OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 7, Iss. 6 — Jun. 1, 1990
  • pp: 1055–1073

Estimating fractal dimension

James Theiler  »View Author Affiliations

JOSA A, Vol. 7, Issue 6, pp. 1055-1073 (1990)

View Full Text Article

Enhanced HTML    Acrobat PDF (2667 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

Original Manuscript: September 13, 1989
Manuscript Accepted: November 21, 1989
Published: June 1, 1990

James Theiler, "Estimating fractal dimension," J. Opt. Soc. Am. A 7, 1055-1073 (1990)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited