Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Fractals Meet Chaos

  • Tony CrillyEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_225-3

Definition of the Subject

Though “Chaos” and “Fractals” are yoked together to form “Fractals and Chaos” they have had separate lines of development. And though the names are modern, the mathematical ideas which lie behind them have taken more than a century to gain the prominence they enjoy today. Chaos carries an applied connotation and is linked to differential equations which model physical phenomena. Fractals is directly linked to subsets of Euclidean space which have a fractional dimension, which may be obtained by the iteration of functions.

This brief survey seeks to highlight some of the significant points in the history of both of these subjects. There are brief academic histories of the field. A history of Chaos has been shown attention (Aubin and Dahan Dalmedico 2002; Holmes 2005), while an account of the early history of the iteration of complex functions (up to Julia and Fatou) is given in Alexander (1994). A broad survey of the whole field of fractals is given in Chabert...

Keywords

Fractal Dimension Topological Dimension Hausdorff Dimension Strange Attractor Period Doubling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Bibliography

Primary Literature

  1. Abraham RH (1985) In pursuit of Birkhoff’s chaotic attractor. In: Pnevmatikos SN (ed) Singularities and dynamical systems. North Holland, Amsterdam, pp 303–312Google Scholar
  2. Abraham RH, Ueda Y (eds) (2000) The Chaos Avant-Garde: memories of the early days of chaos theory. World Scientific, River EdgeGoogle Scholar
  3. Alexander DS (1994) A history of complex dynamics; from Schröder to Fatou and Julia. Vieweg, BraunschweigzbMATHGoogle Scholar
  4. Aubin D, Dahan Dalmedico A (2002) Writing the history of dynamical systems and chaos: Longue Durée and revolution, disciplines and cultures. Hist Math 29(3):235–362MathSciNetGoogle Scholar
  5. Baker GL, Gollub JP (1996) Chaotic dynamics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  6. Barrow-Green J (1997) Poincaré and the three body problem. American Mathematical Society/London Mathematical Society, Providence/LondonzbMATHGoogle Scholar
  7. Barrow-Green J (2005) Henri Poincaré, Memoir on the three-body problem. In: Grattan-Guinness I (ed) Landmark writings in Western mathematics 1640–1940. Elsevier, Amsterdam, pp 627–638Google Scholar
  8. Batterson S (2000) Stephen Smale: the mathematician who broke the dimension barrier. American Mathematical Society, ProvidenceGoogle Scholar
  9. Berry MV, Percival IC, Weiss NO (1987) Dynamical chaos. Proc R Soc Lond 413(1844):1–199ADSGoogle Scholar
  10. Birkhoff GD (1927) Dynamical systems. American Mathematical Society, New YorkzbMATHGoogle Scholar
  11. Birkhoff GD (1941) Some unsolved problems of theoretical dynamics. Science 94:598–600ADSzbMATHMathSciNetGoogle Scholar
  12. Brandstater A, Swinney HL (1986) Strange attractors in weakly turbulent Couette-Tayler flow. In: Ott E et al (eds) Coping with chaos. Wiley Interscience, New York, pp 142–155Google Scholar
  13. Briggs J (1992) Fractals, the patterns of chaos: discovering a new aesthetic of art, science, and nature. Thames and Hudson, LondonGoogle Scholar
  14. Brindley J, Kapitaniak T, El Naschie MS (1991) Analytical conditions for strange chaotic and nonchaotic attractors of the quasiperiodically forced van der Pol equation. Physica D 51:28–38ADSzbMATHMathSciNetGoogle Scholar
