Skip to main content
SpringerLink
Log in
Book cover

Mathematics of Complexity and Dynamical Systems pp 537–556Cite as

Fractals Meet Chaos

  • Reference work entry

  • 2479 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Dynamical Systems

Curves and Dimension

Chaos Comes of Age

The Advent of Fractals

The Merger

Future Directions

Bibliography

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Abbreviations

Dimension :

The traditional meaning of dimension in modern mathematics is “topological dimension ” and is an extension of the classical Greek meaning. In modern concepts of this dimension can be defined in terms of a separable metric space. For the practicalities of Fractals and Chaos the notion of dimension can be limited to subsets of Euclidean n‑space where n is an integer. The newly arrived “fractal dimension ” is metrically based and can take on fractional values. Just as for topological dimension.

As for topological dimension itself there is a profusion of different (but related) concepts of metrical dimension. These are widely used in the study of fractals, the ones of principal interest being:

  • Hausdorff dimension (more fully, Hausdorff–Besicovitch dimension),

  • Box dimension (often referred to as Minkowski–Bouligand dimension),

  • Correlation dimension (due to A. Rényi, P. Grassberger and I. Procaccia).

Other types of metric dimension are also possible. There is “divider dimension” (based on ideas of an English mathematician/meteorologist L. F. Richardson in the 1920s); the “Kaplan–Yorke dimension” (1979) derived from Lyapunov exponents, known also as the “Lyapunov dimension ”; “packing dimension” introduced by Tricot (1982). In addition there is an overall general dimension due to A. Rényi (1970) which admits box dimension, correlation dimension and information dimension as special cases. With many of the concepts of dimension there are upper and lower refinements, for example, the separate upper and lower box dimensions. Key references to the vast (and highly technical) subject of mathematical dimension include [31,32,33,60,73,92,93].

Hausdorff dimension :

(Hausdorff–Besicovitch dimension). In the study of fractals, the most sophisticated concept of dimension is Hausdorff dimension, developed in the 1920s.

The following definition of Hausdorff dimension is given for a subset A of the real number line. This is readily generalized to subsets of the plane, Euclidean 3-space and Euclidean n‑space, and more abstractly to separable metric spaces by taking neighborhoods as disks instead of intervals. Let \({\{U_i \}}\) be an r‑covering of A, (a covering of A where the width of all intervals U i , satisfies \({w(U_i)\le r}\)). The measure m r is defined by

$$ m_r (A)=\inf \left(\sum\limits_{i=1}^\infty w(U_i) \right)\:, $$

where the infimum (or greatest of the minimum values) is taken over all r‑coverings of A.

The Hausdorff dimension \({D_\mathrm{H}}\) of A is:

$$ D_\mathrm{H} = \lim\limits_{r\to 0} m_r(A)\:, $$

provided the limit exists. The subset \( E=\{1/n\colon n=1,2,,3,\ldots\}\) of the unit interval has \({D_{\text{H}}=0}\) (the Hausdorff dimension of a countable set is always zero). The Hausdorff dimension is the basis of “fractal dimension” but because it takes into account intervals of unequal widths it may be difficult to calculate in practice.

Box dimension :

(or Minkowski–Bouligand dimension, known also as capacity dimension, cover dimension, grid dimension). The box counting dimension is a more direct and practical method for computing dimension in the case of fractals. To define it, we again confine our attention to the real number line in the knowledge that box dimension is readily extended to subsets of more general spaces.

As before, let \({\{U_i \}}\) be an r‑covering of A, and let \({N_r(A)}\) be the least number of sets in such a covering. The box dimension \({D_\mathrm{B}}\) of A is defined by:

$$ D_\mathrm{B} =\lim\limits_{r\to 0} \frac{\log N_r(A)}{\log 1/r}\:. $$

The box dimension of the subset \( E=\{1/n\colon n=1,2,3,\ldots\}\) of the unit interval can be calculated to give \({D_\mathrm{B}=0.5}\).

In general Hausdorff and Box dimensions are related to each other by the inequality \({D_\mathrm{B} \ge D_\mathrm{H}}\), as happens in the above example. The relationship between \({D_\mathrm{H}}\) and \({D_\mathrm{B}}\) is investigated in [49]. For compact, self‐similar fractal sets \({D_\mathrm{B}=D_\mathrm{H}}\) but there are fractal sets for which \({D_\mathrm{B} > D_\mathrm{H}}\) [76]. Though Hausdorff dimension and Box dimension have similar properties, Box dimension is only finitely additive, while Hausdorff dimension is countably additive.

