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In mathematics, a chaotic map is a map (= evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.

Chaotic maps often generate fractals. Although a fractal may be constructed by an iterative procedure, some fractals are studied in and of themselves, as sets rather than in terms of the map that generates them. This is often because there are several different iterative procedures to generate the same fractal.

List of chaotic maps

Map Time domain Space domain Number of space dimensions Number of parameters Also known as
3-cells CNN system continuous real 3
2D circular chaotic map[1] discrete real 2 1
2D Lorenz system [2] discrete real 2 1
2D Rational chaotic map [3] discrete rational 2 2
Van der Pol system [4] continuous real 2 2
ACT chaotic attractor [5] continuous real 3
Aizawa chaotic attractor [6] continuous real 3
Arneodo chaotic system[7] continuous real 3
Arnold's cat map discrete real 2 0
Baker's map discrete real 2 0
Basin chaotic map[8] discrete real 2
Beta Chaotic Map [9] 12
Bogdanov map discrete real 2 3
Brusselator continuous real 3
Burke-Shaw chaotic attractor[10] continuous real 3 2
Chen chaotic attractor[11] continuous real 3
Chen-Celikovsky system [12] continuous real 3
Chen-Lee system continuous real 3
Chossat-Golubitsky symmetry map
Chua circuit[13] continuous real 3 3
Circle map discrete real 1 2
Complex quadratic map discrete complex 1 1 gives rise to the Mandelbrot set
Complex squaring map discrete complex 1 0 acts on the Julia set for the squaring map.
Complex cubic map discrete complex 1 2
Clifford fractal map[14] discrete real 2 4
Degenerate Double Rotor map
De Jong fractal map[15] discrete real 2 4
Delayed-Logistic system[16] discrete real 2 1
Double rotor map
Duffing map discrete real 2 2 Holmes chaotic map
Duffing equation continuous real 2 5 (3 independent)
Dyadic transformation discrete real 1 0 2x mod 1 map, Bernoulli map, doubling map, sawtooth map
Exponential map discrete complex 2 1
Feigenbaum strange nonchaotic map[17] discrete real 3
Finance system[18] continuous real 3
Folded-Towel hyperchaotic map[19] continuous real 3
Fractal-Dream system[20] discrete real 2
Gauss map discrete real 1 mouse map, Gaussian map
Generalized Baker map
Genesio-Tesi chaotic attractor[21] continuous real 3
Gingerbreadman map[22] discrete real 2
Grinch dragon fractal discrete real 2
Gumowski/Mira map[23] discrete real 2
Hadley chaotic circulation continuous real 3 0
Half-inverted Rössler attractor[24]
Halvorsen chaotic attractor[25] continuous real 3
Hénon map discrete real 2 2
Hénon with 5th order polynomial
Hindmarsh-Rose neuronal model continuous real 3 8
Hitzl-Zele map
Horseshoe map discrete real 2 1
Hopa-Jong fractal[26] discrete real 2
Hopalong orbit fractal[27] discrete real 2
Hyper Logistic map[28] discrete real 2
Hyperchaotic Chen system[29] continuous real 3
Hyper Newton-Leipnik system[citation needed] continuous real 4
Hyper-Lorenz chaotic attractor continuous real 4
Hyper-Lu chaotic system[30] continuous real 4
Hyper-Rössler chaotic attractor[31] continuous real 4
Hyperchaotic attractor[32] continuous real 4
Ikeda chaotic attractor[33] continuous real 3
Ikeda map discrete real 2 3 Ikeda fractal map
Interval exchange map discrete real 1 variable
Kaplan-Yorke map discrete real 2 1
Knot fractal map[34] discrete real 2
Knot-Holder chaotic oscillator [35] continuous real 3
Kuramoto–Sivashinsky equation continuous real
Lambić map [36] discrete discrete 1
Li symmetrical toroidal chaos [37] continuous real 3