  15. Brolin H (1965) Invariant sets under iteration of rational functions. Ark Mat 6:103–144zbMATHMathSciNetGoogle Scholar
  16. Cayley A (1879) The Newton-Fourier imaginary problem. Am J Math 2:97zbMATHMathSciNetGoogle Scholar
  17. Chabert J-L (1990) Un demi-siècle de fractales: 1870–1920. Hist Math 17:339–365zbMATHMathSciNetGoogle Scholar
  18. Crilly AJ, Earnshaw RA, Jones H (eds) (1993) Applications of fractals and chaos: the shape of things. Springer, BerlinzbMATHGoogle Scholar
  19. Crilly T (1999) The emergence of topological dimension theory. In: James IM (ed) History of topology. Elsevier, Amsterdam, pp 1–24Google Scholar
  20. Crilly T (2006) Arthur Cayley: mathematician laureate of the victorian age. Johns Hopkins University Press, BaltimoreGoogle Scholar
  21. Crilly T, Moran A (2002) Commentary on Menger’s work on curve theory and topology. In: Schweizer B et al (eds) Karl Menger, selecta. Springer, ViennaGoogle Scholar
  22. Curry J, Garnett L, Sullivan D (1983) On the iteration of rational functions: computer experiments with Newton’s method. Commun Math Phys 91:267–277ADSzbMATHMathSciNetGoogle Scholar
  23. Dahan-Dalmedico A, Gouzevitch I (2004) Early developments in nonlinear science in Soviet Russia: the Andronov School at Gor’kiy. Sci Context 17:235–265zbMATHMathSciNetGoogle Scholar
  24. Devaney RL (1984) Julia sets and bifurcation diagrams for exponential maps. Bull Am Math Soc 11:167–171zbMATHMathSciNetGoogle Scholar
  25. Devaney RL (2003) An introduction to chaotic dynamical systems, 2nd edn. Westview Press, BoulderzbMATHGoogle Scholar
  26. Diacu F, Holmes P (1996) Celestial encounters: the origins of chaos and stability. Princeton University Press, PrincetonzbMATHGoogle Scholar
  27. Dieudonné J (1960) Foundations of modern analysis. Academic, New YorkzbMATHGoogle Scholar
  28. Douady A, Hubbard J (1982) Iteration des polynomes quadratiques complexes. C R Acad Sci Paris Sér 1 Math 294:123–126zbMATHMathSciNetGoogle Scholar
  29. Dyson F (1997) ‘Nature’s numbers’ by Ian Stewart. Math Intell 19(2):65–67Google Scholar
  30. Edgar G (1993) Classics on fractals. Addison-Wesley, ReadingzbMATHGoogle Scholar
  31. Falconer K (1997) Techniques in fractal geometry. Wiley, ChichesterzbMATHGoogle Scholar
  32. Falconer K (2003) Fractal geometry: mathematical foundations and applications. Wiley, ChichesterGoogle Scholar
  33. Farmer JD, Ott E, Yorke JA (1983) The dimension of chaotic attractors. Physica D 7:153–180ADSMathSciNetGoogle Scholar
  34. Fatou P (1919/1920) Sur les equations fonctionelles. Bull Soc Math Fr 47:161–271; 48:33–94, 208–314Google Scholar
  35. Feigenbaum MJ (1978) Quantitative universality for a class of nonlinear transformations. J Stat Phys 19:25–52ADSzbMATHMathSciNetGoogle Scholar
  36. Feigenbaum MJ (1979) The universal metric properties of nonlinear transformations. J Stat Phys 21:669–706ADSzbMATHMathSciNetGoogle Scholar
  37. Gleick J (1988) Chaos. Sphere Books, LondonGoogle Scholar
  38. Grassberger P (1983) Generalised dimensions of strange attractors. Phys Lett A 97:227–230ADSMathSciNetGoogle Scholar
  39. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208ADSzbMATHMathSciNetGoogle Scholar
  40. Hénon M (1976) A two dimensional mapping with a strange attractor. Commun Math Phys 50:69–77ADSzbMATHGoogle Scholar
  41. Haeseler F, Peitgen H-O, Saupe D (1984) Cayley’s problem and Julia sets. Math Intell 6:11–20zbMATHGoogle Scholar
