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In mathematics, a chaotic map is a map (an 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 and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems).

List of chaotic maps

[edit]
Map Time domain Space domain Number of space dimensions Number of parameters Also known as
3-cells CNN system continuous real 3
2D Lorenz system[1] discrete real 2 1 Euler method approximation to (non-chaotic) ODE.
2D Rational chaotic map[2] discrete rational 2 2
ACT chaotic attractor [3] continuous real 3
Aizawa chaotic attractor[4] continuous real 3 5
Arneodo chaotic system[5] continuous real 3
Arnold's cat map discrete real 2 0
Baker's map discrete real 2 0
Basin chaotic map[6] discrete real 2 1
Beta Chaotic Map[7] 12
Bogdanov map discrete real 2 3
Brusselator continuous real 3
Burke-Shaw chaotic attractor[8] continuous real 3 2
Chen chaotic attractor[9] continuous real 3 3 Not topologically conjugate to the Lorenz attractor.
Chen-Celikovsky system[10] continuous real 3 "Generalized Lorenz canonical form of chaotic systems"
Chen-LU system[11] continuous real 3 3 Interpolates between Lorenz-like and Chen-like behavior.
Chen-Lee system continuous real 3
Chossat-Golubitsky symmetry map
Chua circuit[12] 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[13] discrete real 2 4
Degenerate Double Rotor map
De Jong fractal map[14] discrete real 2 4
Delayed-Logistic system[15] discrete real 2 1
Discretized circular Van der Pol system[16] discrete real 2 1 Euler method approximation to 'circular' Van der Pol-like ODE.
Discretized Van der Pol system[17] discrete real 2 2 Euler method approximation to Van der Pol ODE.
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[18] discrete real 3
Finance system[19] continuous real 3
Folded-Towel hyperchaotic map[20] continuous real 3
Fractal-Dream system[21] discrete real 2
Gauss map discrete real 1 mouse map, Gaussian map
Generalized Baker map
Genesio-Tesi chaotic attractor[22] continuous real 3
Gingerbreadman map[23] discrete real 2 0
Grinch dragon fractal discrete real 2
Gumowski/Mira map[24] discrete real 2 1
Hadley chaotic circulation continuous real 3 0
Half-inverted Rössler attractor[25]
Halvorsen chaotic attractor[26] 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[27] discrete real 2
Hopalong orbit fractal[28] discrete real 2
Hyper Logistic map[29] discrete real 2
Hyperchaotic Chen system[30] continuous real 3
Hyper Newton-Leipnik system[citation needed] continuous real 4
Hyper-Lorenz chaotic attractor continuous real 4
Hyper-Lu chaotic system[31] continuous real 4
Hyper-Rössler chaotic attractor[32] continuous real 4
Hyperchaotic attractor[33] continuous real 4
Ikeda chaotic attractor[34] 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[35] discrete real 2
Knot-Holder chaotic oscillator[36] continuous real 3
Kuramoto–Sivashinsky equation continuous real
Lambić map[37] discrete discrete 1
Li symmetrical toroidal chaos[38] 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 9
Lozi map[39] discrete real 2
Moore-Spiegel chaotic oscillator[40] continuous real 3
Scroll-Attractor[41] continuous real 3
Jerk Circuit[42] continuous real 3
Newton-Leipnik system continuous real 3
Nordmark truncated map
Nosé-Hoover system continuous real 3
Novel chaotic system[43] continuous real 3
Pickover fractal map[44] 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[45] continuous real 3 3
Polynom Type-B fractal map[46] continuous real 3 6
Polynom Type-C fractal map[47] continuous real 3 18
Pulsed rotor
Quadrup-Two orbit fractal[48] 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[49] continuous real 3 3
Rössler attractor continuous real 3 3
Rucklidge system[50] continuous real 3 2
Sakarya chaotic attractor[51] continuous real 3 2
Shaw-Pol chaotic oscillator[52][53] continuous real 3 3
Shimizu-Morioka system[54] continuous real 3 2
Shobu-Ose-Mori piecewise-linear map discrete real 1 piecewise-linear approximation for Pomeau-Manneville Type I map
Sinai map - [1][2]
Sprott B chaotic system[55][56] continuous real 3 2
Sprott C chaotic system[57][58] continuous real 3 3
Sprott-Linz A chaotic attractor[59][60][61] continuous real 3 0
Sprott-Linz B chaotic attractor[62][63][64] continuous real 3 0
Sprott-Linz C chaotic attractor[65][66][67] continuous real 3 0
Sprott-Linz D chaotic attractor[68][69][70] continuous real 3 1
Sprott-Linz E chaotic attractor[71][72][73] continuous real 3 1
Sprott-Linz F chaotic attractor[74][75][76] continuous real 3 1
Sprott-Linz G chaotic attractor[77][78][79] continuous real 3 1
Sprott-Linz H chaotic attractor[80][81][82] continuous real 3 1
Sprott-Linz I chaotic attractor[83][84][85] continuous real 3 1
Sprott-Linz J chaotic attractor[86][87][88] continuous real 3 1
Sprott-Linz K chaotic attractor[89][90][91] continuous real 3 1
Sprott-Linz L chaotic attractor[92][93][94] continuous real 3 2
Sprott-Linz M chaotic attractor[95][96][97] continuous real 3 1
Sprott-Linz N chaotic attractor[98][99][100] continuous real 3 1
Sprott-Linz O chaotic attractor[101][102][103] continuous real 3 1
Sprott-Linz P chaotic attractor[104][105][106] continuous real 3 1
Sprott-Linz Q chaotic attractor[107][108][109] continuous real 3 2
Sprott-Linz R chaotic attractor[110][111][112] continuous real 3 2
Sprott-Linz S chaotic attractor[113][114][115] continuous real 3 1
Standard map, Kicked rotor discrete real 2 1 Chirikov standard map, Chirikov-Taylor map
Strizhak-Kawczynski chaotic oscillator[116][117] continuous real 3 9
Symmetric Flow attractor[118] continuous real 3 1
Symplectic map
Tangent map
Tahn map[119] discrete real 1 1 Ring laser map [120]Beta distribution[121]

