log B X ( r ) ,
G q ( N , r ) { 1 N i = 1 N [ 1 N 1 j = 1 j i N Θ ( r ¯ X i X j ¯ ) ] q 1 } 1 / ( q 1 ) .
C q ( N , r ) = 1 N q # { ( X i 1 , , X i q ) : ¯ X n X m ¯ r for all n , m { i 1 , , i q } } ,
D q = 1 q 1 lim r 0 lim N log C q ( N , r ) log r .
r k = 1 N i = 1 N R ( X i , k ) .
r k γ ( k / N ) γ / D ( γ ) .
γ = ( q 1 ) D q , D ( γ ) = D q .
r k γ [ Γ ( k + γ / D ) Γ ( k ) k γ / D ] ( k / N ) γ / D .
1 T X 0 ( r ) B X 0 ( r ) .
[ T X ( r ) 1 q ] 1 / ( 1 q ) [ B X ( r ) q 1 ] 1 / ( 1 q ) r D q .
d L = j + i = 1 j λ i | λ j + 1 | ,
j = sup { k : i = 1 k λ i > 0 } .
D 0 log [ 1 / n ( r ) ] log r .
log [ ( 1 / n 0 ) r D 0 ] log r = D 0 log r + log ( 1 / n 0 ) log r = D 0 + slow to vanish as r 0 log ( 1 / n 0 ) log r D 0 .
D 0 Δ [ log n ( r ) ] Δ ( log r ) .
n ( r ) = lim N n ̂ ( N , r ) .
D 0 = lim r 0 lim N log [ 1 / n ̂ ( N , r ) ] log r .
n ( r ) n ̂ ( N , r ) r α N β
ν = lim r 0 lim N log C ( N , r ) log r .
ν ̂ ( r ) = d log C ( N , r ) d log r = d C ( N , r ) / d r C ( N , r ) / r ,
ν ̂ ( r ) = Δ log C ( N , r ) Δ log r = log C ( N , r 2 ) log C ( N , r 1 ) log r 2 log r 1 .
ν ̂ ( r 0 ) = 1 log ( r i j / r 0 ) ,
ν ̂ ( r 0 ) = C ( r 0 ) 0 r 0 [ C ( r ) / r ] d r
= 0 r 0 [ d C ( r ) / d r ] d r 0 r 0 [ C ( r ) / r ] d r .
N min = N 0 Θ D ,
ν ( r ) = d C / d r C ( r ) / r = m ( 2 2 r ) 2 r m ( 1 r 2 ) ,
N min = 5 m .
C ( W , N , r ) = 2 ( N + 1 W ) ( N W ) × n = W N 1 i = 0 N 1 n Θ ( r ¯ X i X i + n ¯ ) .
C ( W , N , r ) = # of distances less than r except for those from pairs of points closer together in time than W # of distances altogether .

Metrics