  42. Halsey TC, Jensen MH, Kadanoff LP, Procaccia Shraiman I (1986) Fractal measures and their singularities: the characterization of strange sets. Phys Rev A 33:1141–1151ADSzbMATHMathSciNetGoogle Scholar
  43. Harman PM (1998) The natural philosophy of James Clerk Maxwell. Cambridge University Press, CambridgezbMATHGoogle Scholar
  44. Hofstadter DR (1985) Metamathematical themas: questing for the essence of mind and pattern. Penguin, LondonGoogle Scholar
  45. Holmes P (2005) Ninety plus years of nonlinear dynamics: more is different and less is more. Int J Bifurcation Chaos Appl Sci Eng 15(9):2703–2716zbMATHGoogle Scholar
  46. Hurewicz W, Wallman H (1941) Dimension theory. Princeton University Press, PrincetonGoogle Scholar
  47. Hutchinson JE (1981) Fractals and self similarity. Indiana Math J 30:713–747zbMATHMathSciNetGoogle Scholar
  48. Julia G (1918) Sur l’iteration des functions rationnelles. J Math Pures Appl 8:47–245Google Scholar
  49. Kazim Z (2002) The Hausdorff and box dimension of fractals with disjoint projections in two dimensions. Glasg Math J 44:117–123zbMATHMathSciNetGoogle Scholar
  50. Lapidus ML, van Frankenhuysen M (2000) Fractal geometry and number theory: complex dimensions of fractal strings and zeros of zeta functions. Birkhäuser, BostonGoogle Scholar
  51. Lapidus ML, van Frankenhuijsen M (2004) Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Parts, 1, 2. American Mathematical Society, ProvidenceGoogle Scholar
  52. Li YC (2007) On the true nature of turbulence. Math Intell 29(1):45–48Google Scholar
  53. Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82(10):985–992zbMATHMathSciNetGoogle Scholar
  54. Lipscomb SL (2009a) The quest for universal spaces in dimension theory. Not Am Math Soc 56(11):1418–1424zbMATHMathSciNetGoogle Scholar
  55. Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141ADSGoogle Scholar
  56. Lorenz EN (1993) The essence of chaos. University of Washington Press, SeattlezbMATHGoogle Scholar
  57. Mandelbrot BB (1975) Les objets fractals, forme, hazard et dimension. Flammarion, ParisGoogle Scholar
  58. Mandelbrot BB (1980) Fractal aspects of the iteration of z → λ(1 − z) for complex λ, z. Ann N Y Acad Sci 357:249–259ADSGoogle Scholar
  59. Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San FranciscozbMATHGoogle Scholar
  60. Mandelbrot BB (2002) A maverick’s apprenticeship. Imperial College Press, LondonGoogle Scholar
  61. Mandelbrot BB, Hudson RL (2004) The (mis)behavior of markets: a fractal view of risk, ruin, and reward. Basic Books, New YorkGoogle Scholar
  62. Mattila P (1995) Geometry of sets and measures in Euclidean spaces. Cambridge University Press, CambridgezbMATHGoogle Scholar
  63. Mauldin RD, Williams SC (1986) On the Hausdorff dimension of some graphs. Trans Am Math Soc 298:793–803zbMATHMathSciNetGoogle Scholar
  64. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467ADSGoogle Scholar
  65. May RM (1987) Chaos and the dynamics of biological populations. Proc R Soc Ser A 413(1844):27–44ADSzbMATHGoogle Scholar
  66. May RM (2001) Stability and complexity in model ecosystems, 2nd edn, with new introduction. Princeton University Press, PrincetonGoogle Scholar
  67. McMurran S, Tattersall J (1999) Mary Cartwright 1900–1998. Not Am Math Soc 46(2):214–220zbMATHMathSciNetGoogle Scholar
  68. Menger K (1943) What is dimension? Am Math Mon 50:2–7zbMATHMathSciNetGoogle Scholar
  69. Metropolis N, Stein ML, Stein P (1973) On finite limit sets for transformations on the unit interval. J Comb Theory 15:25–44zbMATHMathSciNetGoogle Scholar