[122]

Thomas' cyclically symmetric attractor[123] continuous real 3 1
Tent map discrete real 1
Tinkerbell map discrete real 2 4
Triangle map
Ueda chaotic oscillator[124] continuous real 3 3
Van der Pol oscillator continuous real 2 3
Willamowski-Rössler model[125] continuous real 3 10
WINDMI chaotic attractor[126][127][128] continuous real 1 2
Zaslavskii map discrete real 2 4
Zaslavskii rotation map
Zeraoulia-Sprott map[129] discrete real 2 2
Chialvo map discrete discrete 3

List of fractals

[edit]
See also: List of fractals by Hausdorff dimension

References

[edit]
  1. ^ Chaos from Euler Solution of ODEs
  2. ^ On the dynamics of a new simple 2-D rational discrete mapping
  3. ^ http://www.yangsky.us/ijcc/pdf/ijcc83/IJCC823.pdf[permanent dead link‍]
  4. ^ The Aizawa attractor
  5. ^ Local Stability and Hopf Bifurcation Analysis of the Arneodo’s System
  6. ^ Basin of attraction Archived 2014-07-01 at the Wayback Machine
  7. ^ Zahmoul, Rim; Ejbali, Ridha; Zaied, Mourad (2017). "Image encryption based on new Beta chaotic maps". Optics and Lasers in Engineering. 96: 39–49. Bibcode:2017OptLE..96...39Z. doi:10.1016/j.optlaseng.2017.04.009.
  8. ^ 1981 The Burke & Shaw system
  9. ^ A new chaotic attractor coined
  10. ^ A new chaotic attractor coined
  11. ^ A new chaotic attractor coined
  12. ^ http://www.scholarpedia.org/article/Chua_circuit Chua Circuit
  13. ^ Clifford Attractors
  14. ^ Peter de Jong Attractors
  15. ^ A discrete population model of delayed regulation
  16. ^ Chaos from Euler Solution of ODEs
  17. ^ Chaos from Euler Solution of ODEs
  18. ^ Irregular Attractors
  19. ^ A New Finance Chaotic Attractor
  20. ^ Hyperchaos Archived 2015-12-22 at the Wayback Machine
  21. ^ Visions of Chaos 2D Strange Attractor Tutorial
  22. ^ A new chaotic system and beyond: The generalized Lorenz-like system
  23. ^ Gingerbreadman map
  24. ^ Mira Fractals
  25. ^ Half-inverted tearing
  26. ^ Halvorsen: A tribute to Dr. Edward Norton Lorenz
  27. ^ Peter de Jong Attractors
  28. ^ Hopalong orbit fractal
  29. ^ Irregular Attractors
  30. ^ Global chaos synchronization of hyperchaotic chen system by sliding model control
  31. ^ Hyper-Lu system
  32. ^ The first hyperchaotic system
  33. ^ Hyperchaotic attractor Archived 2015-12-22 at the Wayback Machine
  34. ^ Attractors
  35. ^ Knot fractal map Archived 2015-12-22 at the Wayback Machine
  36. ^ Lefranc, Marc; Letellier, Christophe; Gilmore, Robert (2008). "Chaos topology". Scholarpedia. 3 (7): 4592. Bibcode:2008SchpJ...3.4592G. doi:10.4249/scholarpedia.4592.
  37. ^ Lambić, Dragan (2015). "A new discrete chaotic map based on the composition of permutations". Chaos, Solitons & Fractals. 78: 245–248. Bibcode:2015CSF....78..245L. doi:10.1016/j.chaos.2015.08.001.
  38. ^ A 3D symmetrical toroidal chaos
  39. ^ Lozi maps
  40. ^ Moore-Spiegel Attractor
  41. ^ A new chaotic system and beyond: The generalized lorenz-like system
  42. ^ A New Chaotic Jerk Circuit
  43. ^ Chaos Control and Hybrid Projective Synchronization of a Novel Chaotic System
  44. ^ Pickover
  45. ^ Polynomial Type-A
  46. ^ Polynomial Type-B
  47. ^ Polynomial Type-C
  48. ^ Quadrup Two Orbit Fractal
  49. ^ Rikitake chaotic attractor Archived 2010-06-20 at the Wayback Machine
  50. ^ Description of strange attractors using invariants of phase-plane
  51. ^ Skarya Archived 2015-12-22 at the Wayback Machine
  52. ^ Van der Pol Oscillator Equations
  53. ^ Shaw-Pol chaotic oscillator Archived 2015-12-22 at the Wayback Machine
  54. ^ The Shimiziu-Morioka System
  55. ^ Sprott B chaotic attractor Archived 2007-02-27 at the Wayback Machine
  56. ^ Chaos Blog - Sprott B system Archived 2015-12-22 at the Wayback Machine