Correlation dimension :

For a set of n points A on the real number line, let \({P(r)}\) be the probability that two different points of A chosen at random are closer than r apart. For a large number of points n, the graph of \({v=\log P(r)}\) against \({u=\log r}\) is approximated to its slope for small values of r, and theoretically, a straight line. The correlation dimension \({D_\mathrm{C}}\) is defined as its slope for small values of r, that is,

$$ D_\mathrm{C} = \lim\limits_{r\to 0} \frac{\text{d} v}{\text{d} u}\:. $$

The correlation dimension involves the separation of points into “boxes”, whereas the box dimension merely counts the boxes that cover A.

If \({P_i}\) is the probability of a point of A being in box i (approximately \({{n_i}/n}\) where n i is the number of points in box i and n is the totality of points in A) then an alternative definition of correlation dimension is

$$ D_\mathrm{C} = \lim\limits_{r\to 0} \frac{\log \sum\limits_i {P_i^2} } {\log r}\:. $$
Attractor :

A point set in phase space which “attracts” trajectories in its vicinity. More formally, a bounded set A in phase space is called an attractor for the solution \({x(t)}\) of a differential equation if

  • \({x(0)\in A\Rightarrow x(t)\in A\enskip for\;all\;\enskip t}\). Thus, an attractor is invariant under the dynamics (trajectories which start in A remain in A).

  • There is a neighborhood \({U\supset A}\) such that any trajectory starting in U is attracted to A (the trajectory gets closer and closer to A).

  • If \({B\subset A}\) and if B satisfies the above two properties then \({B=A}\).

An attractor is therefore the minimal set of points A which attracts all orbits starting at some point in a neighborhood of A.

Orbit:

is a sequence of points \({\{x_i\}=x_0 ,x_1 ,x_2 ,\ldots}\) defined by an iteration \({x_n =f^n(x_0)}\). If n is a positive it is called a forwards orbit, and if n is negative a backwards orbit. If \({x_0 = x_n}\) for some finite value n, the orbit is periodic. In this case, the smallest value of n for which this is true is called the period of the orbit.

For an invertible function f, a point x is homoclinic to a if

$$ \lim f^n(x)=\lim f^{-n}(x)=a\quad {\text{as}}\enskip n\to \infty\:. $$

and in this case the orbit \({\{f^n(x_0)\}}\) is called a homoclinic orbit – the orbit which converges to the same saddle point a forwards or backwards. This term was introduced by Poincaré.

The terminology “orbit” may be regarded as applied to the solution of a difference equation, in a similar way the solution of a differential equation \({x(t)}\) is termed a trajectory. Orbit is the term used for discrete dynamical system and trajectory for the continuous time case.

Basin of attraction :

If a is an attractive fixed point of a function f, its basin of attraction \({B(a)}\) is the subset of points defined by

$$ B(a)=\{x\colon f^k(x)\to a\,,\;\text{as}\; k\to\infty\}\:. $$

It is the subset containing all the initial points of orbits attracted to a. The basins of attraction may have a complicated structure. An important example applies to the case where a is a point in the complex plane C.

Julia set :

A set J f is the boundary between the basins of attraction of a function f. For example, in the case where \({z=\pm 1}\) are attracting points (solutions of \({z^2-1=0}\)), the Julia set of the “Newton–Fourier” function \({f(z)=z-((z^2-1)/2z)}\) is the set of complex numbers which lie along the imaginary axis \({x = 0}\) (as proved by Schröder and Cayley in the 1870s). The case of the Julia set involved with the solutions of \({z^3-1=0}\) was beyond these pioneers and is fractal in nature. An alternative definition for a Julia set is the closure of the subset of the complex plane whose orbits of f tend to infinity.