  70. Morse M (1946) George David Birkhoff and his mathematical work. Bull Am Math Soc 52(5, Part 1):357–391zbMATHMathSciNetGoogle Scholar
  71. Nese JM, Dutton JA, Wells R (1987) Calculated attractor dimensions for low-order spectral models. J Atmos Sci 44(15):1950–1972ADSMathSciNetGoogle Scholar
  72. Ott E, Sauer T, Yorke JA (1994) Coping with chaos. Wiley Interscience, New YorkzbMATHGoogle Scholar
  73. Peitgen H-O, Richter PH (1986) The beauty of fractals. Springer, BerlinzbMATHGoogle Scholar
  74. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New YorkGoogle Scholar
  75. Pesin YB (1997) Dimension theory in dynamical systems: contemporary views and applications. University of Chicago Press, ChicagoGoogle Scholar
  76. Poincaré H (1903) L’Espace et ses trois dimensions. Rev Metaphys Morale 11:281–301Google Scholar
  77. Poincaré H, Halsted GB (tr) (1946) Science and method. In: Cattell JM (ed) Foundations of science. The Science Press, LancasterGoogle Scholar
  78. Przytycki F, Urbañski M (1989) On Hausdorff dimension of some fractal sets. Stud Math 93:155–186zbMATHGoogle Scholar
  79. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398ADSGoogle Scholar
  80. Rényi A (1970) Probability theory. North Holland, AmsterdamGoogle Scholar
  81. Richardson LF (1993) Collected papers, 2 vols. Cambridge University Press, CambridgeGoogle Scholar
  82. Ruelle D (1980) Strange attractors. Math Intell 2:126–137zbMATHMathSciNetGoogle Scholar
  83. Ruelle D (2006) What is a strange attractor? Not Am Math Soc 53(7):764–765zbMATHMathSciNetGoogle Scholar
  84. Ruelle D, Takens F (1971) On the nature of turbulence. Commun Math Phys 20:167–192; 23:343–344ADSzbMATHMathSciNetGoogle Scholar
  85. Russell DA, Hanson JD, Ott E (1980) Dimension of strange attractors. Phys Rev Lett 45:1175–1178ADSMathSciNetGoogle Scholar
  86. Shaw R (1984) The dripping faucet as a model chaotic system. Aerial Press, Santa CruzGoogle Scholar
  87. Siegel CL (1942) Iteration of analytic functions. Ann Math 43:607–612zbMATHGoogle Scholar
  88. Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817zbMATHMathSciNetGoogle Scholar
  89. Smale S (1980) The mathematics of time: essays on dynamical systems, economic processes, and related topics. Springer, New YorkzbMATHGoogle Scholar
  90. Smale S (1998) Chaos: finding a horseshoe on the beaches of Rio. Math Intell 20:39–44zbMATHMathSciNetGoogle Scholar
  91. Sparrow C (1982) The Lorenz equations: bifurcations, chaos, and strange attractors. Springer, New YorkzbMATHGoogle Scholar
  92. Sprott JC (2003) Chaos and time-series analysis. Oxford University Press, OxfordzbMATHGoogle Scholar
  93. Takens F (1980) Detecting strange attractors in turbulence. In: Rand DA, Young L-S (eds) Dynamical systems and turbulence, vol 898, Springer lecture notes in mathematics. Springer, New York, pp 366–381Google Scholar
  94. Theiler J (1990) Estimating fractal dimensions. J Opt Soc Am A 7(6):1055–1073ADSMathSciNetGoogle Scholar
  95. Tricot C (1982) Two definitions of fractional dimension. Math Proc Camb Philos Soc 91:57–74zbMATHMathSciNetGoogle Scholar
  96. Tucker W (1999) The Lorenz Attractor exists. C R Acad Sci Paris Sér 1 Math 328:1197–1202zbMATHGoogle Scholar
  97. Viana M (2000) What’s new on Lorenz strange attractors. Math Intell 22(3):6–19zbMATHMathSciNetGoogle Scholar
  98. Winfree AT (1983) Sudden cardiac death – a problem for topology. Sci Am 248:118–131Google Scholar
  99. Zhang S-Y (compiler) (1991) Bibliography on chaos. World Scientific, SingaporeGoogle Scholar