Linear map on unit square
Logistic map discrete real 1 1
Lorenz system continuous real 3 3
Lorenz system's Poincaré return map discrete real 2 3
Lorenz 96 model continuous real arbitrary 1
Lotka-Volterra system continuous real 3 4
Lozi map [38] discrete real 2
Moore-Spiegel chaotic oscillator [39] continuous real 3
Scroll-Attractor [40] continuous real 3
Jerk Circuit [41] continuous real 3
Newton-Leipnik system continuous real 3
Nordmark truncated map
Nosé-Hoover system continuous real 3
Novel chaotic system [42] continuous real 3
Pickover fractal map [43] continuous real 3
Pomeau-Manneville maps for intermittent chaos discrete real 1 or 2 Normal-form maps for intermittency (Types I, II and III)
Polynom Type-A fractal map [44] continuous real 3 3
Polynom Type-B fractal map [45] continuous real 3 6
Polynom Type-C fractal map [46] continuous real 3 18
Pulsed rotor
Quadrup-Two orbit fractal [47] discrete real 2 3
Quasiperiodicity map
Mikhail Anatoly chaotic attractor continuous real 3 2
Random Rotate map
Rayleigh-Benard chaotic oscillator continuous real 3 3
Rikitake chaotic attractor [48] continuous real 3 3
Rössler attractor continuous real 3 3
Rucklidge system [49] continuous real 3 2
Sakarya chaotic attractor [50] continuous real 3 2
Shaw-Pol chaotic oscillator [51][52] continuous real 3 3
Shimizu-Morioka system [53] continuous real 3 2
Shobu-Ose-Mori piecewise-linear map discrete real 1 piecewise-linear approximation for Pomeau-Manneville Type I map
Sinai map - [3][4]
Sprott B chaotic system [54][55] continuous real 3 2
Sprott C chaotic system [56][57] continuous real 3 3
Sprott-Linz A chaotic attractor [58][59][60] continuous real 3 0
Sprott-Linz B chaotic attractor [61][62][63] continuous real 3 0
Sprott-Linz C chaotic attractor [64][65][66] continuous real 3 0
Sprott-Linz D chaotic attractor [67][68][69] continuous real 3 1
Sprott-Linz E chaotic attractor [70][71][72] continuous real 3 1
Sprott-Linz F chaotic attractor [73][74][75] continuous real 3 1
Sprott-Linz G chaotic attractor [76][77][78] continuous real 3 1
Sprott-Linz H chaotic attractor [79][80][81] continuous real 3 1
Sprott-Linz I chaotic attractor [82][83][84] continuous real 3 1
Sprott-Linz J chaotic attractor [85][86][87] continuous real 3 1
Sprott-Linz K chaotic attractor [88][89][90] continuous real 3 1
Sprott-Linz L chaotic attractor [91][92][93] continuous real 3 2
Sprott-Linz M chaotic attractor [94][95][96] continuous real 3 1
Sprott-Linz N chaotic attractor [97][98][99] continuous real 3 1
Sprott-Linz O chaotic attractor [100][101][102] continuous real 3 1
Sprott-Linz P chaotic attractor [103][104][105] continuous real 3 1
Sprott-Linz Q chaotic attractor [106][107][108] continuous real 3 2
Sprott-Linz R chaotic attractor [109][110][111] continuous real 3 2
Sprott-Linz S chaotic attractor [112][113][114] continuous real 3 1
Standard map, Kicked rotor discrete real 2 1 Chirikov standard map, Chirikov-Taylor map
Strizhak-Kawczynski chaotic oscillator [115][116] continuous real 3 9
Symmetric Flow attractor [117] continuous real 3 1
Symplectic map
Tangent map
Thomas' cyclically symmetric attractor [118] continuous real 3 1
Tent map discrete real 1
Tinkerbell map discrete real 2 4
Triangle map
Ueda chaotic oscillator [119] continuous real 3 3
Van der Pol oscillator continuous real 1 3
Willamowski-Rössler model [120] continuous real 3 10
WINDMI chaotic attractor [121][122][123] continuous real 1 2
Zaslavskii map discrete real 2 4
Zaslavskii rotation map
Zeraoulia-Sprott map [124] discrete real 2 2