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 303–312

    Google Scholar 

  2. Abraham RH, Ueda Y (eds) (2000) The Chaos Avant-Garde: memories of the early days of chaos theory. World Scientific, River Edge

    MATH  Google Scholar 

  3. Alexander DS (1994) A History of Complex Dynamics; from Schröder to Fatou and Julia. Vieweg, Braunschweig

    MATH  Google Scholar 

  4. Aubin D, Dahan Dalmedico A (2002) Writing the History of Dynamical Systems and Chaos: Longue Durée and revolution, Disciplines and Cultures. Historia Mathematica 29(3):235–362

    MathSciNet  Google Scholar 

  5. Baker GL, Gollub JP (1996) Chaotic Dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  6. Barrow‐Green J (1997) Poincaré and the three body problem. American Mathematical Society, Providence; London Mathematical Society, London

    Google Scholar 

  7. Barrow‐Green J (2005) Henri Poincaré, Memoir on the Three-Body Problem (1890). In: Grattan‐Guinness I (ed) Landmark Writings in Western Mathematics 1640–1940. Elsevier, Amsterdam, pp 627–638

    Google Scholar 

  8. Batterson S (2000) Stephen Smale: the mathematician who broke the dimension barrier. American Mathematical Society, Providence

    MATH  Google Scholar 

  9. Berry MV, Percival IC, Weiss NO (1987) Dynamical Chaos. Proc Royal Soc London 413(1844):1–199

    Google Scholar 

  10. Birkhoff GD (1927) Dynamical Systems. American Mathematical Society, New York

    MATH  Google Scholar 

  11. Birkhoff GD (1941) Some unsolved problems of theoretical dynamics. Science 94:598–600

    MathSciNet  MATH  Google 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 142–155

    Google Scholar 

  13. Briggs J (1992) Fractals, the patterns of chaos: discovering a new aesthetic of art, science, and nature. Thames and Hudson, London

    Google 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 51:28–38

    MathSciNet  MATH  Google Scholar 

  15. Brolin H (1965) Invariant Sets Under Iteration of Rational Functions. Ark Mat 6:103–144

    MathSciNet  MATH  Google Scholar 

  16. Campbell DK, Ecke R, Hyman JM (eds) (1992) Nonlinear science: the next decade. MIT Press, Cambridge

    MATH  Google Scholar 

  17. Cayley A (1879) The Newton–Fourier Imaginary Problem. Am J Math 2:97

    MathSciNet  Google Scholar 

  18. Chabert J-L (1990) Un demi‐siècle de fractales: 1870–1920. Historia Mathematica 17:339–365

    MathSciNet  MATH  Google Scholar 

  19. Crilly AJ, Earnshaw RA, Jones H (eds) (1993) Applications of fractals and chaos: the shape of things. Springer, Berlin

    MATH  Google Scholar 

  20. Crilly T (1999) The emergence of topological dimension theory. In: James IM (ed) History of Topology. Elsevier, Amsterdam, pp 1–24

    Google Scholar 

  21. Crilly T (2006) Arthur Cayley: Mathematician Laureate of the Victorian Age. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  22. Crilly T, Moran A (2002) Commentary on Menger’s Work on Curve Theory and Topology. In: Schweizer, B. et al Karl Menger, Selecta. Springer, Vienna

    Google Scholar 

  23. Curry J, Garnett L, Sullivan D (1983) On the iteration of rational functions: Computer experiments with Newton’s method. Commun Math Phys 91:267–277

    MathSciNet  MATH  Google Scholar 

  24. Dahan‐Dalmedico A, Gouzevitch I (2004) Early developments in nonlinear science in Soviet Russia: The Andronov School at Gor’kiy. Sci Context 17:235–265

    Google Scholar 

  25. Devaney RL (1984) Julia sets and bifurcation diagrams for exponential maps. Bull Am Math Soc 11:167–171

    MathSciNet  MATH  Google Scholar 

  26. Devaney RL (2003) An Introduction to Chaotic Dynamical Systems, 2nd edn. Westview Press, Boulder

    MATH  Google Scholar 

  27. Diacu F, Holmes P (1996) Celestial Encounters: the Origins of Chaos and Stability. Princeton University Press, Princeton

    MATH  Google Scholar 

  28. Dieudonné J (1960) Foundations of Modern Analysis. Academic Press, New York

    Google Scholar 

  29. Douady A, Hubbard J (1982) Iteration des polynomes quadratiques complexes. Comptes Rendus Acad Sci Paris Sér 1 Math 294:123–126