Books and Reviews

  1. Abraham RH, Shaw CD (1992) Dynamics, the geometry of behavior. Addison-Wesley, Redwood CityzbMATHGoogle Scholar
  2. Barnsley MF (2006) SuperFractals. Cambridge University Press, Cambridge, UKzbMATHGoogle Scholar
  3. Barnsley MF, Devaney R, Mandelbrot BB, Peitgen H-O, Saupe D, Voss R (1988) The science of fractal images. Springer, BerlinzbMATHGoogle Scholar
  4. Barnsley MF, Rising H (1993) Fractals everywhere. Academic, BostonzbMATHGoogle Scholar
  5. Çambel AB (1993) Applied complexity theory: a paradigm for complexity. Academic, San DiegozbMATHGoogle Scholar
  6. Crilly AJ, Earnshaw RA, Jones H (eds) (1991) Fractals and chaos. Springer, BerlinGoogle Scholar
  7. Cvitanovic P (1989) Universality in chaos, 2nd edn. Adam Hilger, BristolzbMATHGoogle Scholar
  8. Elliott EW, Kiel LD (eds) (1997) Chaos theory in the social sciences: foundations and applications. University of Michigan Press, Ann ArborGoogle Scholar
  9. Gilmore R, Lefranc M (2002) The topology of chaos: Alice in stretch and squeezeland. Wiley Interscience, New YorkGoogle Scholar
  10. Gilmore RG, Letellier C (2007) The symmetry of chaos. Oxford University Press, Oxford, UKzbMATHGoogle Scholar
  11. Glass L, MacKey MM (1988) From clocks to chaos. Princeton University Press, PrincetonzbMATHGoogle Scholar
  12. Holden A (ed) (1986) Chaos. Manchester University Press, ManchesterzbMATHGoogle Scholar
  13. Kautz R (2011) Chaos: the science of predictable random motion. Oxford University Press, Oxford, UKGoogle Scholar
  14. Kellert SH (1993) In the wake of chaos. University of Chicago Press, ChicagozbMATHGoogle Scholar
  15. Lauwerier H (1991) Fractals: endlessly repeated geometrical figures. Princeton University Press, PrincetonzbMATHGoogle Scholar
  16. Lipscomb SL (2009b) Fractals and universal spaces in dimension theory. Springer, New YorkzbMATHGoogle Scholar
  17. Mullin T (ed) (1994) The nature of chaos. Oxford University Press, OxfordGoogle Scholar
  18. Ott E (2002) Chaos in dynamical systems. Cambridge University Press, CambridgezbMATHGoogle Scholar
  19. Parker B (1996) Chaos in the cosmos: the stunning complexity of the universe. Plenum, New York/LondonGoogle Scholar
  20. Peitgen H-O, Jürgens H, Saupe D, Zahlten C (1990) Fractals: an animated discussion with Edward Lorenz and Benoit Mandelbrot. A VHS film in color (63 mins). Freeman, New YorkGoogle Scholar
  21. Prigogine I, Stengers I (1985) Order out of chaos: man’s new dialogue with nature. Fontana, LondonGoogle Scholar
  22. Ruelle D (1993) Chance and chaos. Penguin, LondonGoogle Scholar
  23. Schroeder M (1991) Fractals, chaos, power laws. Freeman, New YorkzbMATHGoogle Scholar
  24. Smith P (1998) Explaining chaos. Cambridge University Press, Cambridge, UKzbMATHGoogle Scholar
  25. Strogatz SH (2000) Nonlinear dynamics and chaos. Perseus, New YorkGoogle Scholar
  26. Tél T, Gruiz M (2006) Chaotic dynamics: an introduction based on classical mechanics. Cambridge University Press, Cambridge, UKGoogle Scholar
  27. Thompson JMT, Stewart HB (2002) Nonlinear dynamics and chaos. Wiley, ChichesterzbMATHGoogle Scholar
  28. Wolfram S (2002) A new kind of science. Wolfram Media, ChampaignzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.MathematicsMiddlesex UniversityLondonUK