List of fractals

See also: List of fractals by Hausdorff dimension

References

  1. ^ Chaos from Euler Solution of ODEs
  2. ^ Chaos from Euler Solution of ODEs
  3. ^ On the dynamics of a new simple 2-D rational discrete mapping
  4. ^ Chaos from Euler Solution of ODEs
  5. ^ http://www.yangsky.us/ijcc/pdf/ijcc83/IJCC823.pdf[permanent dead link]
  6. ^ The Aizawa attractor
  7. ^ Local Stability and Hopf Bifurcation Analysis of the Arneodo’s System
  8. ^ Basin of attraction Archived 2014-07-01 at the Wayback Machine
  9. ^ Image encryption based on new Beta chaotic maps
  10. ^ 1981 The Burke & Shaw system
  11. ^ A new chaotic attractor coined
  12. ^ A new chaotic attractor coined
  13. ^ http://www.scholarpedia.org/article/Chua_circuit Chua Circuit
  14. ^ Clifford Attractors
  15. ^ Peter de Jong Attractors
  16. ^ A discrete population model of delayed regulation
  17. ^ Irregular Attractors
  18. ^ A New Finance Chaotic Attractor
  19. ^ Hyperchaos Archived 2015-12-22 at the Wayback Machine
  20. ^ Visions of Chaos 2D Strange Attractor Tutorial
  21. ^ A new chaotic system and beyond: The generalized Lorenz-like system
  22. ^ Gingerbreadman map
  23. ^ Mira Fractals
  24. ^ Half-inverted tearing
  25. ^ Halvorsen: A tribute to Dr. Edward Norton Lorenz
  26. ^ Peter de Jong Attractors
  27. ^ Hopalong orbit fractal
  28. ^ Irregular Attractors
  29. ^ Global chaos synchronization of hyperchaotic chen system by sliding model control
  30. ^ Hyper-Lu system
  31. ^ The first hyperchaotic system
  32. ^ Hyperchaotic attractor Archived 2015-12-22 at the Wayback Machine
  33. ^ Attractors
  34. ^ Knot fractal map Archived 2015-12-22 at the Wayback Machine
  35. ^ [1]
  36. ^ A new discrete chaotic map based on the composition of permutations
  37. ^ A 3D symmetrical toroidal chaos
  38. ^ Lozi maps
  39. ^ Moore-Spiegel Attractor
  40. ^ A new chaotic system and beyond: The generalized lorenz-like system
  41. ^ A New Chaotic Jerk Circuit
  42. ^ Chaos Control and Hybrid Projective Synchronization of a Novel Chaotic System
  43. ^ Pickover
  44. ^ Polynomial Type-A
  45. ^ Polynomial Type-B
  46. ^ Polynomial Type-C
  47. ^ Quadrup Two Orbit Fractal
  48. ^ Rikitake chaotic attractor Archived 2010-06-20 at the Wayback Machine
  49. ^ Description of strange attractors using invariants of phase-plane
  50. ^ Skarya Archived 2015-12-22 at the Wayback Machine
  51. ^ Van der Pol Oscillator Equations
  52. ^ Shaw-Pol chaotic oscillator Archived 2015-12-22 at the Wayback Machine
  53. ^ The Shimiziu-Morioka System
  54. ^ Sprott B chaotic attractor Archived 2007-02-27 at the Wayback Machine
  55. ^ Chaos Blog - Sprott B system Archived 2015-12-22 at the Wayback Machine
  56. ^ Sprott C chaotic attractor Archived 2007-02-27 at the Wayback Machine
  57. ^ Chaos Blog - Sprott C system Archived 2015-12-22 at the Wayback Machine
  58. ^ Sprott's Gateway - Sprott-Linz A chaotic attractor Archived 2007-02-27 at the Wayback Machine
  59. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  60. ^ Chaos Blog - Sprott-Linz A chaotic attractor Archived 2015-12-22 at the Wayback Machine
  61. ^ Sprott's Gateway - Sprott-Linz B chaotic attractor Archived 2007-02-27 at the Wayback Machine
  62. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  63. ^ Chaos Blog - Sprott-Linz B chaotic attractor Archived 2015-12-22 at the Wayback Machine
  64. ^ Sprott's Gateway - Sprott-Linz C chaotic attractor Archived 2007-02-27 at the Wayback Machine
  65. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  66. ^ Chaos Blog - Sprott-Linz C chaotic attractor Archived 2015-12-22 at the Wayback Machine
  67. ^ Sprott's Gateway - Sprott-Linz D chaotic attractor Archived 2007-02-27 at the Wayback Machine
  68. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  69. ^ Chaos Blog - Sprott-Linz D chaotic attractor Archived 2015-12-22 at the Wayback Machine
  70. ^ Sprott's Gateway - Sprott-Linz E chaotic attractor Archived 2007-02-27 at the Wayback Machine
  71. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  72. ^ Chaos Blog - Sprott-Linz E chaotic attractor Archived 2015-12-22 at the Wayback Machine
  73. ^ Sprott's Gateway - Sprott-Linz F chaotic attractor Archived 2007-02-27 at the Wayback Machine
  74. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  75. ^ Chaos Blog - Sprott-Linz F chaotic attractor Archived 2015-12-22 at the Wayback Machine