  57. ^ Sprott C chaotic attractor Archived 2007-02-27 at the Wayback Machine
  58. ^ Chaos Blog - Sprott C system Archived 2015-12-22 at the Wayback Machine
  59. ^ Sprott's Gateway - Sprott-Linz A chaotic attractor Archived 2007-02-27 at the Wayback Machine
  60. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  61. ^ Chaos Blog - Sprott-Linz A chaotic attractor Archived 2015-12-22 at the Wayback Machine
  62. ^ Sprott's Gateway - Sprott-Linz B chaotic attractor Archived 2007-02-27 at the Wayback Machine
  63. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  64. ^ Chaos Blog - Sprott-Linz B chaotic attractor Archived 2015-12-22 at the Wayback Machine
  65. ^ Sprott's Gateway - Sprott-Linz C chaotic attractor Archived 2007-02-27 at the Wayback Machine
  66. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  67. ^ Chaos Blog - Sprott-Linz C chaotic attractor Archived 2015-12-22 at the Wayback Machine
  68. ^ Sprott's Gateway - Sprott-Linz D chaotic attractor Archived 2007-02-27 at the Wayback Machine
  69. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  70. ^ Chaos Blog - Sprott-Linz D chaotic attractor Archived 2015-12-22 at the Wayback Machine
  71. ^ Sprott's Gateway - Sprott-Linz E chaotic attractor Archived 2007-02-27 at the Wayback Machine
  72. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  73. ^ Chaos Blog - Sprott-Linz E chaotic attractor Archived 2015-12-22 at the Wayback Machine
  74. ^ Sprott's Gateway - Sprott-Linz F chaotic attractor Archived 2007-02-27 at the Wayback Machine
  75. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  76. ^ Chaos Blog - Sprott-Linz F chaotic attractor Archived 2015-12-22 at the Wayback Machine
  77. ^ Sprott's Gateway - Sprott-Linz G chaotic attractor Archived 2007-02-27 at the Wayback Machine
  78. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  79. ^ Chaos Blog - Sprott-Linz G chaotic attractor Archived 2015-12-22 at the Wayback Machine
  80. ^ Sprott's Gateway - Sprott-Linz H chaotic attractor Archived 2007-02-27 at the Wayback Machine
  81. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  82. ^ Chaos Blog - Sprott-Linz H chaotic attractor Archived 2015-12-22 at the Wayback Machine
  83. ^ Sprott's Gateway - Sprott-Linz I chaotic attractor Archived 2007-02-27 at the Wayback Machine
  84. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  85. ^ Chaos Blog - Sprott-Linz I chaotic attractor Archived 2015-12-22 at the Wayback Machine
  86. ^ Sprott's Gateway - Sprott-Linz J chaotic attractor Archived 2007-02-27 at the Wayback Machine
  87. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  88. ^ Chaos Blog - Sprott-Linz J chaotic attractor Archived 2015-12-22 at the Wayback Machine
  89. ^ Sprott's Gateway - Sprott-Linz K chaotic attractor Archived 2007-02-27 at the Wayback Machine
  90. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  91. ^ Chaos Blog - Sprott-Linz K chaotic attractor Archived 2015-12-22 at the Wayback Machine
  92. ^ Sprott's Gateway - Sprott-Linz L chaotic attractor Archived 2007-02-27 at the Wayback Machine
  93. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  94. ^ Chaos Blog - Sprott-Linz L chaotic attractor Archived 2015-12-22 at the Wayback Machine
  95. ^ Sprott's Gateway - Sprott-Linz M chaotic attractor Archived 2007-02-27 at the Wayback Machine
  96. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  97. ^ Chaos Blog - Sprott-Linz M chaotic attractor Archived 2015-12-22 at the Wayback Machine
  98. ^ Sprott's Gateway - Sprott-Linz N chaotic attractor Archived 2007-02-27 at the Wayback Machine
  99. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  100. ^ Chaos Blog - Sprott-Linz N chaotic attractor Archived 2015-12-22 at the Wayback Machine