    MathSciNet  Google Scholar 

  30. Dyson F (1997) ‘Nature’s Numbers’ by Ian Stewart. Math Intell 19(2):65–67

    Google Scholar 

  31. Edgar G (1993) Classics on Fractals. Addison‐Wesley, Reading

    MATH  Google Scholar 

  32. Falconer K (1990) Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester

    MATH  Google Scholar 

  33. Falconer K (1997) Techniques in Fractal Geometry. Wiley, Chichester

    MATH  Google Scholar 

  34. Farmer JD, Ott E, Yorke JA (1983) The Dimension of Chaotic Attractors. Physica D 7:153–180

    MathSciNet  Google Scholar 

  35. Fatou P (1919/20) Sur les equations fonctionelles. Bull Soc Math France 47:161–271; 48:33–94, 208–314

    Google Scholar 

  36. Feigenbaum MJ (1978) Quantitative Universality for a Class of Nonlinear Transformations. J Stat Phys 19:25–52

    MathSciNet  MATH  Google Scholar 

  37. Feigenbaum MJ (1979) The Universal Metric Properties of Nonlinear Transformations. J Stat Phys 21:669–706

    MathSciNet  MATH  Google Scholar 

  38. Gleick J (1988) Chaos. Sphere Books, London

    Google Scholar 

  39. Grassberger P (1983) Generalised dimensions of strange attractors. Phys Lett A 97:227–230

    MathSciNet  Google Scholar 

  40. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208

    MathSciNet  MATH  Google Scholar 

  41. Hénon M (1976) A Two dimensional Mapping with a Strange Attractor. Commun Math Phys 50:69–77

    Google Scholar 

  42. Haeseler F, Peitgen H-O, Saupe D (1984) Cayley’s Problem and Julia Sets. Math Intell 6:11–20

    MATH  Google Scholar 

  43. Halsey TC, Jensen MH, Kadanoff LP, Procaccia Shraiman I (1986) Fractal measures and their singularities: the characterization of strange sets. Phys Rev A 33:1141–1151

    MathSciNet  MATH  Google Scholar 

  44. Harman PM (1998) The Natural Philosophy of James Clerk Maxwell. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  45. Hofstadter DR (1985) Metamathematical Themas: questing for the essence of mind and pattern. Penguin, London

    Google Scholar 

  46. Holmes P (2005) Ninety plus years of nonlinear dynamics: More is different and less is more. Int J Bifurc Chaos Appl Sci Eng 15(9):2703–2716

    MATH  Google Scholar 

  47. Hurewicz W, Wallman H (1941) Dimension Theory. Princeton University Press, Princeton

    Google Scholar 

  48. Julia G (1918) Sur l’iteration des functions rationnelles. J Math Pures Appl 8:47–245

    Google Scholar 

  49. Kazim Z (2002) The Hausdorff and box dimension of fractals with disjoint projections in two dimensions. Glasgow Math J 44:117–123

    MathSciNet  MATH  Google Scholar 

  50. Lapidus ML, van Frankenhuijsen M (2004) Fractal geometry and Applications: A Jubilee of Benoît Mandelbrot, Parts, 1, 2. American Mathematical Society, Providence

    Google Scholar 

  51. Lapidus ML, van Frankenhuysen (2000) Fractal geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions. Birkhäuser, Boston

    MATH  Google Scholar 

  52. Li YC (2007) On the True Nature of Turbulence. Math Intell 29(1):45–48

    Google Scholar 

  53. Lorenz EN (1963) Deterministic Nonperiodic Flow. J Atmospheric Sci 20:130–141

    Google Scholar 

  54. Lorenz EN (1993) The Essence of Chaos. University of Washington Press, Seattle

    MATH  Google Scholar 

  55. Mandelbrot BB (1975) Les objets fractals, forme, hazard et dimension. Flammarion, Paris

    Google Scholar 

  56. Mandelbrot BB (1980) Fractal aspects of the iteration of \({z\to \lambda (1-z)}\) for complex \({\lambda ,z}\). Ann New York Acad Sci 357:249–259

    Google Scholar 

  57. Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco

    MATH  Google Scholar 

  58. Mandelbrot BB (2002) A maverick’s apprenticeship. Imperial College Press, London

    Google Scholar 

  59. Mandelbrot BB, Hudson RL (2004) The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. Basic Books, New York

    MATH  Google Scholar 

  60. Mattila P (1995) Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  61. Mauldin RD, Williams SC (1986) On the Hausdorff dimension of some graphs. Trans Am Math Soc 298:793–803