  76. ^ Sprott's Gateway - Sprott-Linz G chaotic attractor Archived 2007-02-27 at the Wayback Machine
  77. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  78. ^ Chaos Blog - Sprott-Linz G chaotic attractor Archived 2015-12-22 at the Wayback Machine
  79. ^ Sprott's Gateway - Sprott-Linz H chaotic attractor Archived 2007-02-27 at the Wayback Machine
  80. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  81. ^ Chaos Blog - Sprott-Linz H chaotic attractor Archived 2015-12-22 at the Wayback Machine
  82. ^ Sprott's Gateway - Sprott-Linz I chaotic attractor Archived 2007-02-27 at the Wayback Machine
  83. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  84. ^ Chaos Blog - Sprott-Linz I chaotic attractor Archived 2015-12-22 at the Wayback Machine
  85. ^ Sprott's Gateway - Sprott-Linz J chaotic attractor Archived 2007-02-27 at the Wayback Machine
  86. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  87. ^ Chaos Blog - Sprott-Linz J chaotic attractor Archived 2015-12-22 at the Wayback Machine
  88. ^ Sprott's Gateway - Sprott-Linz K chaotic attractor Archived 2007-02-27 at the Wayback Machine
  89. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  90. ^ Chaos Blog - Sprott-Linz K chaotic attractor Archived 2015-12-22 at the Wayback Machine
  91. ^ Sprott's Gateway - Sprott-Linz L chaotic attractor Archived 2007-02-27 at the Wayback Machine
  92. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  93. ^ Chaos Blog - Sprott-Linz L chaotic attractor Archived 2015-12-22 at the Wayback Machine
  94. ^ Sprott's Gateway - Sprott-Linz M chaotic attractor Archived 2007-02-27 at the Wayback Machine
  95. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  96. ^ Chaos Blog - Sprott-Linz M chaotic attractor Archived 2015-12-22 at the Wayback Machine
  97. ^ Sprott's Gateway - Sprott-Linz N chaotic attractor Archived 2007-02-27 at the Wayback Machine
  98. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  99. ^ Chaos Blog - Sprott-Linz N chaotic attractor Archived 2015-12-22 at the Wayback Machine
  100. ^ Sprott's Gateway - Sprott-Linz O chaotic attractor Archived 2007-02-27 at the Wayback Machine
  101. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  102. ^ Chaos Blog - Sprott-Linz O chaotic attractor Archived 2015-12-22 at the Wayback Machine
  103. ^ Sprott's Gateway - Sprott-Linz P chaotic attractor Archived 2007-02-27 at the Wayback Machine
  104. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  105. ^ Chaos Blog - Sprott-Linz P chaotic attractor Archived 2015-12-22 at the Wayback Machine
  106. ^ Sprott's Gateway - Sprott-Linz Q chaotic attractor Archived 2007-02-27 at the Wayback Machine
  107. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  108. ^ Chaos Blog - Sprott-Linz Q chaotic attractor Archived 2015-12-22 at the Wayback Machine
  109. ^ Sprott's Gateway - Sprott-Linz R chaotic attractor Archived 2007-02-27 at the Wayback Machine
  110. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  111. ^ Chaos Blog - Sprott-Linz R chaotic attractor Archived 2015-12-22 at the Wayback Machine
  112. ^ Sprott's Gateway - Sprott-Linz S chaotic attractor Archived 2007-02-27 at the Wayback Machine
  113. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  114. ^ Chaos Blog - Sprott-Linz S chaotic attractor Archived 2015-12-22 at the Wayback Machine
  115. ^ Strizhak-Kawczynski chaotic oscillator[permanent dead link]
  116. ^ Chaos Blog - Strizhak-Kawczynski chaotic oscillator Archived 2015-12-22 at the Wayback Machine
  117. ^ Sprott's Gateway - A symmetric chaotic flow
  118. ^ http://sprott.physics.wisc.edu/chaostsa/ Sprott's Gateway - Chaos and Time-Series Analysis
  119. ^ Oscillator of Ueda
  120. ^ Internal fluctuations in a model of chemical chaos
  121. ^ [2]
  122. ^ Synchronization of Chaotic Fractional-Order WINDMI Systems via Linear State Error Feedback Control
  123. ^ Adaptive Backstepping Controller Design for the Anti-Synchronization of Identical WINDMI Chaotic Systems with Unknown Parameters and its SPICE Implementation
  124. ^ Chen, Guanrong; Kudryashova, Elena V.; Kuznetsov, Nikolay V.; Leonov, Gennady A. (2016). "Dynamics of the Zeraoulia–Sprott Map Revisited". International Journal of Bifurcation and Chaos. 26 (7): 1650126–21. arXiv:1602.08632. Bibcode:2016IJBC...2650126C. doi:10.1142/S0218127416501261.
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