  101. ^ Sprott's Gateway - Sprott-Linz O chaotic attractor Archived 2007-02-27 at the Wayback Machine
  102. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  103. ^ Chaos Blog - Sprott-Linz O chaotic attractor Archived 2015-12-22 at the Wayback Machine
  104. ^ Sprott's Gateway - Sprott-Linz P chaotic attractor Archived 2007-02-27 at the Wayback Machine
  105. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  106. ^ Chaos Blog - Sprott-Linz P chaotic attractor Archived 2015-12-22 at the Wayback Machine
  107. ^ Sprott's Gateway - Sprott-Linz Q chaotic attractor Archived 2007-02-27 at the Wayback Machine
  108. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  109. ^ Chaos Blog - Sprott-Linz Q chaotic attractor Archived 2015-12-22 at the Wayback Machine
  110. ^ Sprott's Gateway - Sprott-Linz R chaotic attractor Archived 2007-02-27 at the Wayback Machine
  111. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  112. ^ Chaos Blog - Sprott-Linz R chaotic attractor Archived 2015-12-22 at the Wayback Machine
  113. ^ Sprott's Gateway - Sprott-Linz S chaotic attractor Archived 2007-02-27 at the Wayback Machine
  114. ^ A new chaotic system and beyond: The generalized Lorenz-like System
  115. ^ Chaos Blog - Sprott-Linz S chaotic attractor Archived 2015-12-22 at the Wayback Machine
  116. ^ Strizhak-Kawczynski chaotic oscillator[permanent dead link‍]
  117. ^ Chaos Blog - Strizhak-Kawczynski chaotic oscillator Archived 2015-12-22 at the Wayback Machine
  118. ^ Sprott's Gateway - A symmetric chaotic flow
  119. ^ Okulov, A. Yu (2020). "Structured light entities, chaos and nonlocal maps". Chaos, Solitons & Fractals. 133: 109638. arXiv:1901.09274. Bibcode:2020CSF...13309638O. doi:10.1016/j.chaos.2020.109638. S2CID 247759987. {{cite journal}}: Check |url= value (help)[permanent dead link‍]
  120. ^ Okulov, A. Yu.; Oraevsky, A. N. (1986). "Space–temporal behavior of a light pulse propagating in a nonlinear nondispersive medium". Journal of the Optical Society of America B. 3 (5): 741. Bibcode:1986JOSAB...3..741O. doi:10.1364/JOSAB.3.000741. S2CID 124347430.
  121. ^ Okulov, A Yu; Oraevskiĭ, A. N. (1984). "Regular and stochastic self-modulation of radiation in a ring laser with a nonlinear element". Soviet Journal of Quantum Electronics. 14 (9): 1235–1237. doi:10.1070/QE1984v014n09ABEH006171.
  122. ^ Okulov, Alexey Yurievich (2020). "Numerical investigation of coherent and turbulent structures of light via nonlinear integral mappings". Computer Research and Modeling. 12 (5): 979–992. arXiv:1911.10694. doi:10.20537/2076-7633-2020-12-5-979-992. S2CID 211133329. {{cite journal}}: Check |url= value (help)[permanent dead link‍]
  123. ^ http://sprott.physics.wisc.edu/chaostsa/ Sprott's Gateway - Chaos and Time-Series Analysis
  124. ^ Oscillator of Ueda
  125. ^ Internal fluctuations in a model of chemical chaos
  126. ^ "Main Page - Weigel's Research and Teaching Page". aurora.gmu.edu. Archived from the original on 10 April 2011. Retrieved 17 January 2022.
  127. ^ Synchronization of Chaotic Fractional-Order WINDMI Systems via Linear State Error Feedback Control
  128. ^ Vaidyanathan, S.; Volos, Ch. K.; Rajagopal, K.; Kyprianidis, I. M.; Stouboulos, I. N. (2015). "Adaptive Backstepping Controller Design for the Anti-Synchronization of Identical WINDMI Chaotic Systems with Unknown Parameters and its SPICE Implementation" (PDF). Journal of Engineering Science and Technology Review. 8 (2): 74–82. doi:10.25103/jestr.082.11.
  129. ^ 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. S2CID 11406449.
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