    MathSciNet  MATH  Google Scholar 

  62. May RM (1976) Simple Mathematical models with very complicated dynamics. Nature 261:459–467

    Google Scholar 

  63. May RM (1987) Chaos and the dynamics of biological populations. Proc Royal Soc Ser A 413(1844):27–44

    MATH  Google Scholar 

  64. May RM (2001) Stability and Complexity in Model Ecosystems, 2nd edn, with new introduction. Princeton University Press, Princeton

    Google Scholar 

  65. McMurran S, Tattersall J (1999) Mary Cartwright 1900–1998. Notices Am Math Soc 46(2):214–220

    MathSciNet  MATH  Google Scholar 

  66. Menger K (1943) What is dimension? Am Math Mon 50:2–7

    MathSciNet  MATH  Google Scholar 

  67. Metropolis, Stein ML, Stein P (1973) On finite limit sets for transformations on the unit interval. J Comb Theory 15:25–44

    MathSciNet  MATH  Google Scholar 

  68. Morse M (1946) George David Birkhoff and his mathematical work. Bull Am Math Soc 52(5, Part 1):357–391

    MathSciNet  MATH  Google Scholar 

  69. Nese JM, Dutton JA, Wells R (1987) Calculated Attractor Dimensions for low-Order Spectral Models. J Atmospheric Sci 44(15):1950–1972

    MathSciNet  Google Scholar 

  70. Ott E, Sauer T, Yorke JA (1994) Coping with Chaos. Wiley Interscience, New York

    MATH  Google Scholar 

  71. Peitgen H-O, Richter PH (1986) The Beauty of Fractals. Springer, Berlin

    MATH  Google Scholar 

  72. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New York

    Google Scholar 

  73. Pesin YB (1997) Dimension Theory in Dynamical Systems: contemporary views and applications. University of Chicago Press, Chicago

    Google Scholar 

  74. Poincaré H (1903) L’Espace et ses trois dimensions. Rev Métaphys Morale 11:281–301

    Google Scholar 

  75. Poincaré H, Halsted GB (tr) (1946) Science and Method. In: Cattell JM (ed) Foundations of Science. The Science Press, Lancaster

    Google Scholar 

  76. Przytycki F, Urbañski M (1989) On Hausdorff dimension of some fractal sets. Studia Mathematica 93:155–186

    Google Scholar 

  77. Rössler OE (1976) An Equation for Continuous Chaos. Phys Lett A 57:397–398

    Google Scholar 

  78. Rényi A (1970) Probability Theory. North Holland, Amsterdam

    Google Scholar 

  79. Richardson LF (1993) Collected Papers, 2 vols. Cambridge University Press, Cambridge

    Google Scholar 

  80. Ruelle D (1980) Strange Attractors. Math Intell 2:126–137

    MathSciNet  MATH  Google Scholar 

  81. Ruelle D (2006) What is a Strange Attractor? Notices Am Math Soc 53(7):764–765

    MathSciNet  MATH  Google Scholar 

  82. Ruelle D, Takens F (1971) On the nature of turbulence. Commun Math Phys 20:167–192; 23:343–344

    MathSciNet  MATH  Google Scholar 

  83. Russell DA, Hanson JD, Ott E (1980) Dimension of strange attractors. Phys Rev Lett 45:1175–1178

    MathSciNet  Google Scholar 

  84. Shaw R (1984) The Dripping Faucet as a Model Chaotic System. Aerial Press, Santa Cruz

    Google Scholar 

  85. Siegel CL (1942) Iteration of Analytic Functions. Ann Math 43:607–612

    MATH  Google Scholar 

  86. Smale S (1967) Differentiable Dynamical Systems. Bull Am Math Soc 73:747–817

    MathSciNet  MATH  Google Scholar 

  87. Smale S (1980) The Mathematics of Time: essays on dynamical systems, economic processes, and related topics. Springer, New York

    MATH  Google Scholar 

  88. Smale S (1998) Chaos: Finding a horseshoe on the Beaches of Rio. Math Intell 20:39–44

    MathSciNet  MATH  Google Scholar 

  89. Sparrow C (1982) The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York

    MATH  Google Scholar 

  90. Sprott JC (2003) Chaos and Time‐Series Analysis. Oxford University Press, Oxford

    MATH  Google Scholar 

  91. Takens F (1980) Detecting Strange Attractors in Turbulence. In: Rand DA, Young L-S (eds) Dynamical Systems and Turbulence. Springer Lecture Notes in Mathematics, vol 898. Springer, New York, pp 366–381

    Google Scholar 

  92. Theiler J (1990) Estimating fractal dimensions. J Opt Soc Am A 7(6):1055–1073

    MathSciNet  Google Scholar 

  93. Tricot C (1982) Two definitions of fractional dimension. Math Proc Cambridge Philos Soc 91:57–74

    MathSciNet  MATH  Google Scholar 

  94. Tucker W (1999) The Lorenz Attractor exists. Comptes Rendus Acad Sci Paris Sér 1, Mathematique 328:1197–1202

    Google Scholar 

  95. Viana M (2000) What’s New on Lorenz Strange Attractors. Math Intell 22(3):6–19

    MathSciNet  MATH  Google Scholar 

  96. Winfree AT (1983) Sudden Cardiac death- a problem for topology. Sci Am 248:118–131

    Google Scholar 

  97. Zhang S-Y (compiler) (1991) Bibliography on Chaos. World Scientific, Singapore

    Google Scholar 

Books and Reviews

  1. Abraham RH, Shaw CD (1992) Dynamics, The Geometry of Behavior. Addison‐Wesley, Redwood City

    MATH  Google Scholar 

  2. Barnsley MF, Devaney R, Mandelbrot BB, Peitgen H-O, Saupe D, Voss R (1988) The Science of Fractal Images. Springer, Berlin

    MATH  Google Scholar 

  3. Barnsley MF, Rising H (1993) Fractals Everywhere. Academic Press, Boston

    MATH  Google Scholar 

  4. Çambel AB (1993) Applied Complexity Theory: A Paradigm for Complexity. Academic Press, San Diego

    Google Scholar 

  5. Crilly AJ, Earnshaw RA, Jones H (eds) (1991) Fractals and Chaos. Springer, Berlin

    Google Scholar 

  6. Cvitanovic P (1989) Universality in Chaos, 2nd edn. Adam Hilger, Bristol

    MATH  Google Scholar 

  7. Elliott EW, Kiel LD (eds) (1997) Chaos Theory in the Social Sciences: foundations and applications. University of Michigan Press, Ann Arbor

    Google Scholar 

  8. Gilmore R, Lefranc M (2002) The Topology of Chaos: Alice in Stretch and Squeezeland. Wiley Interscience, New York

    MATH  Google Scholar 

  9. Glass L, MacKey MM (1988) From Clocks to Chaos. Princeton University Press, Princeton

    MATH  Google Scholar 

  10. Holden A (ed) (1986) Chaos. Manchester University Press, Manchester

    MATH  Google Scholar 

  11. Kellert SH (1993) In the Wake of Chaos. University of Chicago Press, Chicago

    MATH  Google Scholar 

  12. Lauwerier H (1991) Fractals: Endlessly Repeated Geometrical Figures. Princeton University Press, Princeton

    MATH  Google Scholar 

  13. Mullin T (ed) (1994) The Nature of Chaos. Oxford University Press, Oxford

    Google Scholar 

  14. Ott E (2002) Chaos in Dynamical Systems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  15. Parker B (1996) Chaos in the Cosmos: The Stunning Complexity of the Universe. Plenum, New York, London

    Google Scholar 

  16. Peitgen H-O, Jürgens, Saupe D, Zahlten C (1990) Fractals: An animated discussion with Edward Lorenz and Benoit Mandelbrot. A VHS film in color (63 mins). Freeman, New York

    Google Scholar 

  17. Prigogine I, Stengers I (1985) Order out of Chaos: man’s new dialogue with Nature. Fontana, London

    Google Scholar 

  18. Ruelle D (1993) Chance and Chaos. Penguin, London

    Google Scholar 

  19. Schroeder M (1991) Fractals, Chaos, Power Laws. Freeman, New York

    MATH  Google Scholar 

  20. Smith P (1998) Explaining Chaos. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  21. Thompson JMT, Stewart HB (1988) Nonlinear Dynamics and Chaos. Wiley, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Middlesex University, London, UK

    Tony Crilly

Authors
  1. Tony Crilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Crilly, T. (2012). Fractals Meet Chaos. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_34

Download citation

Publish with us

Policies and ethics

Search

Navigation